9 Trigonometric. Functions

Transcription

1 9 Trigonometric Functions In this chapter, ou will stud trigonometric functions. Trigonometr is used to find relationships between the sides and angles of triangles, and to write trigonometric functions as models of real-life quantities. Trigonometric functions are also used to model quantities that are periodic. In this chapter, ou should learn the following. How to describe angles, use radian measure, and use degree measure. (9.) How to evaluate trigonometric functions using the unit circle. (9.) How to evaluate trigonometric functions of acute angles and use the fundamental trigonometric identities. (9.) How to use reference angles to evaluate trigonometric functions of an angle. (9.) How to sketch the graphs of sine and cosine functions. (9.5) How to sketch the graphs of tangent, cotangent, secant, and cosecant functions. (9.) How to evaluate inverse trigonometric functions. (9.7) How to solve real-life problems involving right triangles, directional bearings, and harmonic motion. (9.8) Andre Jenn / Alam Throughout the da, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tide. How can the depth be modeled b a trigonometric function? (See Section 9.5, Eample 7.) A reference angle is the acute angle formed b the terminal side of and the horizontal ais. You will learn how to use reference angles to calculate the values of trigonometric functions of angles greater than 90 degrees. (See Section 9..) 55 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

2 5 Chapter 9 Trigonometric Functions 9. Radian and Degree Measure Describe angles. Use radian measure. Use degree measure. Use angles to model and solve real-life problems. Angles As derived from the Greek language, the word trigonometr means measurement of triangles. Initiall, trigonometr dealt with relationships among the sides and angles of triangles and was used in the development of astronom, navigation, and surveing. With the development of calculus and the phsical sciences in the 7th centur, a different perspective arose one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains. Consequentl, the applications of trigonometr epanded to include a vast number of phsical phenomena involving rotations and vibrations. These phenomena include sound waves, light ras, planetar orbits, vibrating strings, pendulums, and orbits of atomic particles. The approach in this tet incorporates both perspectives, starting with angles and their measure. Terminal side Terminal side Verte Initial side Initial side (a) Angle Figure 9. (b) Angle in standard position An angle is determined b rotating a ra (half-line) about its endpoint. The starting position of the ra is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure 9.(a). The endpoint of the ra is the verte of the angle. This perception of an angle fits a coordinate sstem in which the origin is the verte and the initial side coincides with the positive -ais. Such an angle is in standard position, as shown in Figure 9.(b). Positive angles are generated b counterclockwise rotation, and negative angles b clockwise rotation, as shown in Figure 9.. Angles are labeled with Greek letters (alpha), (beta), and (theta), as well as uppercase letters A, B, and C. In Figure 9., note that angles and have the same initial and terminal sides. Such angles are coterminal. Positive angle (counterclockwise) Negative angle (clockwise) α β β α Positive and negative angles Coterminal angles Figure 9. Figure 9. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

3 9. Radian and Degree Measure 57 Radian Measure r r s = r The measure of an angle is determined b the amount of rotation from the initial side to the terminal side. One wa to measure angles is in radians. This tpe of measure is especiall useful in calculus. To define a radian, ou can use a central angle of a circle, one whose verte is the center of the circle, as shown in Figure 9.. Arc length radius when Figure 9. radians r r radians Figure 9.5 radians r r radian radian r 5 radians r radians DEFINITION OF RADIAN One radian is the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle. See Figure 9.. Because the circumference of a circle is r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of s r. Moreover, because there are just over si radius lengths in a full circle, as shown in Figure 9.5. In general, the radian measure of a central angle is obtained b dividing the arc length s b r. That is, sr, where is measured in radians. Because the units of measure for s and r are the same, this ratio is unitless it is simpl a real number. Because the radian measure of an angle of one full revolution is, ou can obtain revolution radians revolution radians revolution radians. These and other common angles are shown in Figure 9...8, Figure 9. Quadrant II Quadrant I < < 0 < < = = 0 Quadrant III < < Figure 9.7 = = Quadrant IV < < Recall that the four quadrants in a coordinate sstem are numbered I, II, III, and IV. Figure 9.7 shows which angles between 0 and lie in each of the four quadrants. Note that angles between 0 and are acute and that angles between and are obtuse. Because two angles are coterminal if the have the same initial and terminal sides, the angles 0 and are coterminal, as are the angles and. You can find an angle that is coterminal to a given angle b adding or subtracting (one revolution), as demonstrated in Eample. A given angle has infinitel man coterminal angles. For instance, is coterminal with n where n is an integer. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

4 58 Chapter 9 Trigonometric Functions STUDY TIP The phrase the terminal side of lies in a quadrant is often abbreviated b simpl saing that lies in a quadrant. The terminal sides of the quadrant angles 0,,, and do not lie within quadrants. EXAMPLE Sketching and Finding Coterminal Angles a. For the positive angle, subtract to obtain a coterminal angle See Figure 9.8(a). b. For the positive angle, subtract to obtain a coterminal angle. 5. See Figure 9.8(b). c. For the negative angle, add to obtain a coterminal angle. See Figure 9.8(c). = 0 = 0 0 (a) (b) (c) Figure = Two positive angles and are complementar (complements of each other) if their sum is. Two positive angles are supplementar (supplements of each other) if their sum is. See Figure 9.9. β α β α Complementar angles: Figure 9.9 Supplementar angles: EXAMPLE Complementar and Supplementar Angles a. The complement of 5 is 5 The supplement of 5 is 5 b. Because 5 is greater than, it has no complement. (Remember that complements are positive angles.) The supplement is Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

5 9. Radian and Degree Measure = (0 ) 0 = (0) 5 = (0 ) 8 0 = (0 ) Figure Degree Measure A second wa to measure angles is in terms of degrees, denoted b the smbol. A measure of one degree is equivalent to a rotation of 0 of a complete revolution about the verte. To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure 9.0. So, a full revolution (counterclockwise) corresponds to 0, a half revolution to 80, a quarter revolution to 90, and so on. Because radians corresponds to one complete revolution, degrees and radians are related b the equations 0 rad and 80 rad. From the latter equation, ou obtain 80 rad and rad 80 which lead to the following conversion rules. CONVERSIONS BETWEEN DEGREES AND RADIANS rad. To convert degrees to radians, multipl degrees b To convert radians to degrees, multipl radians b rad. To appl these two conversion rules, use the basic relationship (See Figure 9..) rad 80. STUDY TIP When no units of angle measure are specified, radian measure is implied. For instance, if ou write or ou impl that radians or radians., 0 5 Figure TECHNOLOGY With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historicall, however, fractional parts of degrees were epressed in minutes and seconds, using the prime and double prime notations, respectivel. That is, one minute 0 one second 00. Consequentl, an angle of degrees, minutes, and 7 seconds is represented b Man calculators have special kes for converting an angle in degrees, minutes, and seconds D M S to decimal degree form, and vice versa. 7. EXAMPLE Converting from Degrees to Radians a. 5 5 deg Multipl b deg radians b deg Multipl b 80. c deg Multipl b deg radians EXAMPLE Converting from Radians to Degrees a. Multipl b 80. rad rad 80 deg 9 rad rad 80 deg radians b. Multipl b 80. rad 9 rad 80 deg 80 c. rad rad 80 deg 0.59 Multipl b 80. rad rad rad 90 rad Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

6 570 Chapter 9 Trigonometric Functions Applications The radian measure formula, sr, can be used to measure arc length along a circle. ARC LENGTH For a circle of radius r, a central angle intercepts an arc of length s given b s r Length of circular arc where is measured in radians. Note that if r, then s, and the radian measure of equals the arc length. EXAMPLE 5 Finding Arc Length s Figure 9. = 0 r = A circle has a radius of inches. Find the length of the arc intercepted b a central angle of 0, as shown in Figure 9.. Solution To use the formula s r, first convert 0 to radian measure. 0 0 deg 80 deg radians Then, using a radius of r inches, ou can find the arc length to be s r.7 inches. r rad Note that the units for are determined b the units for r because is given in radian measure, which has no units. The formula for the length of a circular arc can be used to analze the motion of a particle moving at a constant speed along a circular path. STUDY TIP Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes. B dividing the formula for arc length b t, ou can establish a relationship between linear speed v and angular speed, as shown. s r s r t t v r LINEAR AND ANGULAR SPEEDS Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is Linear speed v Moreover, if is the angle (in radian measure) corresponding to the arc length s, then the angular speed (the lowercase Greek letter omega) of the particle is Angular speed arc length time s t. central angle time t. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

7 9. Radian and Degree Measure 57 EXAMPLE Finding Linear Speed 0. cm The second hand of a clock is 0. centimeters long, as shown in Figure 9.. Find the linear speed of the tip of this second hand as it passes around the clock face. Solution In one revolution, the arc length traveled is Figure 9. s r 0. Substitute for r. 0. centimeters. The time required for the second hand to travel this distance is t minute 0 seconds. So, the linear speed of the tip of the second hand is Linear speed s t 0. centimeters 0 seconds.08 centimeters per second. Figure 9. ft EXAMPLE 7 Finding Angular and Linear Speeds The blades of a wind turbine are feet long (see Figure 9.). The propeller rotates at 5 revolutions per minute. a. Find the angular speed of the propeller in radians per minute. b. Find the linear speed of the tips of the blades. Solution a. Because each revolution generates radians, it follows that the propeller turns 5 0 radians per minute. In other words, the angular speed is Angular speed t 0 radians minute 0 radians per minute. b. The linear speed is Linear speed s t r t 0 feet minute 0,9 feet per minute. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

8 57 Chapter 9 Trigonometric Functions 9. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 0, fill in the blanks.. means measurement of triangles.. An is determined b rotating a ra about its endpoint.. Two angles that have the same initial and terminal sides are.. One is the measure of a central angle that intercepts an arc equal to the radius of the circle. 5. Angles that measure between 0 and are angles, and angles that measure between and are angles.. Two positive angles that have a sum of are angles, whereas two positive angles that have a sum of are angles. 7. The angle measure that is equivalent to a rotation of of a complete revolution about an angle s verte is one degrees radians. 9. The radian measure formula, can be used to measure along a circle. 0. The speed of a particle is the ratio of arc length to time traveled, and the speed of a particle is the ratio of central angle to time traveled. In Eercises, estimate the angle to the nearest one-half radian sr, 0 In Eercises, sketch each angle in standard position.. (a) (b). (a) 7 (b) 5. (a) (b). (a) (b) In Eercises 7 0, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in radians. 7. (a) (b) 8. (a) (b) 9. (a) (b) 9 0. (a) (b) = 0 7 = 0 In Eercises, find (if possible) the complement and supplement of each angle.. (a) (b). (a) (b). (a) (b). (a) (b).5 In Eercises 5 0, estimate the number of degrees in the angle. Use a protractor to check our answer = = In Eercises 7, determine the quadrant in which each angle lies. (The angle measure is given in radians.) 7. (a) (b) 8. (a) (b) (a) (b) 0. (a) 5 (b) 9. (a).5 (b).5. (a).0 (b) Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

9 9. Radian and Degree Measure 57 In Eercises, determine the quadrant in which each angle lies.. (a) 0 (b) 85. (a) 8. (b) (a) 50 (b). (a) 0 (b). In Eercises 5 8, sketch each angle in standard position. 5. (a) 90 (b) 80. (a) 70 (b) 0 7. (a) 0 (b) 5 8. (a) 750 (b) 00 In Eercises 9 5, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in degrees. 9. (a) 90 (b) (a) 90 (b) (a) (b) 5. (a) (b) In Eercises 5 5, find (if possible) the complement and supplement of each angle. 5. (a) 8 (b) (a) (b) (a) 50 (b) (a) 0 (b) 70 In Eercises 57 0, rewrite each angle in radian measure as a multiple of. (Do not use a calculator.) 57. (a) 0 (b) (a) 5 (b) (a) 0 (b) 0 0. (a) 70 (b) In Eercises, rewrite each angle in degree measure. (Do not use a calculator.). (a) (b). (a) 7 (b) 9. (a) (b) 7. (a) (b) = 5 = = = 70 0 In Eercises 5 7, convert the angle measure from degrees to radians. Round to three decimal places In Eercises 7 80, convert the angle measure from radians to degrees. Round to three decimal places In Eercises 8 8, convert each angle measure to decimal degree form without using a calculator. Then check our answers using a calculator. 8. (a) 5 5 (b) (a) 5 0 (b) 8. (a) (b) (a) 5 (b) 08 0 In Eercises 85 88, convert each angle measure to degrees, minutes, and seconds without using a calculator. Then check our answers using a calculator. 85. (a) 0. (b) (a) 5. (b) (a).5 (b) (a) 0. (b) 0.79 In Eercises 89 9, find the length of the arc on a circle of radius r intercepted b a central angle. Radius r inches feet 9. meters 9. 0 centimeters In Eercises 9 9, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius r inches 8 inches 9. feet 8 feet.8 Central Angle rc Length s centimeters 0.5 centimeters kilometers 50 kilometers Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

10 57 Chapter 9 Trigonometric Functions In Eercises 97 00, use the given arc length and radius to find the angle (in radians) Distance Between Cities In Eercises 0 and 05, find the distance between the cities. Assume that Earth is a sphere of radius 000 miles and that the cities are on the same longitude (one cit is due north of the other). Cit 0. Dallas, Teas Omaha, Nebraska 05. San Francisco, California Seattle, Washington 7 Latitude 75 WRITING ABOUT CONCEPTS 0. A fan motor turns at a given angular speed. How does the speed of the tips of the blades change if a fan of greater diameter is installed on the motor? Eplain. 0. Is a degree or a radian the larger unit of measure? Eplain our reasoning. 0. If the radius of a circle is increasing and the magnitude of a central angle is held constant, how is the length of the intercepted arc changing? Eplain our reasoning. 7 9 N 5 50 N 7 7 N N 0. Linear and Angular Speeds A circular power saw has a 7 - inch-diameter blade that rotates at 5000 revolutions per minute. (a) Find the angular speed of the saw blade in radians per minute. (b) Find the linear speed (in feet per minute) of one of the cutting teeth as the contact the wood being cut. 07. Linear and Angular Speeds A carousel with a 50-foot diameter makes revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed (in feet per minute) of the platform rim of the carousel Angular Speed A two-inch-diameter pulle on an electric motor that runs at 700 revolutions per minute is connected b a belt to a four-inch-diameter pulle on a saw arbor. (a) Find the angular speed (in radians per minute) of each pulle. (b) Find the revolutions per minute of the saw. 09. Angular Speed A car is moving at a rate of 5 miles per hour, and the diameter of its wheels is feet. (a) Find the number of revolutions per minute the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute. CAPSTONE 0. Write a short paper in our own words eplaining the meaning of each of the following concepts to a classmate. (a) an angle in standard position (b) positive and negative angles (c) coterminal angles (d) angle measure in degrees and radians (e) obtuse and acute angles (f) complementar and supplementar angles. Speed of a Biccle The radii of the pedal sprocket, the wheel sprocket, and the wheel of the biccle in the figure are inches, inches, and inches, respectivel. A cclist is pedaling at a rate of revolution per second. Find the speed of the biccle in feet per second and miles per hour. in. in. in. True or False? In Eercises, determine whether the statement is true or false. Justif our answer.. A measurement of radians corresponds to two complete revolutions from the initial side to the terminal side of an angle.. The difference between the measures of two coterminal angles is alwas a multiple of 0 if epressed in degrees and is alwas a multiple of radians if epressed in radians.. An angle that measures 0 lies in Quadrant III. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

11 9. Trigonometric Functions: The Unit Circle Trigonometric Functions: The Unit Circle Identif a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle. Use the domain and period to evaluate sine and cosine functions. Use a calculator to evaluate trigonometric functions. The Unit Circle (0, ) The two historical perspectives of trigonometr incorporate different methods for introducing the trigonometric functions. Our first introduction to these functions is based on the unit circle. Consider the unit circle given b (, 0) (0, ) Unit circle Figure 9.5 (, 0) Unit circle as shown in Figure 9.5. Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure 9.. t > 0 (, ) t t t < 0 (, 0) (, 0) t (, ) t Figure 9. As the real number line is wrapped around the unit circle, each real number t corresponds to a point, on the circle. For eample, the real number 0 corresponds to the point, 0. Moreover, because the unit circle has a circumference of, the real number also corresponds to the point, 0. Real Number 0 Point on Unit Circle, 0 0,, 0 0,, 0 In general, each real number t also corresponds to a central angle (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula s r (with r ) indicates that the real number t is the (directional) length of the arc intercepted b the angle, given in radians. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

12 57 Chapter 9 Trigonometric Functions The Trigonometric Functions From the preceding discussion, it follows that the coordinates and are two functions of the real variable t. You can use these coordinates to define the si trigonometric functions of t. sine cosine tangent cotangent secant cosecant These si functions are normall abbreviated sin, cos, tan, cot, sec, and csc, respectivel. NOTE Observe that the functions in the second row are the reciprocals of the corresponding functions in the first row. DEFINITIONS OF TRIGONOMETRIC FUNCTIONS Let t be a real number and let, be the point on the unit circle corresponding to t. sin t cos t tan t, 0 csc t, 0 sec t, 0 cot t, 0 (0, ),, ( ) ( ) (, 0) ( ) (, 0),, (0, ) Figure 9.7 (, ) ( ), ( ), (, 0) ( ), Figure 9.8 (0, ) (0, ) ( ), (, 0), ( ) ( ) ( ), ( ), STUDY TIP On the unit circles in Figures 9.7 and 9.8, each point, can be designated as cos t, sin t. In the definitions of the trigonometric functions, note that the tangent and secant are not defined when 0. For instance, because t corresponds to, 0,, it follows that tan and sec are undefined. Similarl, the cotangent and cosecant are not defined when 0. For instance, because t 0 corresponds to,, 0, cot 0 and csc 0 are undefined. In Figure 9.7, the unit circle has been divided into eight equal arcs, corresponding to t-values of Similarl, in Figure 9.8, the unit circle has been divided into equal arcs, corresponding to t-values of 0, and.,,,, 5,,,,,,, To verif the points on the unit circle in Figure 9.7, note that also, lies on the line. So, substituting for in the equation of the unit circle produces the following. 5 0,,,,,, 7 and.,, 7 5 ± Because the point is in the first quadrant, and because, ou also have You can use similar reasoning to verif the rest of the points in Figure 9.7. and the points in Figure 9.8. Using the, coordinates in Figures 9.7 and 9.8, ou can evaluate the trigonometric functions for common t-values. This procedure is demonstrated in Eamples,, and. You should stud and learn these eact function values for common t-values because the will help ou in later sections to perform calculations. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

13 9. Trigonometric Functions: The Unit Circle 577 EXAMPLE Evaluating Trigonometric Functions Evaluate the si trigonometric functions at each real number. a. t b. t 5 c. t 0 d. t Solution For each t-value, begin b finding the corresponding point, on the unit circle. Then use the definitions of trigonometric functions listed on the previous page. a. t corresponds to the point cos tan 5 b. t corresponds to the point 5 sin 5 cos sin 5 tan c. t 0 corresponds to the point,, 0.,, cos t, sin t. csc sec cot,, cos t, sin t. csc 5 sec 5 cot 5 sin 0 0 cos 0 tan csc 0 is undefined. sec 0 cot 0 is undefined. d. t corresponds to the point,, 0. sin 0 cos tan 0 0 csc is undefined. sec cot is undefined. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

14 578 Chapter 9 Trigonometric Functions EXAMPLE Evaluating Trigonometric Functions Evaluate the si trigonometric functions at t. Solution Moving clockwise around the unit circle, it follows that corresponds to the point,,. t sin cos tan csc sec cot (0, ) (, 0) (, 0) (0, ) Figure 9.9 t =, +,... t =,, t =, +,... Figure 9.0 t =, +, +,... t =, +,... t = 0,, t =, +,... t =, +, +,... Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure 9.9. B definition, sin t and cos t. Because, is on the unit circle, ou know that and. So, the values of sine and cosine also range between and. and sin t cos t Adding to each value of t in the interval 0, completes a second revolution around the unit circle, as shown in Figure 9.0. The values of sint and cost correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result and sint n sin t cost n cos t for an integer n and real number t. Functions that behave in such a repetitive (or cclic) manner are called periodic. NOTE In Figure 9.0, note that positive multiples of are added to the t-values. You could just as well have added negative multiples. For instance, and are also coterminal with. DEFINITION OF PERIODIC FUNCTION A function f is periodic if there eists a positive real number c such that ft c f t for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

15 9. Trigonometric Functions: The Unit Circle 579 Recall from Section. that a function f is even if f t f t, and is odd if f t ft. EVEN AND ODD TRIGONOMETRIC FUNCTIONS The cosine and secant functions are even. cost cos t sect sec t The sine, cosecant, tangent, and cotangent functions are odd. sint sin t csct csc t tant tan t cott cot t STUDY TIP From the definition of periodic function, it follows that the sine and cosine functions are periodic and have a period of. The other four trigonometric functions are also periodic, and will be discussed further in Section 9.. EXAMPLE Using the Period to Evaluate the Sine and Cosine a. Because ou have, b. Because 7 ou have, cos 7 cos sin sin cos 0. sin. c. For sin t sint because the sine function is odd. 5, 5 TECHNOLOGY When evaluating trigonometric functions with a calculator, remember to enclose all fractional angle measures in parentheses. For instance, if ou want to evaluate sin t for t /, ou should enter SIN ENTER. These kestrokes ield the correct value of 0.5. Note that some calculators automaticall place a left parenthesis after trigonometric functions. Check the user s guide for our calculator for specific kestrokes on how to evaluate trigonometric functions. Evaluating Trigonometric Functions with a Calculator When evaluating a trigonometric function with a calculator, ou need to set the calculator to the desired mode of measurement (degree or radian). Most calculators do not have kes for the cosecant, secant, and cotangent functions. To evaluate these functions, ou can use the ke with their respective reciprocal functions sine, cosine, and tangent. For instance, to evaluate csc8, use the fact that csc 8 sin8 and enter the following kestroke sequence in radian mode. SIN 8 ENTER Displa.59 EXAMPLE Using a Calculator Function Mode Calculator Kestrokes Displa a. sin Radian SIN ENTER b. cot.5 Radian TAN.5 ENTER Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

16 580 Chapter 9 Trigonometric Functions 9. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, fill in the blanks.. Each real number t corresponds to a point, on the.. A function f is if there eists a positive real number c such that f t c f t for all t in the domain of f.. The smallest number c for which a function f is periodic is called the of f.. A function f is if f t ft and if f t f t. In Eercises 5 8, determine the eact values of the si trigonometric functions of the real number t In Eercises 9, find the point, on the unit circle that corresponds to the real number t. 9. t 0. t. t.. t t. In Eercises 7, evaluate (if possible) the sine, cosine, and tangent of the real number t. t ( ), 5 5 ( ) 5, t t t 5 7. t 8. t t 8 5 (, 7 7) t t (, 5 ) 9. t 0. t. t 7. t. t. t 5 5. t. t In Eercises 7, evaluate (if possible) the si trigonometric functions of the real number t. 7. t 8. t 5 9. t 0. t 7. t. t. t. t In Eercises 5, evaluate the trigonometric function using its period as an aid. 5. sin. cos 7. cos cos sin 8. In Eercises 8, use the value of the trigonometric function to evaluate the indicated functions.. sin t. sint 8 (a) sint (a) sin t (b) csct (b) csc t 5. cost 5. cos t (a) cos t (a) cost (b) sect (b) sect 7. sin t 5 8. cos t 5 (a) (a) (b) sint (b) cost sin t sin 9 sin 9 cos 9 cos t Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

17 9. Trigonometric Functions: The Unit Circle 58 In Eercises 9 58, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) 9. sin 50. tan 5. cot 5. csc 5. cos.7 5. cos csc sec sec cot Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos t, where is the displacement (in feet) and t is the time (in seconds). Find the displacements when (a) t 0, (b) t, and (c) t. 0. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring and subject to the damping effect of friction is given b t et cos t, where is the displacement (in feet) and t is the time (in seconds). (a) Complete the table. (b) Use the table feature of a graphing utilit to approimate the time when the weight reaches equilibrium. (c) What appears to happen to the displacement as increases? True or False? In Eercises, determine whether the statement is true or false. Justif our answer.. Because sint sin t, it can be said that the sine of a negative angle is a negative number.. Because the sine is an odd function, its graph is smmetric to the -ais.. The real number 0 corresponds to the point 0, on the unit circle.. t 0 cos 7 cos 5. Verif that cos t cos t b approimating cos.5 and cos Verif that sint t sin t sin t b approimating sin 0.5, sin 0.75, and sin. t WRITING ABOUT CONCEPTS 7. Let, and, be points on the unit circle corresponding to t t and t t, respectivel. (a) Identif the smmetr of the points, and,. (b) Make a conjecture about an relationship between sin t and (c) Make a conjecture about an relationship between cos t and 8. Use the unit circle to verif that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd. 9. Because ft sin t is an odd function and gt cos t is an even function, what can be said about the function ht ftgt? 70. Because ft sin t and gt tan t are odd functions, what can be said about the function ht ftgt? sin t. cos t. 7. Graphical Analsis With our graphing utilit in radian and parametric modes, enter the equations X T cos T and Y T sin T and use the following settings. Tmin 0, Tma., Tstep 0. Xmin.5, Xma.5, Xscl Ymin, Yma, Yscl (a) Graph the entered equations and describe the graph. (b) Use the trace feature to move the cursor around the graph. What do the t-values represent? What do the - and -values represent? (c) What are the least and greatest values of and? CAPSTONE 7. A student ou are tutoring has used a unit circle divided into 8 equal parts to complete the table for selected values of t. What is wrong? t 0 0 sin t cos t tan t Undef. 0 Undef. 0 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

18 58 Chapter 9 Trigonometric Functions 9. Right Triangle Trigonometr Evaluate trigonometric functions of acute angles. Use fundamental trigonometric identities. Use trigonometric functions to model and solve real-life problems. Hpotenuse Side adjacent to Figure 9. Side opposite The Si Trigonometric Functions Our second look at the trigonometric functions is from a right triangle perspective. Consider a right triangle, with one acute angle labeled, as shown in Figure 9.. Relative to the angle, the three sides of the triangle are the hpotenuse, the opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ). Using the lengths of these three sides, ou can form si ratios that define the si trigonometric functions of the acute angle. sine cosecant cosine secant tangent cotangent In the following definitions, it is important to see that 0 < < 90 lies in the first quadrant) and that for such angles the value of each trigonometric function is positive. HISTORICAL NOTE Georg Joachim Rhaeticus (5 57) was the leading Teutonic mathematical astronomer of the th centur. He was the first to define the trigonometric functions as ratios of the sides of a right triangle. RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS Let be an acute angle of a right triangle. The si trigonometric functions of the angle are defined as follows. (Note that the functions in the second row are the reciprocals of the corresponding functions in the first row.) sin opp hp csc hp opp cos adj hp sec hp adj The abbreviations opp, adj, and hp represent the lengths of the three sides of a right triangle. opp the length of the side opposite adj the length of the side adjacent to hp the length of the hpotenuse tan opp adj cot adj opp Hpotenuse Figure 9. EXAMPLE Evaluating Trigonometric Functions Use the triangle in Figure 9. to find the values of the si trigonometric functions of. Solution B the Pthagorean Theorem, hp opp adj, it follows that hp 5 5. So, the si trigonometric functions of are sin opp hp 5 csc hp opp 5 cos adj hp 5 sec hp adj 5 tan opp adj cot adj opp. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

19 9. Right Triangle Trigonometr 58 EXAMPLE Evaluating Trigonometric Functions of 5 Find the values of sin 5, cos 5, and tan 5. 5 Solution Construct a right triangle having 5 as one of its acute angles, as shown in Figure 9.. Choose the length of the adjacent side to be. From geometr, ou know that the other acute angle is also 5. So, the triangle is isosceles and the length of the opposite side is also. Using the Pthagorean Theorem, ou find the length of the hpotenuse to be. 5 Figure 9. sin 5 opp hp cos 5 adj hp tan 5 opp adj EXAMPLE Evaluating Trigonometric Functions of 0 and 0 Figure TECHNOLOGY You can use a calculator to convert the answers in Eample to decimals. However, the radical form is the eact value and in most cases, the eact value is preferred. STUDY TIP Because the angles 0, 5, and 0,, and occur frequentl in trigonometr, ou should learn to construct the triangles shown in Figures 9. and 9.. Use the equilateral triangle shown in Figure 9. to find the values of sin 0, cos 0, sin 0, and cos 0. Solution Use the Pthagorean Theorem and the equilateral triangle in Figure 9. to verif the lengths of the sides shown in the figure. For ou have adj, opp, and hp. So, sin 0 opp hp 0, and For adj, opp, and hp. So, sin 0 opp hp and cos 0 adj hp. cos 0 adj hp. SINES, COSINES, AND TANGENTS OF SPECIAL ANGLES sin 0 sin sin 5 sin sin 0 sin cos 5 cos cos 0 cos cos 0 cos 0, tan 0 tan tan 5 tan tan 0 tan In the preceding bo, note that sin 0 cos 0. This occurs because 0 and 0 are complementar angles. In general, it can be shown from the right triangle definitions that cofunctions of complementar angles are equal. That is, if is an acute angle, the following relationships are true. sin90 cos cos90 sin tan90 cot cot90 tan sec90 csc csc90 sec Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

20 58 Chapter 9 Trigonometric Functions Trigonometric Identities In trigonometr, a great deal of time is spent studing relationships between trigonometric functions (identities). FUNDAMENTAL TRIGONOMETRIC IDENTITIES Reciprocal Identities sin csc csc sin Quotient Identities tan sin cos Pthagorean Identities sin cos cos sec sec cos cot cos sin tan sec cot csc tan cot cot tan Note that sin represents sin, cos represents cos, and so on. EXAMPLE Appling Trigonometric Identities Let be an acute angle such that sin 0.. Find the values of (a) cos and (b) tan using trigonometric identities. Solution a. To find the value of cos, use the Pthagorean identit sin cos. So, ou have 0. cos cos Substitute 0. for sin. Subtract 0. from each side. cos Etract the positive square root. 0. b. Now, knowing the sine and cosine of, ou can find the tangent of to be tan sin cos Figure Use the definitions of cos and tan, and the triangle shown in Figure 9.5, to check these results. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

21 9. Right Triangle Trigonometr 585 NOTE Throughout this tet, angles are assumed to be measured in radians unless noted otherwise. For eample, sin means the sine of radian and sin means the sine of degree. Applications Involving Right Triangles To use a calculator to evaluate trigonometric functions of angles measured in degrees, first set the calculator to degree mode and then proceed as demonstrated in Section 9.. For instance, ou can find values of cos 8 and sec 8 as shown. Function cos 8 sec 8 Calculator Kestrokes COS 8 ENTER ENTER.570 COS Displa Man applications of trigonometr involve a process called solving right triangles. In this tpe of application, ou are usuall given one side of a right triangle and one of the acute angles and asked to find one of the other sides, or ou are given two sides and asked to find one of the acute angles. In Eample 5, the angle ou are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression. EXAMPLE 5 Using Trigonometr to Solve a Right Triangle Figure 9. = 5 ft Angle of elevation 78. Not drawn to scale A surveor is standing 5 feet from the base of the Washington Monument, as shown in Figure 9.. The surveor measures the angle of elevation to the top of the monument as 78.. How tall is the Washington Monument? Solution From Figure 9., ou see that tan 78. opp adj where 5 and is the height of the monument. So, the height of the Washington Monument is tan feet. EXAMPLE Using Trigonometr to Solve a Right Triangle You are 00 ards from a river. Rather than walking directl to the river, ou walk 00 ards along a straight path to the river s edge. Find the acute angle between this path and the river s edge, as illustrated in Figure d 00 d Figure 9.7 Solution From Figure 9.7, ou can see that the sine of the angle is So, sin opp 0. hp Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

22 58 Chapter 9 Trigonometric Functions 0 B now ou are able to recognize that is the acute angle that satisfies the equation sin. Suppose, however, that ou were given the equation sin 0. and were asked to find the acute angle. Because sin 0 and sin ou might guess that lies somewhere between 0 and 5. In a later section, ou will stud a method b which a more precise value of can be determined. EXAMPLE 7 Solving a Right Triangle Find the length c of the skateboard ramp shown in Figure c ft Figure 9.8 Solution From Figure 9.8, ou can see that sin 8. opp hp c. So, the length of the skateboard ramp is c sin feet. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

23 9. Right Triangle Trigonometr Eercises See for worked-out solutions to odd-numbered eercises.. Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent (i) (ii) (iii) (iv) (v) (vi) In Eercises, fill in the blanks.. Relative to the angle, the three sides of a right triangle are the side, the side, and the.. Cofunctions of angles are equal.. An angle that measures from the horizontal upward to an object is called the angle of, whereas an angle that measures from the horizontal downward to an object is called the angle of. In Eercises 5 8, find the eact values of the si trigonometric functions of the angle shown in the figure. (Use the Pthagorean Theorem to find the third side of the triangle.) hpotenuse adjacent adjacent opposite hpotenuse opposite adjacent hpotenuse opposite hpotenuse opposite adjacent In Eercises 9, find the eact values of the si trigonometric functions of the angle for each of the two triangles. Eplain wh the function values are the same In Eercises 0, sketch a right triangle corresponding to the trigonometric function of the acute angle. Use the Pthagorean Theorem to determine the third side and then find the other five trigonometric functions of.. tan. cos 5 5. sec. tan 5 7. sin 8. sec cot 0. csc 9 In Eercises 0, construct an appropriate triangle to complete the table. 0 90, 0 /. sin. cos. sec. tan 5. cot. csc 7. csc 8. sin 9. cot 0. tan 5 Function 7.5 deg rad.5 Function Value 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

24 588 Chapter 9 Trigonometric Functions In Eercises, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions.. sin 0, cos 0 (a) sin 0 (b) cos 0 (c) tan 0 (d) cot 0. sin 0 tan 0, (a) csc 0 (b) cot 0 (c) cos 0 (d) cot 0. cos (a) sin (b) tan (c) sec (d) csc90. sec 5 (a) cos (b) cot (c) cot90 (d) sin 5. cot 5 (a) tan (b) csc (c) cot90 (d) cos. cos 7 (a) sec (b) sin (c) cot (d) sin90 In Eercises 7, use trigonometric identities to transform the left side of the equation into the right side 0 < < /. 7. tan cot 8. cos sec 9. tan cos sin 0. cot sin cos. sin sin cos. cos cos sin. sec tan sec tan. sin cos sin sin 5. cos csc sec cos sin tan cot. csc tan In Eercises 7 5, use a calculator to evaluate each function. Round our answers to four decimal places. (Be sure the calculator is in the correct angle mode.) 7. (a) sin 0 (b) cos (a) tan.5 (b) cot.5 9. (a) sin.5 (b) csc (a) cot 79.5 (b) sec (a) cos 50 5 (b) sec (a) sec (b) csc (a) cot 5 (b) tan 5 5. (a) sec (b) cos (a) csc 0 (b) tan 8 5. (a) sec (b) cot In Eercises 57, find the values of 0 < < 90 and radians 0 < < / aid of a calculator. 57. (a) sin (b) csc 58. (a) cos (b) tan 59. (a) sec (b) cot 0. (a) tan (b) cos. (a) csc (b) sin. (a) cot (b) sec in degrees without the In Eercises, solve for,, or r as indicated.. Solve for.. Solve for Solve for.. Solve for r. 0 WRITING ABOUT CONCEPTS 7. In right triangle trigonometr, eplain wh sin 0 regardless of the size of the triangle. 8. You are given onl the value tan. Is it possible to find the value of sec without finding the measure of? Eplain. 0 5 r 0 0 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

25 9. Right Triangle Trigonometr 589 WRITING ABOUT CONCEPTS (continued) 9. (a) Complete the table. sin (b) Is or sin greater for in the interval 0, 0.5? (c) As approaches 0, how do and sin compare? Eplain. 70. (a) Complete the table. sin cos (b) Discuss the behavior of the sine function for in the range from 0 to 90. (c) Discuss the behavior of the cosine function for in the range from 0 to 90. (d) Use the definitions of the sine and cosine functions to eplain the results of parts (b) and (c). 7. Empire State Building You are standing 5 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 8th floor (the observator) is 8. If the total height of the building is another meters above the 8th floor, what is the approimate height of the building? One of our friends is on the 8th floor. What is the distance between ou and our friend? 7. Height A si-foot person walks from the base of a broadcasting tower directl toward the tip of the shadow cast b the tower. When the person is feet from the tower and feet from the tip of the shadow, the person s shadow starts to appear beond the tower s shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the tower? 7. Angle of Elevation You are skiing down a mountain with a vertical height of 500 feet. The distance from the top of the mountain to the base is 000 feet. What is the angle of elevation from the base to the top of the mountain? Width of a River A biologist wants to know the width w of a river so that instruments for studing the pollutants in the water can be set properl. From point A, the biologist walks downstream 00 feet and sights to point C (see figure). From this sighting, it is determined that How wide is the river? 75. Machine Shop Calculations A steel plate has the form of one-fourth of a circle with a radius of 0 centimeters. Two two-centimeter holes are to be drilled in the plate positioned as shown in the figure. Find the coordinates of the center of each hole. 0 5 Figure for 75 Figure for 7 7. Machine Shop Calculations A tapered shaft has a diameter of 5 centimeters at the small end and is 5 centimeters long (see figure). The taper is. Find the diameter d of the large end of the shaft. 77. Geometr Use a compass to sketch a quarter of a circle of radius 0 centimeters. Using a protractor, construct an angle of 0 in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. B actual measurement, calculate the coordinates, of the point of intersection and use these measurements to approimate the si trigonometric functions of a 0 angle (, ) 0 0 cm 0 w A C (, ) = 5 00 ft (, ) d 5 cm 5. 5 cm Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

26 590 Chapter 9 Trigonometric Functions 78. Height A 0-meter line is used to tether a helium-filled balloon. Because of a breeze, the line makes an angle of approimatel 85 with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the balloon? (d) The breeze becomes stronger and the angle the balloon makes with the ground decreases. How does this affect the triangle ou drew in part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures. Angle, Height Angle, Height (f) As the angle the balloon makes with the ground approaches 0, how does this affect the height of the balloon? Draw a right triangle to eplain our reasoning. 79. Length A gu wire runs from the ground to a cell tower. The wire is attached to the cell tower 50 feet above the ground. The angle formed between the wire and the ground is (see figure). 80. Height of a Mountain In traveling across flat land, ou notice a mountain directl in front of ou. Its angle of elevation (to the peak) is.5. After ou drive miles closer to the mountain, the angle of elevation is 9. Approimate the height of the mountain..5 9 mi Not drawn to scale True or False? In Eercises 8 8, determine whether the statement is true or false. Justif our answer. 8. sin 0 csc 0 8. sec 0 csc 0 8. sin 5 cos 5 8. cot 0 csc 0 sin sin sin 0 8. tan5 tan Geometr Use the equilateral triangle shown in Figure 9. and similar triangles to verif the points in Figure 9.8 (in Section 9.) that do not lie on the aes. CAPSTONE 88. The Johnstown Inclined Plane in Pennslvania is one of the longest and steepest hoists in the world. The railwa cars travel a distance of 89.5 feet at an angle of approimatel 5., rising to a height of 9.5 feet above sea level. 50 ft ft 9.5 feet above sea level Not drawn to scale = (a) How long is the gu wire? (b) How far from the base of the tower is the gu wire anchored to the ground? (a) Find the vertical rise of the inclined plane. (b) Find the elevation of the lower end of the inclined plane. (c) The cars move up the mountain at a rate of 00 feet per minute. Find the rate at which the rise verticall. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

27 9. Trigonometric Functions of An Angle Trigonometric Functions of An Angle Evaluate trigonometric functions of an angle. Use reference angles to evaluate trigonometric functions. Introduction In Section 9., the definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are etended to cover an angle. If is an acute angle, these definitions coincide with those given in the preceding section. DEFINITIONS OF TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Let be an angle in standard position with, a point on the terminal side of and r 0. sin r cos r (, ) tan, 0 sec r, 0 cot, 0 csc r, 0 r Because r cannot be zero, it follows that the sine and cosine functions are defined for an real value of. However, if 0, the tangent and secant of are undefined. For eample, the tangent of 90 is undefined. Similarl, if 0, the cotangent and cosecant of are undefined. EXAMPLE Evaluating Trigonometric Functions Let, be a point on the terminal side of. Find each trigonometric function. a. sin b. csc c. cos d. sec e. tan f. cot (, ) r Solution Referring to Figure 9.9, ou see that,, and r 5 5. So, ou have Figure 9.9 a. sin b. csc r Positive values 5 r 5 c. cos d. sec r Negative values r 5 5 e. tan f. cot Negative values. Notice that the sign of each trigonometric function is the same as the sign of its reciprocal. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

28 59 Chapter 9 Trigonometric Functions < < < 0 > 0 0 < < > 0 > 0 The signs of the trigonometric functions in the four quadrants can be determined from the definitions of the functions. For instance, because cos r, it follows that cos is positive wherever > 0, which is in Quadrants I and IV. (Remember, r is alwas positive.) In a similar manner, ou can verif the results shown in Figure 9.0. EXAMPLE Evaluating Trigonometric Functions < 0 < 0 < < > 0 < 0 < < Given tan 5 and cos > 0, find sin and sec. Solution Note that lies in Quadrant IV because that is the onl quadrant in which the tangent is negative and the cosine is positive. Moreover, using tan Quadrant II sin : cos : tan : + Quadrant I sin : + cos : ++ tan : 5 and the fact that is negative in Quadrant IV, ou can let 5 and. So, r 5 and ou have Quadrant III sin : cos : tan : + Figure 9.0 Quadrant IV sin : cos : + tan : sin sec r r EXAMPLE Trigonometric Functions of Quadrant Angles Evaluate the cosine and tangent functions at the four quadrant angles 0, and,,. Solution To begin, choose a point on the terminal side of each angle, as shown in Figure 9.. For each of the four points, r, and ou have the following. cos 0 r cos r 0 0 cos r cos r 0 0 tan tan 0 undefined tan 0 0 tan 0 undefined,, 0, 0,,, 0, 0, (0, ) (, 0) (, 0) 0 (0, ) Figure 9. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

29 9. Trigonometric Functions of An Angle 59 Reference Angles The values of the trigonometric functions of angles greater than 90 (or less than 0) can be determined from their values at corresponding acute angles called reference angles. DEFINITION OF REFERENCE ANGLE Let be an angle in standard position. Its reference angle is the acute angle formed b the terminal side of and the horizontal ais. Figure 9. shows the reference angles for in Quadrants II, III, and IV. Quadrant II Reference angle: Reference angle: Reference angle: Quadrant III Quadrant IV = (radians) = 80 (degrees) = (radians) = 80 (degrees) = (radians) = 0 (degrees) Figure 9. = 00 (a) =. (b) = 5 5 (c) Figure 9. = 0 =. = 5 5 and 5 are coterminal. EXAMPLE Finding Reference Angles Find the reference angle. 00 a. b. c. Solution a. Because 00 lies in Quadrant IV, the angle it makes with the -ais is Degrees Figure 9.(a) shows the angle and its reference angle b. Because. lies between.5708 and it follows that it is in Quadrant II and its reference angle is Radians Figure 9.(b) shows the angle and its reference angle c. First, determine that 5 is coterminal with 5, which lies in Quadrant III. So, the reference angle is Degrees Figure 9.(c) shows the angle , and its reference angle Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

30 59 Chapter 9 Trigonometric Functions (, ) Trigonometric Functions of An Angle To see how a reference angle is used to evaluate a trigonometric function, consider the point, on the terminal side of, as shown in Figure 9.. B definition, ou know that opp r = hp adj opp, adj Figure 9. sin r and For the right triangle with acute angle and sides of lengths and ou have sin opp hp r tan. and tan opp adj So, it follows that sin and sin are equal, ecept possibl in sign. The same is true for tan and tan and for the other four trigonometric functions. In all cases, the sign of the function value can be determined b the quadrant in which lies.., EVALUATING TRIGONOMETRIC FUNCTIONS OF ANY ANGLE To find the value of a trigonometric function of an angle :. Determine the function value for the associated reference angle.. Depending on the quadrant in which lies, affi the appropriate sign to the function value. B using reference angles and the special angles discussed in the preceding section, ou can greatl etend the scope of eact trigonometric values. For instance, knowing the function values of 0 means that ou know the function values of all angles for which 0 is a reference angle. For convenience, the table below shows the eact values of the trigonometric functions of special angles and quadrant angles. Trigonometric Values of Common Angles degrees radians 0 sin cos tan Undef. 0 Undef. STUDY TIP Learning the table of values above is worth the effort. Doing so will increase both our efficienc and our confidence, especiall with calculus. Here are patterns for the sine and cosine functions that ma help ou remember the values sin 0 cos 0 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

31 9. Trigonometric Functions of An Angle 595 EXAMPLE 5 Trigonometric Functions of Nonacute Angles Evaluate each trigonometric function. a. cos b. tan0 c. csc Solution a. Because lies in Quadrant III, the reference angle is, as shown in Figure 9.5(a). Moreover, the cosine is negative in Quadrant III, so cos cos. b. Because , it follows that 0 is coterminal with the second-quadrant angle 50. So, the reference angle is , as shown in Figure 9.5(b). Finall, because the tangent is negative in Quadrant II, ou have tan0 tan 0. c. Because, it follows that is coterminal with the second-quadrant angle. So, the reference angle is as shown in Figure 9.5(c). Because the cosecant is positive in Quadrant II, ou have csc csc. sin, = = 0 = = = = 0 (a) (b) (c) Figure 9.5 EXAMPLE Using Trigonometric Identities Let be an angle in Quadrant II such that sin. Find cos. Solution Using the Pthagorean identit sin cos, ou obtain cos Substitute for sin. cos Because cos cos < 0 8 in Quadrant II, ou can use the negative root to obtain 9. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

32 59 Chapter 9 Trigonometric Functions 9. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, fill in the blanks. Let be an angle in standard position, with, a point on the terminal side of and r 0.. sin. r. tan. sec 5.. r In Eercises 7 and 8, fill in the blanks. 7. Because r cannot be, the sine and cosine functions are for an real value of. 8. The acute positive angle that is formed b the terminal side of the angle and the horizontal ais is called the angle of and is denoted b. In Eercises 9, determine the eact values of the si trigonometric functions of the angle. 9. (a) (b) (, ) 0. (a) (b) (, 5). (a) (b) (, ). (a) (b) (, ) ( 8, 5) (, ) In Eercises 8, the point is on the terminal side of an angle in standard position. Determine the eact values of the si trigonometric functions of the angle.. 5,. 8, 5 (, ) (, ) 5. 5,., , 7. 8., 7 In Eercises 9, state the quadrant in which 9. sin > 0 and cos > 0 0. sin < 0 and cos < 0. sin > 0 and cos < 0. sec > 0 and cot < 0 lies. In Eercises, find the values of the si trigonometric functions of with the given constraint. Function Value Constraint. tan 5 8. cos 8 7 sin > 0 tan < 0 5. sin 5 lies in Quadrant II.. cos 5 lies in Quadrant III. 7. cot 8. csc 9. sec cos > 0 cot < 0 sin < 0 0. sin 0 sec. cot is undefined.. tan is undefined. In Eercises, the terminal side of lies on the given line in the specified quadrant. Find the values of the si trigonometric functions of b finding a point on the line. Line Quadrant. II. III 5. 0 III. 0 IV In Eercises 7, evaluate the trigonometric function of the quadrant angle. 7. sin 8. csc 9. sec 0. sec. sin. cot. csc. cot Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

33 9. Trigonometric Functions of An Angle 597 In Eercises 5 5, find the reference angle, and sketch and in standard position In Eercises 5 8, evaluate the sine, cosine, and tangent of the angle without using a calculator In Eercises 9 7, find the indicated trigonometric value in the specified quadrant. Function Quadrant Trigonometric Value 9. sin 5 IV cos 70. cot II sin 7. tan III sec 7. csc IV cot 7. cos 5 8 I sec 7. sec 9 III tan In Eercises 75 8, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) 75. sin 0 7. sec cos0 78. csc0 79. tan cot.5 8. tan9 8. tan9 8. sin sec cot8 8. csc5 7 In Eercises 87 9, find two solutions of the equation. Give our answers in degrees 0 < 0 and in radians 0 <. Do not use a calculator. 87. (a) sin (b) sin 88. (a) cos (b) cos 89. (a) csc (b) cot 90. (a) sec (b) sec 9. (a) tan (b) cot 9. (a) sin (b) sin WRITING ABOUT CONCEPTS 9. Consider an angle in standard position with r centimeters, as shown in the figure. Write a short paragraph describing the changes in the values of,, sin, cos, and tan as increases continuousl from 0 to The figure shows point P, on a unit circle and right triangle OAP. O r cm P(, ) A (a) Find sin t and cos t using the unit circle definitions of sine and cosine (from Section 9.). (b) What is the value of r? Eplain. (c) Use the definitions of sine and cosine given in this section to find sin and cos. Write our answers in terms of and. (d) Based on our answers to parts (a) and (c), what can ou conclude? t (, ) Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

34 598 Chapter 9 Trigonometric Functions 95. Sales A compan that produces snowboards, which are seasonal products, forecasts monthl sales over the net ears to be S. 0.t. cost, where S is measured in thousands of units and t is the time in months, with t representing Januar 00. Predict sales for each of the following months. (a) Februar 00 (b) Februar 0 (c) June 00 (d) June 0 9. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos t, where is the displacement (in centimeters) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t and (c) t,. (a) Use the regression feature of a graphing utilit to find a model of the form a sinbt c d for each cit. Let t represent the month, with t corresponding to Januar. (b) Use the models from part (a) to find the monthl normal temperatures for the two cities in Februar, March, Ma, June, August, September, and November. (c) Compare the models for the two cities. 00. Distance An airplane, fling at an altitude of miles, is on a flight path that passes directl over an observer (see figure). If is the angle of elevation from the observer to the plane, find the distance d from the observer to the plane when (a) (b) and (c) 0. 0, 90, Equilibrium Displacement d mi Not drawn to scale Figure for 9 and Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring and subject to the damping effect of friction is given b t e t cos t, where is the displacement (in centimeters) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t and (c) t,. 98. Electric Circuits The current I (in amperes) when 00 volts is applied to a circuit is given b I 5e t sin t, where t is the time (in seconds) after the voltage is applied. Approimate the current at t 0.7 second after the voltage is applied. 99. Data Analsis: Meteorolog The table shows the monthl normal temperatures (in degrees Fahrenheit) for selected months in New York Cit N and Fairbanks, Alaska F. (Source: National Climatic Data Center) Month New York Cit, N Fairbanks, F Januar 0 April 5 Jul 77 October 58 True or False? In Eercises 0 0, determine whether the statement is true or false. Justif our answer. 0. In each of the four quadrants, the signs of the secant function and sine function will be the same. 0. To find the reference angle for an angle (given in degrees), find the integer n such that 0 0n 0. The difference 0n is the reference angle. 0. If sin and cos < 0, then tan 5 5. CAPSTONE 0. Write a short paper in our own words eplaining to a classmate how to evaluate the si trigonometric functions of an angle in standard position. Include an eplanation of reference angles and how to use them, the signs of the functions in each of the four quadrants, and the trigonometric values of common angles. Be sure to include figures or diagrams in our paper. December 8 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

35 9.5 Graphs of Sine and Cosine Functions Graphs of Sine and Cosine Functions Sketch the graphs of basic sine and cosine functions. Use amplitude and period to help sketch the graphs of sine and cosine functions. Sketch translations of the graphs of sine and cosine functions. Use sine and cosine functions to model real-life data. Basic Sine and Cosine Curves In this section, ou will stud techniques for sketching the graphs of the sine and cosine functions. The graph of the sine function is a sine curve. In Figure 9., the black portion of the graph represents one period of the function and is called one ccle of the sine curve. The gra portion of the graph indicates that the basic sine curve repeats indefinitel in the positive and negative directions. The graph of the cosine function is shown in Figure 9.7. Recall from Section 9. that the domain of the sine and cosine functions is the set of all real numbers. Moreover, the range of each function is the interval,, and each function has a period of. Do ou see how this information is consistent with the basic graphs shown in Figures 9. and 9.7? = sin Range: 5 Figure 9. Period: = cos Range: 5 Figure 9.7 Period: Note in Figures 9. and 9.7 that the sine curve is smmetric with respect to the origin, whereas the cosine curve is smmetric with respect to the -ais. These properties of smmetr follow from the fact that the sine function is odd and the cosine function is even. Note also that the cosine curve appears to be a left shift of of the sine curve. More will be said about this later in the section. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

36 00 Chapter 9 Trigonometric Functions To sketch the graphs of the basic sine and cosine functions b hand, it helps to note five ke points in one period of each graph: the intercepts, maimum points, and minimum points (see Figure 9.8). Maimum Intercept Minimum Intercept Intercept (, ) = sin (0, 0) Figure 9.8 Quarter period Period: Half period (, 0) (, ) Three-quarter period (, 0) Full period (0, ) Intercept Minimum Maimum = cos Quarter period Period: (, 0 ) (, 0) (, ) Half period Intercept Maimum (, ) Three-quarter period Full period EXAMPLE Using Ke Points to Sketch a Sine Curve Sketch the graph of sin on the interval,. Solution Note that sin sin indicates that the -values for the ke points will have twice the magnitude of those on the graph of sin. Divide the period into four equal parts to get the ke points for sin. Intercept 0, 0, B connecting these ke points with a smooth curve and etending the curve in both directions over the interval,, ou obtain the graph shown in Figure 9.9. Maimum,, Intercept, 0, Minimum,, and Intercept, 0 = sin = sin 5 7 Figure 9.9 TECHNOLOGY When using a graphing utilit to graph trigonometric functions, pa special attention to the viewing window ou use. For instance, tr graphing sin0/0 in the standard viewing window in radian mode. What do ou observe? Use the zoom feature to find a viewing window that displas a good view of the graph. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

37 9.5 Graphs of Sine and Cosine Functions 0 Amplitude and Period In the remainder of this section ou will stud the graphic effect of each of the constants a, b, c, and d in equations of the forms d a sinb c and d a cosb c. A quick review of the transformations ou studied in Section. should help in this investigation. The constant factor a in a sin acts as a scaling factor a vertical stretch or vertical shrink of the basic sine curve. If the basic sine curve is stretched, and if a a >, <, the basic sine curve is shrunk. The result is that the graph of a sin ranges between a and a instead of between and. The absolute value of a is the amplitude of the function a sin. The range of the function a sin for a > 0 is a a. DEFINITION OF AMPLITUDE OF SINE AND COSINE CURVES The amplitude of a sin and a cos represents half the distance between the maimum and minimum values of the function and is given b Amplitude a. EXAMPLE Scaling: Vertical Shrinking and Stretching On the same coordinate aes, sketch the graph of each function. a. cos Figure 9.0 = cos = cos = cos b. cos Solution a. Because the amplitude of is the maimum value is and the minimum value is cos,. Divide one ccle, 0, into four equal parts to get the ke points Maimum 0,,, and, 0,, b. A similar analsis shows that the amplitude of cos is, and the ke points are Maimum Intercept Intercept Minimum Minimum Intercept, 0, Intercept, 0, Maimum,. Maimum 0,,,, and,., 0, The graphs of these two functions are shown in Figure 9.0. Notice that the graph of cos is a vertical shrink of the graph of cos and the graph of cos is a vertical stretch of the graph of cos. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

38 0 Chapter 9 Trigonometric Functions = cos = cos You know from Section. that the graph of f is a reflection in the -ais of the graph of f. For instance, the graph of cos is a reflection of the graph of cos, as shown in Figure 9.. Because a sin completes one ccle from 0 to, it follows that a sin b completes one ccle from 0 to b. PERIOD OF SINE AND COSINE FUNCTIONS Let b be a positive real number. The period of a sin b and a cos b is given b Figure 9. Period b. Note that if 0 < b <, the period of a sin b is greater than and represents a horizontal stretching of the graph of a sin. Similarl, if b >, the period of a sin b is less than and represents a horizontal shrinking of the graph of a sin. If b is negative, the identities sin sin and cos cos are used to rewrite the function. EXAMPLE Scaling: Horizontal Stretching Sketch the graph of sin. STUDY TIP In general, to divide a period-interval into four equal parts, successivel add period, starting with the left endpoint of the interval. For instance, for the period-interval, of length, ou would successivel add to get, 0,,, and as the -values for the ke points on the graph. Solution The amplitude is. Moreover, because b, the period is Substitute for b. Now, divide the period-interval 0, into four equal parts with the values,, and to obtain the ke points on the graph. b Intercept 0, 0, The graph is shown in Figure 9.. Figure 9.. = sin = sin Maimum,, Intercept, 0, Period: Minimum,, and Intercept, 0 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

39 9.5 Graphs of Sine and Cosine Functions 0 Translations of Sine and Cosine Curves The constant c in the general equations a sinb c and a cosb c creates a horizontal translation (shift) of the basic sine and cosine curves. Comparing a sin b with a sinb c, ou find that the graph of a sinb c completes one ccle from b c 0 to b c. B solving for, ou can find the interval for one ccle to be Left endpoint Right endpoint c b c b b. Period This implies that the period of a sinb c is b, and the graph of a sin b is shifted b an amount cb. The number cb is the phase shift. GRAPHS OF SINE AND COSINE FUNCTIONS The graphs of a sinb c and a cosb c have the following characteristics. (Assume b > 0. ) Amplitude a Period b The left and right endpoints of a one-ccle interval can be determined b solving the equations b c 0 and b c. EXAMPLE Horizontal Translation Analze the graph of Algebraic Solution The amplitude is and the period is. B solving the equations and sin 0 ou see that the interval, 7 corresponds to one ccle of the graph. Dividing this interval into four equal parts produces the ke points Intercept, 0,. Maimum 5,, 7 Intercept, 0, Minimum,, Intercept and 7, 0. Graphical Solution Use a graphing utilit set in radian mode to graph sin, as shown in Figure 9.. Use the minimum, maimum, and zero or root features of the graphing utilit to approimate the ke points.05, 0,., 0.5,.9, 0, 5.7, 0.5, and 7., 0. = sin ( ) 5 Figure 9. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

40 0 Chapter 9 Trigonometric Functions = cos( + ) EXAMPLE 5 Horizontal Translation Sketch the graph of cos. Solution The amplitude is and the period is B solving the equations. 0 and Period: Figure 9. ou see that the interval, corresponds to one ccle of the graph. Dividing this interval into four equal parts produces the ke points Minimum Intercept Maimum Intercept Minimum,, 7, 0,,, 5, 0, and,. The graph is shown in Figure 9.. The final tpe of transformation is the vertical translation caused b the constant in the equations d d a sinb c and d a cosb c. The shift is d units upward for d > 0 and d units downward for d < 0. In other words, the graph oscillates about the horizontal line d instead of about the -ais. EXAMPLE Vertical Translation Sketch the graph of cos. Solution The amplitude is and the period is. The ke points over the interval 0, are 0, 5,,,,,,, and, 5. The graph is shown in Figure 9.5. Compared with the graph of f cos, the graph of cos is shifted upward two units. 5 = + cos Period Figure 9.5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

41 9.5 Graphs of Sine and Cosine Functions 05 Mathematical Modeling Sine and cosine functions can be used to model man real-life situations, including electric currents, musical tones, radio waves, tides, and weather patterns. EXAMPLE 7 Finding a Trigonometric Model Throughout the da, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tides. The table shows the depths (in feet) at various times during the morning. (Source: Nautical Software, Inc.) Time, t Midnight A.M. A.M. A.M. 8 A.M. 0 A.M. Noon Depth, Depth (in feet) 0 8 Figure 9. A.M. 8 A.M. Noon Time 0 0 = 5. cos(0.5t.09) Figure 9.7 (.7, 0) (7., 0) = 0 t a. Use a trigonometric function to model the data. b. Find the depths at 9 A.M. and P.M. c. A boat needs at least 0 feet of water to moor at the dock. During what times in the afternoon can it safel dock? Solution a. Begin b graphing the data, as shown in Figure 9.. You can use either a sine or a cosine model. Use a cosine model of the form a cosbt c d. The difference between the maimum height and the minimum height of the graph is twice the amplitude of the function. So, the amplitude is a maimum depth minimum depth The cosine function completes one half of a ccle between the times at which the maimum and minimum depths occur. So, the period is p time of min. depth time of ma. depth 0 which implies that b p 0.5. Because high tide occurs hours after midnight, consider the left endpoint to be cb, so c.09. Moreover, because the average depth is , it follows that d 5.7. So, ou can model the depth with the function given b 5. cos0.5t b. The depths at 9 A.M. and P.M. are as follows. 5. cos foot 5. cos feet A.M. P.M. 5.. c. To find out when the depth is at least 0 feet, ou can graph the model with the line 0 using a graphing utilit, as shown in Figure 9.7. Using the intersect feature, ou can determine that the depth is at least 0 feet between : P.M. t.7 and 5:8 P.M. t 7.. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

42 0 Chapter 9 Trigonometric Functions 9.5 Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, fill in the blanks.. One period of a sine or cosine function is called one of the sine or cosine curve.. The of a sine or cosine curve represents half the distance between the maimum and minimum values of the function. c. For the function given b a sinb c, b represents the of the graph of the function.. For the function given b d a cosb c, d represents a of the graph of the function. In Eercises 5 8, find the period and amplitude. 5. sin 5. cos 0. sin.. sin 0. 5 sin 5. 5 cos. 5 cos sin cos sin cos cos cos sin 0 In Eercises 9, describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts. 9. f sin 0. f cos g sin g cos. f cos. f sin g cos g sin. f cos. f sin g cos g sin 5. f sin. f cos g sin g cos In Eercises 7 0, describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts In Eercises 8, graph f and g on the same set of coordinate aes. (Include two full periods.). f sin. f sin g sin g. f cos. f cos g cos g cos 5. f. f sin sin g sin f f g g sin g sin 7. f cos 8. f cos g cos g cos g g f f Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

43 9.5 Graphs of Sine and Cosine Functions 07 In Eercises 9 0, sketch the graph of the function. (Include two full periods.) 9. 5 sin 0. sin. cos. cos. cos. sin 5. cos. sin 57. cos 58. cos In Eercises, g is related to a parent function f sin or f cos. (a) Describe the sequence of transformations from f to g. (b) Sketch the graph of g. (c) Use function notation to write g in terms of f.. g sin. g sin. g cos. g cos 5. g sin. g sin In Eercises 7 7, use a graphing utilit to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. 7. sin cos cos sin 0 7. Graphical Reasoning In Eercises 7 7, find a and d for the function f a cos d such that the graph of f matches the figure. 7. sin 8. 0 cos 5. cos 5. cos 9. sin 50. sin t 5. sin 5. 5 cos cos 0 5. cos cos cos sin sin 0t Graphical Reasoning In Eercises 77 80, find a, b, and c for the function f a sinb c such that the graph of f matches the figure f 0 8 f f f In Eercises 8 and 8, use a graphing utilit to graph and in the interval [, ]. Use the graphs to find real numbers such that. 8. sin ; 8. cos ; In Eercises 8 8, write an equation for the function that is described b the given characteristics. 8. A sine curve with a period of, an amplitude of, a right phase shift of, and a vertical translation up unit 8. A sine curve with a period of, an amplitude of, a left phase shift of, and a vertical translation down unit 85. A cosine curve with a period of, an amplitude of, a left phase shift of, and a vertical translation down units 8. A cosine curve with a period of, an amplitude of, a right phase shift of, and a vertical translation up units 5 f f f Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

44 08 Chapter 9 Trigonometric Functions WRITING ABOUT CONCEPTS 87. Sketch the graph of cos b for b,, and. How does the value of b affect the graph? How man complete ccles occur between 0 and for each value of b? 88. Sketch the graph of sin c for c, 0, and. How does the value of c affect the graph? 89. Use a graphing utilit to graph h, and use the graph to decide whether h is even, odd, or neither. (a) h cos (b) h sin 90. If f is an even function and g is an odd function, use the results of Eercise 89 to make a conjecture about h, where (a) h f. (b) h g. 9. Respirator Ccle For a person at rest, the velocit v (in liters per second) of airflow during a respirator ccle (the time from the beginning of one breath to t the beginning of the net) is given b v 0.85 sin, where t is the time (in seconds). (Inhalation occurs when v > 0, and ehalation occurs when v < 0. ) (a) Find the time for one full respirator ccle. (b) Find the number of ccles per minute. (c) Sketch the graph of the velocit function. 9. Respirator Ccle After eercising for a few minutes, a person has a respirator ccle for which the velocit of t airflow is approimated b v.75 sin where t is, the time (in seconds). (Inhalation occurs when v > 0, and ehalation occurs when v < 0. ) (a) Find the time for one full respirator ccle. (b) Find the number of ccles per minute. (c) Sketch the graph of the velocit function. 9. Data Analsis: Meteorolog The table shows the maimum dail high temperatures in Las Vegas L and International Falls I (in degrees Fahrenheit) for month t, with t corresponding to Januar. (Source: National Climatic Data Center) t 5 L I t L I (a) A model for the temperature in Las Vegas is given b t Lt cos.7. Find a trigonometric model for International Falls. (b) Use a graphing utilit to graph the data points and the model for the temperatures in Las Vegas. How well does the model fit the data? (c) Use a graphing utilit to graph the data points and the model for the temperatures in International Falls. How well does the model fit the data? (d) Use the models to estimate the average maimum temperature in each cit. Which term of the models did ou use? Eplain. (e) What is the period of each model? Are the periods what ou epected? Eplain. (f) Which cit has the greater variabilit in temperature throughout the ear? Which factor of the models determines this variabilit? Eplain. 9. Health The function given b P 00 0 cos 5t approimates the blood pressure P (in millimeters of mercur) at time t (in seconds) for a person at rest. (a) Find the period of the function. (b) Find the number of heartbeats per minute. 95. Piano Tuning When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion that can be approimated b 0.00 sin 880t, where t is the time (in seconds). (a) What is the period of the function? (b) The frequenc f is given b f p. What is the frequenc of the note? 9. Data Analsis: Astronom The percents (in decimal form) of the moon s face that was illuminated on da in the ear 009, where represents Januar, are shown in the table. (Source: U.S. Naval Observator) (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. (c) Add the graph of our model in part (b) to the scatter plot. How well does the model fit the data? (d) What is the period of the model? (e) Estimate the moon s percent illumination for March, 009. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

45 9.5 Graphs of Sine and Cosine Functions Fuel Consumption The dail consumption C (in gallons) of diesel fuel on a farm is modeled b C 0.. sin t where t is the time (in das), with t corresponding to Januar. (a) What is the period of the model? Is it what ou epected? Eplain. (b) What is the average dail fuel consumption? Which term of the model did ou use? Eplain. (c) Use a graphing utilit to graph the model. Use the graph to approimate the time of the ear when consumption eceeds 0 gallons per da. 98. Ferris Wheel A Ferris wheel is built such that the height h (in feet) above ground of a seat on the wheel at time t (in seconds) can be modeled b ht 5 50 sin 0 t (a) Find the period of the model. What does the period tell ou about the ride? (b) Find the amplitude of the model. What does the amplitude tell ou about the ride? (c) Use a graphing utilit to graph one ccle of the model.. True or False? In Eercises 99 0, determine whether the statement is true or false. Justif our answer. 99. The graph of the function given b f sin translates the graph of f sin eactl one period to the right so that the two graphs look identical. 00. The function given b cos has an amplitude that is twice that of the function given b cos. 0. The graph of cos is a reflection of the graph of sin in the -ais. CAPSTONE 0. Use a graphing utilit to graph the function given b d a sinb c, for several different values of a, b, c, and d. Write a paragraph describing the changes in the graph corresponding to changes in each constant. Conjecture In Eercises 0 and 0, graph f and g on the same set of coordinate aes. Include two full periods. Make a conjecture about the functions. 0. f sin, g cos 0. f sin, g cos SECTION PROJECT Approimating Sine and Cosine Functions Using calculus, it can be shown that the sine and cosine functions can be approimated b the polnomials sin 5! 5! and cos!! where is in radians. (a) Use a graphing utilit to graph the sine function and its polnomial approimation in the same viewing window. How do the graphs compare? (b) Use a graphing utilit to graph the cosine function and its polnomial approimation in the same viewing window. How do the graphs compare? (c) Stud the patterns in the polnomial approimations of the sine and cosine functions and guess the net term in each. Then repeat parts (a) and (b). How did the accurac of the approimations change when additional terms were added? (d) Use the polnomial approimation for the sine function to approimate the following functional values. Compare the results with those given b a calculator. Is the error in the approimation the same in each case? Eplain our reasoning. (i) sin (ii) sin (iii) sin (e) Use the polnomial approimation for the cosine function to approimate the following functional values. Compare the results with those given b a calculator. Is the error in the approimation the same in each case? (i) cos0.5 (ii) cos (iii) cos Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

46 0 Chapter 9 Trigonometric Functions 9. Graphs of Other Trigonometric Functions Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch the graphs of secant and cosecant functions. Sketch the graphs of damped trigonometric functions. Graph of the Tangent Function Recall that the tangent function is odd. That is, tan tan. Consequentl, the graph of tan is smmetric with respect to the origin. You also know from the identit tan sin cos that the tangent is undefined for values at which cos 0. Two such values are ± ± tan Undef Undef. As indicated in the table, tan increases without bound as approaches from the left, and decreases without bound as approaches from the right. So, the graph of tan has vertical asmptotes at and, as shown in Figure 9.8. Moreover, because the period of the tangent function is, vertical asmptotes also occur when n, where n is an integer. The domain of the tangent function is the set of all real numbers other than n, and the range is the set of all real numbers. = tan Period: Domain: all n Range: (, ) Vertical asmptotes: n Smmetr: Origin Figure 9.8 Sketching the graph of a tanb c is similar to sketching the graph of a sinb c in that ou locate ke points that identif the intercepts and asmptotes. Two consecutive vertical asmptotes can be found b solving the equations b c and b c. The midpoint between two consecutive vertical asmptotes is an -intercept of the graph. The period of the function a tanb c is the distance between two consecutive vertical asmptotes. The amplitude of a tangent function is not defined. After plotting the asmptotes and the -intercept, plot a few additional points between the two asmptotes and sketch one ccle. Finall, sketch one or two additional ccles to the left and right. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

47 9. Graphs of Other Trigonometric Functions = tan EXAMPLE Sketching the Graph of a Tangent Function Sketch the graph of Figure 9.9 Solution tan. B solving the equations and ou can see that two consecutive vertical asmptotes occur at and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure tan Undef. 0 Undef. EXAMPLE Sketching the Graph of a Tangent Function = tan Sketch the graph of tan. Figure 9.50 Solution B solving the equations and ou can see that two consecutive vertical asmptotes occur at and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure tan Undef. 0 Undef. B comparing the graphs in Eamples and, ou can see that the graph of a tanb c increases between consecutive vertical asmptotes when a > 0, and decreases between consecutive vertical asmptotes when a < 0. In other words, the graph for a < 0 is a reflection in the -ais of the graph for a > 0. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

48 Chapter 9 Trigonometric Functions TECHNOLOGY Some graphing utilities have difficult graphing trigonometric functions that have vertical asmptotes. Your graphing utilit ma connect parts of the graphs of tangent, cotangent, secant, and cosecant functions that are not supposed to be connected. To eliminate this problem, change the mode of the graphing utilit to dot mode. Graph of the Cotangent Function The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of. However, from the identit cot cos sin ou can see that the cotangent function has vertical asmptotes when sin is zero, which occurs at n, where n is an integer. The graph of the cotangent function is shown in Figure 9.5. Note that two consecutive vertical asmptotes of the graph of a cotb c can be found b solving the equations b c 0 and b c. = cot Period: Domain: all n Range: (, ) Vertical asmptotes: n Smmetr: Origin Figure 9.5 EXAMPLE Sketching the Graph of a Cotangent Function Sketch the graph of = cot cot. Solution B solving the equations 0 0 and ou can see that two consecutive vertical asmptotes occur at 0 and. Figure 9.5 Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure 9.5. Note that the period is, the distance between consecutive asmptotes. 0 9 cot Undef. 0 Undef. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

49 9. Graphs of Other Trigonometric Functions Graphs of the Reciprocal Functions The graphs of the two remaining trigonometric functions can be obtained from the graphs of the sine and cosine functions using the reciprocal identities csc sin and For instance, at a given value of, the -coordinate of sec is the reciprocal of the -coordinate of cos. Of course, when cos 0, the reciprocal does not eist. Near such values of, the behavior of the secant function is similar to that of the tangent function. In other words, the graphs of tan sin cos and have vertical asmptotes at n, where n is an integer, and the cosine is zero at these -values. Similarl, cot cos sin and sec cos. sec cos csc sin have vertical asmptotes where sin 0 that is, at n. To sketch the graph of a secant or cosecant function, ou should first make a sketch of its reciprocal function. For instance, to sketch the graph of csc, first sketch the graph of sin. Then take reciprocals of the -coordinates to obtain points on the graph of csc. This procedure is used to obtain the graphs shown in Figure 9.5. = csc = sec = sin = cos Cosecant: relative minimum Sine: maimum Figure 9.5 Sine: minimum Cosecant: relative maimum Period: Period: Domain: All n Domain: All n Range: (,, ) Range: (,, ) Vertical asmptotes: n Vertical asmptotes: n Smmetr: Origin Smmetr: -ais Figure 9.5 In comparing the graphs of the cosecant and secant functions with those of the sine and cosine functions, note that the hills and valles are interchanged. For eample, a hill (or maimum point) on the sine curve corresponds to a valle (a relative minimum) on the cosecant curve, and a valle (or minimum point) on the sine curve corresponds to a hill (a relative maimum) on the cosecant curve, as shown in Figure 9.5. Additionall, -intercepts of the sine and cosine functions become vertical asmptotes of the cosecant and secant functions, respectivel (see Figure 9.5). Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

50 Chapter 9 Trigonometric Functions = csc ( + ) Figure 9.55 = sin ( + ) EXAMPLE Sketching the Graph of a Cosecant Function Sketch the graph of Solution csc Begin b sketching the graph of sin For this function, the amplitude is and the period is. B solving the equations 0.. and 7 ou can see that one ccle of the sine function corresponds to the interval from to 7. The graph of this sine function is represented b the gra curve in Figure Because the sine function is zero at the midpoint and endpoints of this interval, the corresponding cosecant function csc sin has vertical asmptotes at 7,, etc., The graph of the cosecant function is represented b the black curve in Figure EXAMPLE 5 Sketching the Graph of a Secant Function Sketch the graph of = sec = cos sec. Solution Begin b sketching the graph of cos, as indicated b the gra curve in Figure 9.5. Then, form the graph of sec as the black curve in the figure. Note that the -intercepts of cos Figure 9.5, 0, correspond to the vertical asmptotes,, 0,,, 0,...,... of the graph of sec. Moreover, notice that the period of cos and sec is. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

51 9. Graphs of Other Trigonometric Functions 5 Damped Trigonometric Graphs A product of two functions can be graphed using properties of the individual functions. For instance, consider the function = = f sin as the product of the functions and sin. Using properties of absolute value and the fact that ou have 0 sin. Consequentl, sin, sin Figure 9.57 f() = sin which means that the graph of f sin lies between the lines and. Furthermore, because and f sin ± f sin 0 at at n n the graph of f touches the line or the line at n and has -intercepts at n. A sketch of f is shown in Figure In the function f sin, the factor is called the damping factor. EXAMPLE Damped Sine Wave Sketch the graph of f e sin. Solution Consider f as the product of the two functions e and sin each of which has the set of real numbers as its domain. For an real number, ou know that e 0 and So, which means that sin. e e sin e. e sin e, Furthermore, because f e sin ±e at n and f() = e sin = e = e at the graph of f touches the curves at f e sin 0 e and e and has intercepts at n. n n Figure 9.58 A sketch is shown in Figure Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

52 Chapter 9 Trigonometric Functions 9. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 8, fill in the blanks.. The tangent, cotangent, and cosecant functions are, so the graphs of these functions have smmetr with respect to the.. The graphs of the tangent, cotangent, secant, and cosecant functions all have asmptotes.. To sketch the graph of a secant or cosecant function, first make a sketch of its corresponding function.. For the functions given b f g sin, g is called the factor of the function f. 5. The period of tan is.. The domain of cot is all real numbers such that. 7. The range of sec is. 8. The period of csc is. In Eercises 9, match the function with its graph. State the period of the function. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) In Eercises 5 8, sketch the graph of the function. Include two full periods. 5. tan. tan 7. tan 8. tan 9. sec 0. sec. csc. csc. sec. sec 5. csc. csc 7. cot 8. cot 9. sec 0. tan. tan. tan. csc. csc 5. sec. sec cot csc In Eercises 9 8, use a graphing utilit to graph the function. Include two full periods. 9. tan 0. tan (c) (e) (d) (f) 9. sec 0.. cot.. sec. tan csc sec. sec. sec. tan. cot 5. csc tan 8. sec sec In Eercises 9 5, use a graph to solve the equation on the interval [, ]. 9. tan 50. tan 5. cot 5. cot 5. sec 5. sec 55. csc 5. csc Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

53 9. Graphs of Other Trigonometric Functions 7 In Eercises 57, use the graph of the function to determine whether the function is even, odd, or neither. Verif our answer algebraicall. 57. f sec 58. f tan 59. g cot 0. g csc. f tan. f sec. g csc. g cot In Eercises 5 70, use a graphing utilit to graph the two equations in the same viewing window. Use the graphs to determine whether the epressions are equivalent. Verif the results algebraicall. 5.. sin csc, sin sec, tan tan cot, 9. cot, 70. sec, In Eercises 7 7, match the function with its graph. Describe the behavior of the function as approaches zero. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) Conjecture In Eercises 75 78, graph the functions f and g. Use the graphs to make a conjecture about the relationship between the functions cos sin, 5 f cos g sin cot f sin cos f sin cos cot csc tan (b) (d) f sin 7. 7.,, g cos g 0 g sin f sin, g cos f cos, g cos In Eercises 79 8, use a graphing utilit to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as increases without bound. 79. g e sin 80. f e cos 8. f cos 8. h sin In Eercises 8 88, use a graphing utilit to graph the function. Describe the behavior of the function as approaches zero. 8. cos, 8. sin, > 0 > 0 cos 85. g sin 8. f 87. f sin 88. h sin WRITING ABOUT CONCEPTS 89. Consider the functions given b f sin and g csc on the interval 0,. (a) Graph f and g in the same coordinate plane. (b) Approimate the interval in which f > g. (c) Describe the behavior of each of the functions as approaches. How is the behavior of g related to the behavior of f as approaches? 90. Consider the functions given b f tan and g sec on the interval,. (a) Use a graphing utilit to graph f and g in the same viewing window. (b) Approimate the interval in which f < g. (c) Approimate the interval in which f < g. How does the result compare with that of part (b)? Eplain. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

54 8 Chapter 9 Trigonometric Functions 9. Distance A plane fling at an altitude of 7 miles above a radar antenna will pass directl over the radar antenna (see figure). Let d be the ground distance from the antenna to the point directl under the plane and let be the angle of elevation to the plane from the antenna. ( d is positive as the plane approaches the antenna.) Write d as a function of and graph the function over the interval 0 < <. Temperature (in degrees Fahrenheit) H(t) Month of ear L(t) t 9. Television Coverage A television camera is on a reviewing platform 7 meters from the street on which a parade will be passing from left to right (see figure). Write the distance d from the camera to a particular unit in the parade as a function of the angle, and graph the function over the interval < <. (Consider as negative when a unit in the parade approaches from the left.) Not drawn to scale 7 m d Not drawn to scale d 7 mi (a) What is the period of each function? (b) During what part of the ear is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sk around June, but the graph shows the warmest temperatures at a later date. Approimate the lag time of the temperatures relative to the position of the sun. 9. Sales The projected monthl sales S (in thousands of units) of lawn mowers (a seasonal product) are modeled b S 7 t 0 cost, where t is the time (in months), with t corresponding to Januar. Graph the sales function over ear. 95. Harmonic Motion An object weighing W pounds is suspended from the ceiling b a steel spring (see figure). The weight is pulled downward (positive direction) from its equilibrium position and released. The resulting motion of the weight is described b the function et cos t, t > 0, where is the distance (in feet) and t is the time (in seconds). Camera 9. Meteorolog The normal monthl high temperatures H (in degrees Fahrenheit) in Erie, Pennslvania are approimated b Ht cost.58 sint and the normal monthl low temperatures L are approimated b Lt cost.9 sint where t is the time (in months), with t corresponding to Januar (see figure). (Source: National Climatic Data Center) Equilibrium (a) Use a graphing utilit to graph the function. (b) Describe the behavior of the displacement function for increasing values of time t. True or False? In Eercises 9 and 97, determine whether the statement is true or false. Justif our answer. 9. The graph of csc can be obtained on a calculator b graphing the reciprocal of sin. 97. The graph of sec can be obtained on a calculator b graphing a translation of the reciprocal of sin. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

55 9. Graphs of Other Trigonometric Functions 9 CAPSTONE 98. Determine which function is represented b the graph. Do not use a calculator. Eplain our reasoning. (a) (b) (i) f tan (i) f sec (ii) f tan (ii) f csc (iii) f tan (iii) f csc (iv) f tan (iv) f sec (v) f tan (v) f csc In Eercises 99 and 00, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) as approaches from the right (b) as approaches from the left (c) as approaches from the right (d) as approaches from the left 99. f tan 00. f sec In Eercises 0 and 0, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) As 0, the value of f. (b) As 0, the value of f. (c) As, the value of f. (d) As, the value of f. 0. f cot 0. f csc 0. Think About It Consider the function given b f cos. (a) Use a graphing utilit to graph the function and verif that there eists a zero between 0 and. Use the graph to approimate the zero. (b) Starting with 0, generate a sequence,,,..., where n cos n. For eample, 0 cos 0 cos cos What value does the sequence approach? 0. Approimation Using calculus, it can be shown that the tangent function can be approimated b the polnomial tan! where is in radians. Use a graphing utilit to graph the tangent function and its polnomial approimation in the same viewing window. How do the graphs compare? 05. Approimation Using calculus, it can be shown that the secant function can be approimated b the polnomial sec where is in radians. Use a graphing utilit to graph the secant function and its polnomial approimation in the same viewing window. How do the graphs compare? 0. Pattern Recognition (a) Use a graphing utilit to graph each function. sin sin sin sin 5 sin 5 (b) Identif the pattern started in part (a) and find a function that continues the pattern one more term. Use a graphing utilit to graph. (c) The graphs in parts (a) and (b) approimate the periodic function in the figure. Find a function that is a better approimation.! 5 5! 5! Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

56 0 Chapter 9 Trigonometric Functions 9.7 Inverse Trigonometric Functions Evaluate and graph the inverse sine function. Evaluate and graph the other inverse trigonometric functions. Evaluate and graph the compositions of trigonometric functions. Inverse Sine Function Recall from Section.5 that, for a function to have an inverse function, it must be one-to-one that is, it must pass the Horizontal Line Test. From Figure 9.59, ou can see that sin does not pass the test because different values of ield the same -value. = sin sin has an inverse function on this interval. Figure 9.59 However, if ou restrict the domain to the interval (corresponding to the black portion of the graph in Figure 9.59), the following properties hold.. On the interval,, the function sin is increasing.. On the interval,, sin takes on its full range of values, sin.. On the interval,, sin is one-to-one. So, on the restricted domain, sin has a unique inverse function called the inverse sine function. It is denoted b arcsin or sin. The notation sin is consistent with the inverse function notation f. The arcsin notation (read as the arcsine of ) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin means the angle (or arc) whose sine is. Both notations, arcsin and sin, are commonl used in mathematics, so remember that sin denotes the inverse sine function rather than sin. The values of arcsin lie in the interval arcsin. The graph of arcsin is shown in Eample. DEFINITION OF THE INVERSE SINE FUNCTION The inverse sine function is defined b arcsin if and onl if sin where and. The domain of arcsin is,, and the range is,. NOTE When evaluating the inverse sine function, it helps to remember the phrase the arcsine of is the angle (or number) whose sine is. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

57 9.7 Inverse Trigonometric Functions EXAMPLE Evaluating the Inverse Sine Function If possible, find the eact value. a. b. sin c. sin Solution arcsin a. Because sin for it follows that, arcsin Angle whose sine is b. Because sin for it follows that, sin Angle whose sine is. c. It is not possible to evaluate sin when because there is no angle whose sine is. Remember that the domain of the inverse sine function is,.. STUDY TIP As with the trigonometric functions, much of the work with the inverse trigonometric functions can be done b eact calculations rather than b calculator approimations. Eact calculations help to increase our understanding of the inverse functions b relating them to the right triangle definitions of the trigonometric functions. ( ), (, ) Figure 9.0 (0, 0) = arcsin (, ),, ( ) ( ), ( ) EXAMPLE Graphing the Arcsine Function Sketch a graph of Solution arcsin. arcsin B definition, the equations and are equivalent for. So, their graphs are the same. From the interval,, ou can assign values to in the second equation to make a table of values. Then plot the points and draw a smooth curve through the points. The resulting graph for arcsin sin sin 0 0 is shown in Figure 9.0. Note that it is the reflection (in the line ) of the black portion of the graph in Figure Be sure ou see that Figure 9.0 shows the entire graph of the inverse sine function. Remember that the domain of arcsin is the closed interval, and the range is the closed interval,. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

58 Chapter 9 Trigonometric Functions Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval shown in Figure 9.. 0, as = cos cos has an inverse function on this interval. Figure 9. Consequentl, on this interval the cosine function has an inverse function the inverse cosine function denoted b arccos or cos. Similarl, ou can define an inverse tangent function b restricting the domain of tan to the interval,. The following list summarizes the definitions of the three most common inverse trigonometric functions. The remaining three are defined in Eercises DEFINITIONS OF THE INVERSE TRIGONOMETRIC FUNCTIONS Function Domain Range arcsin if and onl if sin arccos if and onl if cos 0 arctan if and onl if tan < < < < The graphs of these three inverse trigonometric functions are shown in Figure 9.. = arcsin = arccos = arctan Domain:, Domain:, Domain:, Range:, Range: 0, Range:, Figure 9. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

59 9.7 Inverse Trigonometric Functions EXAMPLE Evaluating Inverse Trigonometric Functions Find the eact value. a. arccos b. cos c. arctan 0 d. tan Solution a. Because cos, and lies in 0,, it follows that arccos Angle whose cosine is. b. Because cos, and lies in 0,, it follows that cos. Angle whose cosine is c. Because tan 0 0, and 0 lies in,, it follows that arctan 0 0. Angle whose tangent is 0 d. Because tan, and lies in,, it follows that tan. Angle whose tangent is EXAMPLE Calculators and Inverse Trigonometric Functions STUDY TIP Remember that the domain of the inverse sine function and the inverse cosine function is,, as indicated in Eample (c). Use a calculator to approimate the value (if possible). a. arctan8.5 b. sin 0.7 c. arccos Solution Function Mode Calculator Kestrokes a. arctan8.5 Radian 8.5 From the displa, it follows that arctan b. sin 0.7 Radian SIN 0.7 From the displa, it follows that sin c. arccos Radian COS TAN ENTER ENTER ENTER In real number mode, the calculator should displa an error message because the domain of the inverse cosine function is,. In Eample, if ou had set the calculator to degree mode, the displas would have been in degrees rather than radians. This convention is peculiar to calculators. B definition, the values of inverse trigonometric functions are alwas in radians. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

60 Chapter 9 Trigonometric Functions Compositions of Functions Recall from Section.5 that for all in the domains of f and f, inverse functions have the properties f f and f f. INVERSE PROPERTIES OF TRIGONOMETRIC FUNCTIONS If and, then sinarcsin and arcsinsin. If and 0, then cosarccos and arccoscos. If is a real number and < <, then tanarctan and arctantan. Keep in mind that these inverse properties do not appl for arbitrar values of and. For instance, arcsin sin In other words, the propert arcsinsin arcsin is not valid for values of outside the interval,.. EXAMPLE 5 Using Inverse Properties If possible, find the eact value. a. tanarctan5 b. arcsin sin 5 c. coscos Solution a. Because 5 lies in the domain of the arctan function, the inverse propert applies, and ou have tanarctan5 5. b. In this case, 5 does not lie within the range of the arcsine function,. However, 5 is coterminal with 5 which does lie in the range of the arcsine function, and ou have arcsin sin 5 arcsin sin. c. The epression coscos is not defined because cos is not defined. Remember that the domain of the inverse cosine function is,. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

61 9.7 Inverse Trigonometric Functions 5 = 5 Eample shows how to use right triangles to find eact values of compositions of inverse functions. Then, Eample 7 shows how to use right triangles to convert a trigonometric epression into an algebraic epression. This conversion technique is used frequentl in calculus. u = arccos (a) Angle whose cosine is EXAMPLE Evaluating Compositions of Functions Find the eact value. a. tan arccos 5 ( ) = 5 ( ) u = arcsin 5 b. cos arcsin 5 Solution a. If ou let u arccos then cos u,. Because cos u is positive, u is a first-quadrant angle. You can sketch and label angle u as shown in Figure 9.(a). Consequentl, (b) Angle whose sine is 5 Figure 9. tan arccos tan u opp adj 5. b. If ou let u arcsin 5, then sin u 5. Because sin u is negative, u is a fourth-quadrant angle. You can sketch and label angle u as shown in Figure 9.(b). Consequentl, cos arcsin 5 cos u adj hp 5. EXAMPLE 7 Some Problems from Calculus Write each of the following as an algebraic epression in. a. sinarccos, 0 () b. cotarccos, 0 < u = arccos Angle whose cosine is Figure 9. Solution If ou let u arccos, then cos u, where. Because cos u adj hp ou can sketch a right triangle with acute angle u, as shown in Figure 9.. From this triangle, ou can easil convert each epression to algebraic form. a. sinarccos sin u opp hp 9, 0 b. cotarccos cot u adj 0 < opp 9, In Eample 7, similar arguments can be made for -values ling in the interval, 0. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

62 Chapter 9 Trigonometric Functions 9.7 Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, fill in the blanks. Function Alternative Notation. arcsin. cos. arctan. Without restrictions, no trigonometric function has a(n) function. In Eercises 5 0, evaluate the epression without using a calculator. 5.. arccos Domain 5. arcsin. arcsin 0 7. arccos 8. arccos 0 9. arctan 0. arctan. cos.. arctan. arctan sin arcsin 7. sin tan 0 0. cos Range In Eercises and, use a graphing utilit to graph f, g, and in the same viewing window to verif geometricall that g is the inverse function of f. (Be sure to restrict the domain of f properl.). f sin,. f tan, g arcsin g arctan In Eercises 0, use a calculator to evaluate the epression. Round our result to two decimal places.. arccos 0.7. arcsin arcsin0.75. arccos arctan 8. arctan 5 9. sin cos 0.. arccos0.. arcsin0.5. arctan 0.9. arctan.8 5. arcsin 7 8. arccos 7. tan 9 8. tan tan 7 0. tan 5 tan In Eercises and, determine the missing coordinates of the points on the graph of the function... = arctan (, ) In Eercises 8, use an inverse trigonometric function to write as a function of (, ) + ( ),, ( ) + In Eercises 9 5, use the properties of inverse trigonometric functions to evaluate the epression. 9. sinarcsin tanarctan 5 5. cosarccos0. 5. sinarcsin0. 5. arcsinsin 5. In Eercises 55, find the eact value of the epression. (Hint: Sketch a right triangle.) 55. sinarctan 5. secarcsin costan 58. (, ) (, ) arccos cos 7 5 sin cos cosarcsin 5 0. cscarctan 5. secarctan 5. tanarcsin. sinarccos. cotarctan = arccos + Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

63 9.7 Inverse Trigonometric Functions 7 In Eercises 5 7, write an algebraic epression that is equivalent to the epression. (Hint: Sketch a right triangle, as demonstrated in Eample 7.) 5. cotarctan. sinarctan 7. cosarcsin 8. secarctan 9. sinarccos 70. secarcsin csc arctan 7. In Eercises 75 and 7, use a graphing utilit to graph f and g in the same viewing window to verif that the two functions are equal. Eplain wh the are equal. Identif an asmptotes of the graphs In Eercises 77 80, fill in the blank In Eercises 8 8, sketch a graph of the function. 8. arccos 8. gt arccost 8. f ) arctan 8. f arctan 85. hv tanarccos v 8. f arccos In Eercises 87 9, use a graphing utilit to graph the function. 87. f arccos 88. f arcsin 89. f arctan 90. f arctan tan arccos f sinarctan, f tan arccos, arctan 9 arcsin, arcsin arccos, 0 arccos 0 arcsin arccos arctan, f sin f cos g g 0 cot arctan cos arcsin h r WRITING ABOUT CONCEPTS 9. Define the inverse cotangent function b restricting the domain of the cotangent function to the interval 0,, and sketch its graph. 9. Define the inverse secant function b restricting the domain of the secant function to the intervals 0, and,, and sketch its graph. 95. Define the inverse cosecant function b restricting the domain of the cosecant function to the intervals, 0 and 0,, and sketch its graph. 9. Use the results of Eercises 9 95 to evaluate the following without using a calculator. (a) arcsec (b) arcsec (c) arccot (d) arccsc In Eercises 97 99, prove the identit. 97. arcsin arcsin 98. arctan arctan 99. arccos arccos 00. Consider the functions given b f sin and f arcsin. (a) Use a graphing utilit to graph the composite functions f f and f f. (b) Eplain wh the graphs in part (a) are not the graph of the line. Wh do the graphs of f f and f f differ? In Eercises 0 and 0, write the function in terms of the sine function b using the identit A cos t B sin t A B sin t arctan A B. Use a graphing utilit to graph both forms of the function. What does the graph impl? 0. f t cos t sin t 0. f t cos t sin t In Eercises 0 08, fill in the blank. If not possible, state the reason. 0. As, the value of arcsin. 0. As, the value of arccos. 05. As, the value of arctan. 0. As, the value of arcsin. 07. As, the value of arccos. 08. As, the value of arctan. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it

64 8 Chapter 9 Trigonometric Functions 09. Docking a Boat A boat is pulled in b means of a winch located on a dock 5 feet above the deck of the boat (see figure). Let be the angle of elevation from the boat to the winch and let s be the length of the rope from the winch to the boat. s 5 ft (a) Use a graphing utilit to graph as a function of. (b) Move the cursor along the graph to approimate the distance from the picture when is maimum. (c) Identif the asmptote of the graph and discuss its meaning in the contet of the problem.. Angle of Elevation An airplane flies at an altitude of miles toward a point directl over an observer. Consider and as shown in the figure. (a) Write as a function of s. (b) Find when s 0 feet and s 0 feet. 0. Photograph A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let s be the height of the shuttle. (a) Write mi Not drawn to scale as a function of. (b) Find when 7 miles and mile.. Securit Patrol A securit car with its spotlight on is parked 0 meters from a warehouse. Consider and as shown in the figure. s 0 m (a) Write as a function of s. (b) Find when s 00 meters and s 00 meters. 750 m Not drawn to scale. Photograph A photographer is taking a picture of a three-foot-tall painting hung in an art galler. The camera lens is foot below the lower edge of the painting (see figure). The angle subtended b the camera lens feet from the painting is > 0. arctan[ )], Not drawn to scale (a) Write as a function of. (b) Find when 5 meters and meters. CAPSTONE. Use the results of Eercises 9 95 to eplain how to graph (a) the inverse cotangent function, (b) the inverse secant function, and (c) the inverse cosecant function on a graphing utilit. ft ft α β Not drawn to scale True or False? In Eercises 5 7, determine whether the statement is true or false. Justif our answer. 5.. sin 5 tan 5 7. arctan arcsin arccos arcsin 5 arctan 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it