Trigonometry SELECTED APPLICATIONS

Transcription

1 Trigonometr. Radian and Degree Measure. Trigonometric Functions: The Unit Circle. Right Triangle Trigonometr. Trigonometric Functions of An Angle.5 Graphs of Sine and Cosine Functions.6 Graphs of Other Trigonometric Functions.7 Inverse Trigonometric Functions.8 Applications and Models Airport runwas are named on the basis of the angles the form with due north, measured in a clockwise direction. These angles are called bearings and can be determined using trigonometr. Rajs/Photonica/Gett Images SELECTED APPLICATIONS Trigonometric functions have man real-life applications. The applications listed below represent a small sample of the applications in this chapter. Speed of a Biccle, Eercise 08, page 9 Machine Shop Calculations, Eercise 69, page 0 Sales, Eercise 88, page 0 Respirator Ccle, Eercise 7, page 0 Data Analsis: Meteorolog, Eercise 75, page 0 Predator-Pre Model, Eercise 77, page Securit Patrol, Eercise 97, page 5 Navigation, Eercise 9, page 60 Wave Motion, Eercise 60, page 6 8

2 8 Chapter Trigonometr. Radian and Degree Measure What ou should learn Describe angles. Use radian measure. Use degree measure. Use angles to model and solve real-life problems. Wh ou should learn it You can use angles to model and solve real-life problems. For instance, in Eercise 08 on page 9, ou are asked to use angles to find the speed of a biccle. 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 Te rminal side Verte Initial side Initial side Angle Angle in Standard Position FIGURE. FIGURE. Wolfgang Ratta/Reuters/Corbis 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.. 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.. Positive angles are generated b counterclockwise rotation, and negative angles b clockwise rotation, as shown in Figure.. Angles are labeled with Greek letters (alpha), (beta), and (theta), as well as uppercase letters A, B, and C. In Figure., 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 FIGURE. FIGURE. Coterminal Angles

3 Section. Radian and Degree Measure 8 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.5. Arc length radius when radian FIGURE.5 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.5. Algebraicall, this means that s r where is measured in radians. 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 radians radians r radians FIGURE.6 r r r radian r r 5 radians 6 radians s r. Moreover, because there are just over si radius lengths in a full circle, as shown in Figure.6. Because the units of measure for s and r are the same, the ratio sr has no units it is simpl a real number. Because the radian measure of an angle of one full revolution is, ou can obtain the following. revolution radians revolution 6.8, radians 6 revolution 6 radians These and other common angles are shown in Figure.7. One revolution around a circle of radius r corresponds to an angle of radians because s r r r radians. 6 FIGURE.7 Recall that the four quadrants in a coordinate sstem are numbered I, II, III, and IV. Figure.8 on page 8 shows which angles between 0 and lie in each of the four quadrants. Note that angles between 0 and are acute angles and angles between and are obtuse angles.

4 8 Chapter Trigonometr = Quadrant II < < Quadrant I 0 < < = = 0 Quadrant III Quadrant IV < < < < 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. FIGURE.8 Two angles are coterminal if the have the same initial and terminal sides. For instance, the angles 0 and are coterminal, as are the angles 6 and 6. 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 6 where n is an integer. = 6 Eample Sketching and Finding Coterminal Angles a. For the positive angle 6, subtract to obtain a coterminal angle 6 6. See Figure.9. b. For the positive angle, subtract to obtain a coterminal angle 5. See Figure.0. c. For the negative angle, add to obtain a coterminal angle. See Figure.. = FIGURE.9 FIGURE.0 FIGURE. Now tr Eercise 7. = 5 0 = 0

5 Section. Radian and Degree Measure 85 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.. β α β α Complementar Angles FIGURE. Supplementar Angles Eample Complementar and Supplementar Angles 90 = (60 ) 0 60 = (60 ) = (60 ) = (60 ) FIGURE. If possible, find the complement and the supplement of (a) 5 and (b) 5. a. The complement of 5 is 5 The supplement of 5 is b. Because 5 is greater than, it has no complement. (Remember that complements are positive angles.) The supplement is Degree Measure Now tr Eercise. 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 60 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.. So, a full revolution (counterclockwise) corresponds to 60, 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 60 rad and From the latter equation, ou obtain rad and 80 rad. rad 80 which lead to the conversion rules at the top of the net page

6 86 Chapter Trigonometr Conversions Between Degrees and Radians. To convert degrees to radians, multipl degrees b. To convert radians to degrees, multipl radians b rad rad. To appl these two conversion rules, use the basic relationship (See Figure..) rad FIGURE. When no units of angle measure are specified, radian measure is implied. For instance, if ou write ou impl that radians., Eample Converting from Degrees to Radians Technolog 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 60 one second Consequentl, an angle of 6 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 a. 5 5 deg Multipl b deg radians b deg Multipl b 80. c deg Multipl b deg radians Eample Now tr Eercise 7. Converting from Radians to Degrees a. rad rad 80 deg Multipl b 80. b. rad 9 rad 80 deg 80 Multipl b rad rad 80 deg radians rad rad 90 rad rad c. rad rad 80 deg Multipl b 80. Now tr Eercise 5. If ou have a calculator with a radian-to-degree conversion ke, tr using it to verif the result shown in part (c) of Eample.

7 Applications The radian measure formula, a circle. Section. Radian and Degree Measure 87 sr, can be used to measure arc length along s = 0 r = 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. Eample 5 Finding Arc Length FIGURE.5 A circle has a radius of inches. Find the length of the arc intercepted b a central angle of 0, as shown in Figure.5. To use the formula s r, first convert 0 to radian measure. 0 0 deg 80 deg Then, using a radius of r inches, ou can find the arc length to be s r 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. Now tr Eercise 87. radians 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. 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.

8 88 Chapter Trigonometr Eample 6 Finding Linear Speed FIGURE.6 0. cm The second hand of a clock is 0. centimeters long, as shown in Figure.6. Find the linear speed of the tip of this second hand as it passes around the clock face. In one revolution, the arc length traveled is s r centimeters. Substitute for r. The time required for the second hand to travel this distance is t minute 60 seconds. So, the linear speed of the tip of the second hand is Linear speed s t 0. centimeters 60 seconds.068 centimeters per second. Now tr Eercise ft FIGURE.7 Eample 7 Finding Angular and Linear Speeds A Ferris wheel with a 50-foot radius (see Figure.7) makes.5 revolutions per minute. a. Find the angular speed of the Ferris wheel in radians per minute. b. Find the linear speed of the Ferris wheel. a. Because each revolution generates radians, it follows that the wheel turns.5 radians per minute. In other words, the angular speed is Angular speed t b. The linear speed is Linear speed s t radians minute r t 50 feet minute Now tr Eercise 05. radians per minute. 7. feet per minute.

9 Section. Radian and Degree Measure 89 A sector of a circle is the region bounded b two radii of the circle and their intercepted arc (see Figure.8). r FIGURE.8 Area of a Sector of a Circle For a circle of radius r, the area A of a sector of the circle with central angle is given b where A r is measured in radians. Eample 8 Area of a Sector of a Circle A sprinkler on a golf course fairwa is set to spra water over a distance of 70 feet and rotates through an angle of 0 (see Figure.9). Find the area of the fairwa watered b the sprinkler. First convert 0 to radian measure as follows. FIGURE ft 0 0 deg 80 deg radians rad Multipl b 80. Then, using and r 70, the area is A r square feet. Formula for the area of a sector of a circle Substitute for r and. Simplif. Simplif. Now tr Eercise 07.

10 90 Chapter Trigonometr. Eercises VOCABULARY CHECK: 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. 6. Two positive angles that have a sum of are angles, whereas two positive angles that have a sum of are angles The angle measure that is equivalent to of a complete revolution about an angle s verte is one. 8. The speed of a particle is the ratio of the arc length traveled to the time traveled. 9. The speed of a particle is the ratio of the change in the central angle to time. 0. The area of a sector of a circle with radius r and central angle, where is measured in radians, is given b the formula. In Eercises 6, estimate the angle to the nearest one-half radian..... In Eercises 6, sketch each angle in standard position.. (a) (b). (a) 7 (b) 5 5. (a) (b) 6. (a) (b) 6 In Eercises 7 0, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in radians (a) (b) = = 6 0 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) (b) 7 9. (a).5 (b).5. (a) 6.0 (b) (a) (b) 7 = 6 9. (a) (b) 9 0. (a) (b) = 6

11 Section. Radian and Degree Measure 9 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 In Eercises, determine the quadrant in which each angle lies. In Eercises 5 8, sketch each angle in standard position. In Eercises 9, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in degrees. 9. (a) (b) = 5. (a) 0 (b) 85. (a) 8. (b) (a) 50 (b) 6. (a) 60 (b). 5. (a) 0 (b) (a) 70 (b) 0 7. (a) 05 (b) (a) 750 (b) (a) (b) = 0 = 6 = 0. (a) (b). (a) (b) In Eercises 6, find (if possible) the complement and supplement of each angle.. (a) 8 (b) 5. (a) (b) 6 5. (a) 79 (b) (a) 0 (b) 70 In Eercises 7 50, rewrite each angle in radian measure as a multiple of. (Do not use a calculator.) 7. (a) 0 (b) (a) 5 (b) 0 9. (a) 0 (b) (a) 70 (b) In Eercises 5 5, rewrite each angle in degree measure. (Do not use a calculator.) 5. (a) (b) 5. (a) 7 (b) (a) (b) 5. (a) (b) In Eercises 55 6, convert the angle measure from degrees to radians. Round to three decimal places In Eercises 6 70, convert the angle measure from radians to degrees. Round to three decimal places In Eercises 7 7, convert each angle measure to decimal degree form (a) 5 5 (b) (a) 5 0 (b) 7. (a) (b) (a) 5 6 (b) In Eercises 75 78, convert each angle measure to form. 75. (a) 0.6 (b) (a) 5. (b) (a).5 (b) (a) 0.55 (b) D M S

12 9 Chapter Trigonometr In Eercises 79 8, find the angle in radians Cit 96. San Francisco, California Seattle, Washington Latitude N N In Eercises 8 86, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius r Arc Length s 8. 7 inches 6 inches 8. feet 8 feet centimeters 5 centimeters kilometers 60 kilometers In Eercises 87 90, find the length of the arc on a circle of radius r intercepted b a central angle. Radius r Central Angle inches feet meters radian centimeters radian In Eercises 9 9, find the area of the sector of the circle with radius r and central angle. Radius r 9. inches 9. millimeters 9..5 feet 9.. miles 5 7 Central Angle Distance Between Cities In Eercises 95 and 96, 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 95. Dallas, Teas Omaha, Nebraska Latitude 7 9 N 5 50 N Difference in Latitudes Assuming that Earth is a sphere of radius 678 kilometers, what is the difference in the latitudes of Sracuse, New York and Annapolis, Marland, where Sracuse is 50 kilometers due north of Annapolis? 98. Difference in Latitudes Assuming that Earth is a sphere of radius 678 kilometers, what is the difference in the latitudes of Lnchburg, Virginia and Mrtle Beach, South Carolina, where Lnchburg is 00 kilometers due north of Mrtle Beach? 99. Instrumentation The pointer on a voltmeter is 6 centimeters in length (see figure). Find the angle through which the pointer rotates when it moves.5 centimeters on the scale. 6 cm ft FIGURE FOR 99 FIGURE FOR 00 0 in. Not drawn to scale 00. Electric Hoist An electric hoist is being used to lift a beam (see figure). The diameter of the drum on the hoist is 0 inches, and the beam must be raised feet. Find the number of degrees through which the drum must rotate. 0. Angular Speed A car is moving at a rate of 65 miles per hour, and the diameter of its wheels is.5 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. 0. 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.

13 Section. Radian and Degree Measure 9 0. Linear and Angular Speeds A -inch circular power saw rotates at 500 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. 0. 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 of the platform rim of the carousel. 05. Linear and Angular Speeds The diameter of a DVD is approimatel centimeters. The drive motor of the DVD plaer is controlled to rotate precisel between 00 and 500 revolutions per minute, depending on what track is being read. (a) Find an interval for the angular speed of a DVD as it rotates. (b) Find an interval for the linear speed of a point on the outermost track as the DVD rotates. 06. Area A car s rear windshield wiper rotates 5. The total length of the wiper mechanism is 5 inches and wipes the windshield over a distance of inches. Find the area covered b the wiper. 07. Area A sprinkler sstem on a farm is set to spra water over a distance of 5 meters and to rotate through an angle of 0. Draw a diagram that shows the region that can be irrigated with the sprinkler. Find the area of the region. 08. 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. in. Model It 7 Snthesis True or False? In Eercises 09, determine whether the statement is true or false. Justif our answer. 09. A measurement of radians corresponds to two complete revolutions from the initial side to the terminal side of an angle. 0. The difference between the measures of two coterminal angles is alwas a multiple of 60 if epressed in degrees and is alwas a multiple of radians if epressed in radians.. An angle that measures 60 lies in Quadrant III.. Writing In our own words, eplain the meanings of (a) an angle in standard position, (b) a negative angle, (c) coterminal angles, and (d) an obtuse angle.. Think About It 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.. Think About It Is a degree or a radian the larger unit of measure? Eplain. 5. Writing 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. 6. Proof Prove that the area of a circular sector of radius r with central angle is A r, where is measured in radians. Skills Review Model It (continued) (c) Write a function for the distance d (in miles) a cclist travels in terms of the time t (in seconds). Compare this function with the function from part (b). (d) Classif the tpes of functions ou found in parts (b) and (c). Eplain our reasoning. In Eercises 7 0, simplif the radical epression. in. in. (a) Find the speed of the biccle in feet per second and miles per hour. (b) Use our result from part (a) to write a function for the distance d (in miles) a cclist travels in terms of the number n of revolutions of the pedal sprocket In Eercises, sketch the graphs of 5 and the specified transformation.. f 5. f 5. f 5. f 5

14 9 Chapter Trigonometr. Trigonometric Functions: The Unit Circle What ou should learn 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. Wh ou should learn it Trigonometric functions are used to model the movement of an oscillating weight. For instance, in Eercise 57 on page 00, the displacement from equilibrium of an oscillating weight suspended b a spring is modeled as a function of time. The Unit Circle 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 as shown in Figure.0. Unit circle (, 0) (0, ) (0, ) (, 0) FIGURE.0 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.. t > 0 (, ) t t t < 0 (, 0) (, 0) Richard Megna/Fundamental Photographs t (, ) t FIGURE. 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. 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 length of the arc intercepted b the angle, given in radians.

15 Section. Trigonometric Functions: The Unit Circle 95 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 cosecant cosine secant tangent cotangent These si functions are normall abbreviated sin, csc, cos, sec, tan, and cot, respectivel. Note in the definition at the right 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 csc t, 0 cos t sec t, 0 tan t, cot t, 0 0 (0, ),, ( ), (, 0) ( ) FIGURE., (, ) (, ) (, 0) (0, ) (, ) FIGURE. ( ) (0, ) (0, ) ( ) (, 0) ( ), (, ) (, 0) (, ) (, ) (, ) 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., the unit circle has been divided into eight equal arcs, corresponding to t-values of Similarl, in Figure., the unit circle has been divided into equal arcs, corresponding to t-values of 5 0,,,,,, 7 and.,, 7 0, and. 6,,,, 5 6,, 6,,,, 6, To verif the points on the unit circle in Figure., note that, also lies on the line. So, substituting for in the equation of the unit circle produces the following. ± 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. and the points in Figure.. Using the, coordinates in Figures. and., ou can easil 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 quickl and easil. 5

16 96 Chapter Trigonometr Eample Evaluating Trigonometric Functions Evaluate the si trigonometric functions at each real number. a. t b. t 5 6 c. t 0 d. t For each t-value, begin b finding the corresponding point, on the unit circle. Then use the definitions of trigonometric functions listed on page 95. a. t corresponds to the point 6 cos 6 tan 6 5 b. t corresponds to the point sin 6 5 sin 5 cos 5 tan,,.,,. c. t 0 corresponds to the point,, 0. csc 6 sec 6 cot 6 csc 5 sec 5 cot 5 sin 0 0 csc 0 is undefined. cos 0 sec 0 tan cot 0 is undefined. d. t corresponds to the point,, 0. sin 0 cos tan 0 0 Now tr Eercise. csc is undefined. sec cot is undefined.

17 Section. Trigonometric Functions: The Unit Circle 97 Eploration With our graphing utilit in radian and parametric modes, enter the equations XT = cos T and YT = sin T and use the following settings. Tmin = 0, Tma = 6., Tstep = 0. Xmin = -.5, Xma =.5, Xscl = Ymin = -, Yma =, Yscl =. Graph the entered equations and describe the graph.. Use the trace feature to move the cursor around the graph. What do the t-values represent? What do the - and -values represent?. What are the least and greatest values of and? (0, ) (, 0) (, 0) (0, ) FIGURE. t = t = t =, +, +,..., +,... t =,, , +,... t =, +, t = 0,,... t =, +, +,... t =, +, +,... Eample Evaluating Trigonometric Functions Evaluate the si trigonometric functions at t. Moving clockwise around the unit circle, it follows that t corresponds to the point,,. Now tr Eercise 5. 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.. Because r, it follows that sin t and cos t. Moreover, because, is on the unit circle, ou know that and. So, the values of sine and cosine also range between and. and Adding to each value of t in the interval 0, completes a second revolution around the unit circle, as shown in Figure.5. 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 sin cos tan sin t sint n sin t cost n cos t csc sec cot cos t for an integer n and real number t. Functions that behave in such a repetitive (or cclic) manner are called periodic. 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. FIGURE.5

18 98 Chapter Trigonometr if Recall from Section.5 that a function f is even if f t f t, and is odd 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 Eample Using the Period to Evaluate the Sine and Cosine 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.6. a. Because ou have 6 6, b. Because 7 ou have, cos 7 cos c. For sin t sint because the sine function is odd. 5, 5 sin 6 sin cos 0. 6 sin 6. Now tr Eercise. When evaluating trigonometric functions with a calculator, remember to enclose all fractional angle measures in parentheses. For instance, if ou want to evaluate sin for ou should enter SIN Technolog 6, 6 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 eample, to evaluate csc8, use the fact that csc 8 sin8 and enter the following kestroke sequence in radian mode. SIN 8 ENTER Displa.659 Eample Using a Calculator Function Mode Calculator Kestrokes Displa a. sin Radian SIN ENTER b. cot.5 Radian TAN.5 ENTER Now tr Eercise 5.

19 Section. Trigonometric Functions: The Unit Circle 99. Eercises VOCABULARY CHECK: 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, determine the eact values of the si trigonometric functions of the angle , ( In Eercises 5, find the point, on the unit circle that corresponds to the real number t. 5. t 6. t 7. t 7 8. t t 0. t 5. t. t In Eercises, evaluate (if possible) the sine, cosine, and tangent of the real number.. t. t ( 5. t t 7 8. ( 5, ( ( 5, 5 t t ( ( 5, ( 9. t 0. t 5 6. t. t In Eercises 8, evaluate (if possible) the si trigonometric functions of the real number.. t. 5. t t 8. In Eercises 9 6, evaluate the trigonometric function using its period as an aid. 9. sin cos 8.. cos sin 9 6. In Eercises 7, use the value of the trigonometric function to evaluate the indicated functions. 7. sin t 8. sint 8 (a) sint (a) sin t (b) csct (b) csc t 9. cost 5 0. cos t (a) cos t (a) cost (b) sect (b) sect. sin t 5. cos t 5 (a) (a) (b) sint (b) cost sin t t 5 6 t t 7 cos 5 sin 9 sin 9 6 cos 8 cos t

20 00 Chapter Trigonometr In Eercises 5, 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.) Estimation In Eercises 5 and 5, use the figure and a straightedge to approimate the value of each trigonometric function. To print an enlarged cop of the graph, go to the website 5. (a) sin 5 (b) cos 5. (a) sin 0.75 (b) cos sin. tan 5. csc. 6. cot 7. cos.7 8. cos.5 9. csc sec.8 5. sec.8 5. sin FIGURE FOR Estimation In Eercises 55 and 56, use the figure and a straightedge to approimate the solution of each equation, where 0 t <. To print an enlarged cop of the graph, go to the website (a) sin t 0.5 (b) cos t (a) sin t 0.75 (b) cos t 0.75 Model It 57. 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 6t where is the displacement (in feet) and t is the time (in seconds). 58. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos 6t, where is the displacement (in feet) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t and (c) t,. Snthesis True or False? In Eercises 59 and 60, determine whether the statement is true or false. Justif our answer. 59. Because sint sin t, it can be said that the sine of a negative angle is a negative number. 60. tan a tana 6 6. Eploration 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 6. 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. Skills Review Model It (continued) (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 t increases? sin t. t 0 cos t. In Eercises 6 66, find the inverse function one-to-one function f. 6. f 6. f 65. f, 66. f f of the In Eercises 67 70, sketch the graph of the rational function b hand. Show all asmptotes f 68. f f 6 f

21 Section. Right Triangle Trigonometr 0. Right Triangle Trigonometr What ou should learn Evaluate trigonometric functions of acute angles. Use the fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real-life problems. Wh ou should learn it Trigonometric functions are often used to analze real-life situations. For instance, in Eercise 7 on page, ou can use trigonometric functions to find the height of a helium-filled balloon. 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.6. 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 ). FIGURE.6 Hpotenuse Side adjacent to Side opposite 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. Joseph Sohm; Chromosohm 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

22 0 Chapter Trigonometr Eample Evaluating Trigonometric Functions FIGURE.7 Hpotenuse Use the triangle in Figure.7 to find the values of the si trigonometric functions of. B the Pthagorean Theorem, hp opp adj, it follows that hp 5 5. So, the si trigonometric functions of sin opp hp 5 are csc hp opp 5 cos adj hp 5 sec hp adj 5 Historical Note Georg Joachim Rhaeticus (5 576) was the leading Teutonic mathematical astronomer of the 6th centur. He was the first to define the trigonometric functions as ratios of the sides of a right triangle. 5 FIGURE.8 5 tan opp adj Now tr Eercise. In Eample, ou were given the lengths of two sides of the right triangle, but not the angle. Often, ou will be asked to find the trigonometric functions of a given acute angle. To do this, construct a right triangle having as one of its angles. Eample Evaluating Trigonometric Functions of 5 Find the values of sin 5, cos 5, and tan 5. Construct a right triangle having 5 as one of its acute angles, as shown in Figure.8. 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. sin 5 opp hp cos 5 adj hp tan 5 opp adj Now tr Eercise 7. cot adj opp.

23 Section. Right Triangle Trigonometr 0 Eample Evaluating Trigonometric Functions of 0 and 60 Because the angles 0, 5, and 60 6,, and occur frequentl in trigonometr, ou should learn to construct the triangles shown in Figures.8 and.9. Use the equilateral triangle shown in Figure.9 to find the values of sin 60, cos 60, sin 0, and cos 0. 0 Technolog 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. FIGURE.9 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 60 opp hp 0, and For adj, opp, and hp. So, sin 0 opp hp and Now tr Eercise , cos 60 adj hp. cos 0 adj hp. Sines, Cosines, and Tangents of Special Angles sin 0 sin cos 0 cos tan 0 tan sin 5 sin sin 60 sin cos 5 cos cos 60 cos tan 5 tan tan 60 tan In the bo, note that sin 0 cos 60. This occurs because 0 and 60 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

24 0 Chapter Trigonometr 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. Eample Appling Trigonometric Identities FIGURE Let be an acute angle such that sin 0.6. Find the values of (a) cos and (b) tan using trigonometric identities. a. To find the value of cos, use the Pthagorean identit sin cos. So, ou have 0.6 cos Substitute 0.6 for sin. Subtract 0.6 from each side. Etract the positive square root. b. Now, knowing the sine and cosine of, ou can find the tangent of to be tan sin cos cos cos Use the definitions of cos and tan, and the triangle shown in Figure.0, to check these results. Now tr Eercise 9.

25 Section. Right Triangle Trigonometr 05 Eample 5 Appling Trigonometric Identities Let be an acute angle such that tan. Find the values of (a) cot and (b) sec using trigonometric identities. a. cot Reciprocal identit tan 0 cot FIGURE. You can also use the reciprocal identities for sine, cosine, and tangent to evaluate the cosecant, secant, and cotangent functions with a calculator. For instance, ou could use the following kestroke sequence to evaluate sec 8. COS 8 ENTER The calculator should displa.570. b. sec tan Pthagorean identit sec sec 0 Use the definitions of cot and sec, and the triangle shown in Figure., to check these results. Now tr Eercise. Evaluating Trigonometric Functions with a Calculator 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.. For instance, ou can find values of cos 8 and sec 8 as follows. Function Mode Calculator Kestrokes Displa a. cos 8 Degree COS 8 ENTER b. sec 8 Degree COS 8 ENTER.570 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. Using a Calculator Use a calculator to evaluate sec5 0. Begin b converting to decimal degree form. [Recall that sec 0 Eample Then, use a calculator to evaluate sec Function Calculator Kestrokes Displa sec5 0 sec 5.67 Now tr Eercise 7. COS 60 and 5.67 ENTER.00966

26 06 Chapter Trigonometr Observer Observer FIGURE. Object Angle of elevation Horizontal Horizontal Angle of depression Object Angle of elevation 78. Applications Involving Right Triangles 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 are asked to find one of the other sides, or ou are given two sides and are asked to find one of the acute angles. In Eample 7, 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, as shown in Figure.. Eample 7 Using Trigonometr to Solve a Right Triangle A surveor is standing 5 feet from the base of the Washington Monument, as shown in Figure.. The surveor measures the angle of elevation to the top of the monument as 78.. How tall is the Washington Monument? From Figure., ou can 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. Now tr Eercise 6. FIGURE. = 5 ft Not drawn to scale Eample 8 Using Trigonometr to Solve a Right Triangle An historic lighthouse is 00 ards from a bike path along the edge of a lake. A walkwa to the lighthouse is 00 ards long. Find the acute angle between the bike path and the walkwa, as illustrated in Figure.. 00 d 00 d FIGURE. From Figure., ou can see that the sine of the angle sin opp 00 hp 00. Now ou should recognize that 0. Now tr Eercise 65. is

27 Section. Right Triangle Trigonometr 07 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 0.6 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. Eample 9 Solving a Right Triangle Find the length c of the skateboard ramp shown in Figure c ft FIGURE.5 From Figure.5, ou can see that sin 8. opp hp c. So, the length of the skateboard ramp is c sin feet. Now tr Eercise 67.

28 08 Chapter Trigonometr. Eercises VOCABULARY CHECK:. Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent hpotenuse adjacent hpotenuse adjacent opposite opposite (i) (ii) (iii) (iv) (v) (vi) adjacent opposite opposite hpotenuse hpotenuse adjacent In Eercises and, fill in the blanks.. Relative to the angle, the three sides of a right triangle are the side, the side, and the.. 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, 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.) In Eercises 5 8, 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 9 6, 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. 9. sin 0. cos 5 7. sec. cot 5. tan. sec 6 5. cot 6. csc 7 In Eercises 7 6, construct an appropriate triangle to complete the table. 0 90, 0 / Function (deg) (rad) Function Value 7. sin 0 8. cos 5 9. tan 0. sec. cot. csc. cos. sin 5. cot 6. tan 6

29 Section. Right Triangle Trigonometr 09 In Eercises 7, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. 7. sin 60, cos 60 (a) tan 60 (b) sin 0 (c) cos 0 (d) cot sin 0 tan 0, (a) csc 0 (b) cot 60 (c) cos 0 (d) cot 0 9. (a) sin (b) cos (c) tan (d) sec90 0. sec 5, tan 6 (a) cos (b) cot (c) cot90 (d) sin. cos (a) sec (b) sin (c) cot (d) sin90. tan 5 (a) cot (b) cos (c) tan90 (d) csc In Eercises, use trigonometric identities to transform the left side of the equation into the right side 0 < < /.. tan cot. cos sec 5. tan cos sin 6. cot sin cos 7. cos cos sin 8. sin sin cos 9. sec tan sec tan 0. sin cos sin.. csc, sec sin cos csc sec cos sin tan cot csc tan In Eercises 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.). (a) sin 0 (b) cos 80. (a) tan.5 (b) cot (a) sin 6.5 (b) csc (a) cos 6 8 (b) sin (a) sec (b) csc (a) cos 50 5 (b) sec (a) cot 5 (b) tan (a) sec (b) cos (a) csc 0 (b) tan (a) sec (b) cot In Eercises 5 58, find the values of in degrees 0 < < 90 and radians 0 < < / without the aid of a calculator. 5. (a) sin (b) csc 5. (a) cos (b) tan 55. (a) sec (b) cot 56. (a) tan (b) cos 57. (a) csc (b) sin 58. (a) cot (b) sec In Eercises 59 6, solve for,, or r as indicated. 59. Solve for. 60. Solve for Solve for. 6. Solve for r 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 86th floor (the observator) is 8. If the total height of the building is another meters above the 86th floor, what is the approimate height of the building? One of our friends is on the 86th floor. What is the distance between ou and our friend? r 0

30 0 Chapter Trigonometr 6. 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? 65. 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? 66. Width of a River A biologist wants to know the width w of a river so in order to properl set instruments for studing the pollutants in the water. 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? 5. w C = 5 A 00 ft 67. Length A steel cable zip-line is being constructed for a competition on a realit television show. One end of the zip-line is attached to a platform on top of a 50-foot pole. The other end of the zip-line is attached to the top of a 5-foot stake. The angle of elevation to the platform is (see figure). 68. 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. 69. Machine Shop Calculations A steel plate has the form of one-fourth of a circle with a radius of 60 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 mi (, ) (, ) 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. Not drawn to scale 5 cm d 5 ft = 50 ft 5 cm (a) How long is the zip-line? (b) How far is the stake from the pole? (c) Contestants take an average of 6 seconds to reach the ground from the top of the zip-line. At what rate are contestants moving down the line? At what rate are the dropping verticall?

31 Section. Right Triangle Trigonometr 7. Height A 0-meter line is used to tether a heliumfilled 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 Model It 7. 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 (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. Snthesis True or False? In Eercises 7 78, determine whether the statement is true or false. Justif our answer. 7. sin 60 csc sec 0 csc sin 5 cos cot 0 csc 0 sin sin 78. tan5 tan 5 sin Writing In right triangle trigonometr, eplain wh sin 0 regardless of the size of the triangle. 80. Think About It You are given onl the value tan. Is it possible to find the value of sec without finding the measure of? Eplain. 8. Eploration (a) Complete the table. (b) Is or sin greater for in the interval 0, 0.5? (c) As approaches 0, how do and sin compare? Eplain. 8. Eploration (a) Complete the table. sin cos sin (b) Discuss the behavior of the sine function for range from 0 to 90. (c) Discuss the behavior of the cosine function for range from 0 to 90. in the in the (d) Use the definitions of the sine and cosine functions to eplain the results of parts (b) and (c). Skills Review In Eercises 8 86, perform the operations and simplif. 0 cm 0 (, ) t 5t t 6 9 t t t

32 Chapter Trigonometr. Trigonometric Functions of An Angle What ou should learn Evaluate trigonometric functions of an angle. Use reference angles to evaluate trigonometric functions. Evaluate trigonometric functions of real numbers. Wh ou should learn it You can use trigonometric functions to model and solve real-life problems. For instance, in Eercise 87 on page 9, ou can use trigonometric functions to model the monthl normal temperatures in New York Cit and Fairbanks, Alaska. Introduction In Section., 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 An Angle Let be an angle in standard position with, a point on the terminal side of and r 0. sin cos r r (, ) tan cot, 0, 0 sec r, 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. Eample Evaluating Trigonometric Functions James Urbach/SuperStock Let, be a point on the terminal side of. Find the sine, cosine, and tangent of. Referring to Figure.6, ou can see that,, and r 5 5. So, ou have the following. (, ) sin r 5 r FIGURE.6 cos r 5 tan Now tr Eercise.

33 Section. Trigonometric Functions of An Angle < < < 0 > 0 0 < < > 0 > 0 The signs of the trigonometric functions in the four quadrants can be determined easil 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.7. < 0 < 0 < < > 0 < 0 < < Eample Evaluating Trigonometric Functions Given tan 5 and cos > 0, find sin and sec. 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 Quadrant II sin : + cos : tan : Quadrant I sin : + cos : + tan : + tan 5 and the fact that is negative in Quadrant IV, ou can let 5and. So, r 6 5 and ou have Quadrant III sin : cos : tan : + Quadrant IV sin : cos : + tan : sin r FIGURE.7 sec r Now tr Eercise 7. Eample Trigonometric Functions of Quadrant Angles (0, ) Evaluate the cosine and tangent functions at the four quadrant angles 0, and,,. To begin, choose a point on the terminal side of each angle, as shown in Figure.8. For each of the four points, r, and ou have the following. cos 0 tan 0,, 0 0 r 0 (, 0) (, 0) 0 cos r 0 0 tan 0 undefined, 0, FIGURE.8 (0, ) cos r cos r 0 0 tan 0 0 tan 0 undefined,, 0, 0, Now tr Eercise 9.

34 Chapter Trigonometr 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: = (radians) = 80 (degrees) Quadrant III = (radians) = 80 (degrees) Quadrant IV = (radians) = 60 (degrees) FIGURE.9 = 00 FIGURE.0 =. FIGURE. = 5 5 FIGURE. = 60 =. 5 and 5 are coterminal. = 5 Eample Finding Reference Angles Find the reference angle. 00 a. b. c. a. Because 00 lies in Quadrant IV, the angle it makes with the -ais is Degrees Figure.0 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. 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. shows the angle 00. Now tr Eercise , 5 and its reference angle

35 (, ) Section. Trigonometric Functions of An Angle 5 Trigonometric Functions of Real Numbers 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.. B definition, ou know that opp r = hp adj opp, adj FIGURE. and For the right triangle with acute angle and sides of lengths and ou have sin opp hp r and 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. sin r tan opp adj. tan., Evaluating Trigonometric Functions of An 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. Learning the table of values at the right is worth the effort because doing so will increase both our efficienc and our confidence. Here is a pattern for the sine function that ma help ou remember the values. sin Reverse the order to get cosine values of the same angles. 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 6 sin 0 0 cos 0 0 tan 0 Undef. 0 Undef

36 6 Chapter Trigonometr Eample 5 Using Reference Angles Evaluate each trigonometric function. a. cos b. tan0 c. csc a. Because lies in Quadrant III, the reference angle is, as shown in Figure.. 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.5. Finall, because the tangent is negative in Quadrant II, ou have c. Because, it follows that is coterminal with the second-quadrant angle. So, the reference angle is as shown in Figure.6. Because the cosecant is positive in Quadrant II, ou have, csc csc sin. tan0 tan 0. = = = 0 = 0 = = FIGURE. FIGURE.5 FIGURE.6 Now tr Eercise 5.

37 Section. Trigonometric Functions of An Angle 7 Eample 6 Using Trigonometric Identities Let be an angle in Quadrant II such that sin. Find (a) cos and (b) tan b using trigonometric identities. a. Using the Pthagorean identit sin cos, ou obtain cos Substitute for sin. Because cos < 0 cos 8 9 cos in Quadrant II, ou can use the negative root to obtain b. Using the trigonometric identit tan sin, ou obtain cos tan. Now tr Eercise 59. Substitute for sin and cos. You can use a calculator to evaluate trigonometric functions, as shown in the net eample. Eample 7 Using a Calculator Use a calculator to evaluate each trigonometric function. a. cot 0 b. sin7 c. sec 9 Function Mode Calculator Kestrokes Displa a. cot 0 Degree TAN 0 ENTER b. sin7 Radian SIN 7 ENTER c. sec Radian COS ENTER Now tr Eercise 69.

38 8 Chapter Trigonometr. Eercises VOCABULARY CHECK: In Eercises 6, let be an angle in standard position, with, a point on the terminal side of and r 0.. sin. r. tan. sec r 7. 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, determine the eact values of the si trigonometric functions of the angle.. (a) (b). (a) (b) (, 5). (a) (b) (, ). (a) (b) (, ) (, ) In Eercises 5 0, 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. 7, 6. 8, 5 7., , (8, 5) (, ) (, ) (, ) 9..5, , 7 In Eercises, state the quadrant in which. sin < 0 and. sin > 0 and. sin > 0 and. sec > 0 and lies. In Eercises 5, find the values of the si trigonometric functions of with the given constraint. cos < 0 cos > 0 tan < 0 cot < 0 Function Value Constraint 5. sin 5 lies in Quadrant II. 6. cos 5 lies in Quadrant III. 7. tan cos cot 0. csc. sec. sin 0 sin < 0 tan < 0 cos > 0 cot < 0 sin > 0 sec. cot is undefined.. tan is undefined. In Eercises 5 8, 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 5. II 6. III 7. 0 III 8. 0 IV

39 Section. Trigonometric Functions of An Angle 9 In Eercises 9 6, evaluate the trigonometric function of the quadrant angle. 9. sin 0.. sec. sec. sin. cot 5. csc 6. cot In Eercises 7, find the reference angle, and sketch and in standard position In Eercises 5 58, evaluate the sine, cosine, and tangent of the angle without using a calculator csc In Eercises 59 6, find the indicated trigonometric value in the specified quadrant. Function Quadrant Trigonometric Value 59. sin 5 IV cos 60. cot II sin 6. tan III sec 6. csc IV cot 6. cos 5 8 I sec 6. sec 9 III tan In Eercises 65 80, 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.) 65. sin sec cos0 68. csc0 69. tan cot sec 7 7. tan88 7. tan.5 7. cot tan 76. tan sin sec cot In Eercises 8 86, find two solutions of the equation. Give our answers in degrees 0 < 60 and in radians 0 <. Do not use a calculator. 8. (a) sin (b) sin 8. (a) cos (b) cos 8. (a) csc (b) cot 8. (a) sec (b) sec 85. (a) tan (b) cot 86. (a) sin (b) sin Model It 9 csc Data Analsis: Meteorolog The table shows the monthl normal temperatures (in degrees Fahrenheit) for selected months for New York Cit N and Fairbanks, Alaska F. (Source: National Climatic Data Center) Month New York Fairbanks, Cit, N F Januar 0 April 5 Jul 77 6 October 58 December 8 6 (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.

40 0 Chapter Trigonometr Model It (continued) (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. d 6 mi Not drawn to scale 88. Sales A compan that produces snowboards, which are seasonal products, forecasts monthl sales over the net ears to be t S. 0.t. cos 6 where S is measured in thousands of units and t is the time in months, with t representing Januar 006. Predict sales for each of the following months. (a) Februar 006 (b) Februar 007 (c) June 006 (d) June Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos 6t 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,. 90. 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 6t 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,. 9. 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. 9. Distance An airplane, fling at an altitude of 6 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, 90, 0. FIGURE FOR 9 Snthesis True or False? In Eercises 9 and 9, determine whether the statement is true or false. Justif our answer. 9. In each of the four quadrants, the signs of the secant function and sine function will be the same. 9. To find the reference angle for an angle (given in degrees), find the integer n such that 0 60n 60. The difference 60n is the reference angle. 95. Writing 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 Writing Eplain how reference angles are used to find the trigonometric functions of obtuse angles. Skills Review cm (, ) In Eercises 97 06, graph the function. Identif the domain and an intercepts and asmptotes of the function f g h f ln 06. log 0

41 Section.5 Graphs of Sine and Cosine Functions.5 Graphs of Sine and Cosine Functions What ou should learn 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. Wh ou should learn it Sine and cosine functions are often used in scientific calculations. For instance, in Eercise 7 on page 0, ou can use a trigonometric function to model the airflow of our respirator ccle. 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.7, 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.8. Recall from Section. 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.7 and.8? Range: = sin 5 Period: FIGURE.7 = cos Range: 5 Karl Weatherl/Corbis Period: FIGURE.8 Note in Figures.7 and.8 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.

42 Chapter Trigonometr 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). Maimum Intercept Minimum Intercept Intercept (, ) = sin (0, 0) Quarter period Half period Period: (, 0) (, ) Three-quarter period (, 0) Full period (0, ) Intercept Minimum Maimum = cos Quarter period Period: (, 0 ) (, 0) (, ) Half period Intercept (, ) Three-quarter period Maimum Full period FIGURE.9 Eample Using Ke Points to Sketch a Sine Curve Sketch the graph of sin on the interval,. 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 Maimum Intercept Minimum 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.50., 0,,, and, 0 Technolog 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. FIGURE.50 = sin = sin Now tr Eercise

43 Section.5 Graphs of Sine and Cosine Functions 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.7 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 the basic sine curve is shrunk. The result is that the graph of a sin a a >, <, 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. = cos = cos Eample Scaling: Vertical Shrinking and Stretching On the same coordinate aes, sketch the graph of each function. a. b. cos cos FIGURE.5 Eploration = cos 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? 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 Intercept Minimum Intercept Maimum 0,, b. A similar analsis shows that the amplitude of cos is, and the ke points are and Maimum Intercept Minimum Intercept Maimum 0,,, 0,, 0,,,,, The graphs of these two functions are shown in Figure.5. 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. Now tr Eercise 7., 0,, 0, and,.,.

44 Chapter Trigonometr = cos = cos You know from Section.7 that the graph of fis 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.5. 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.5 Period b. Eploration Sketch the graph of sin c where c, 0, and. How does the value of c affect the graph? 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 6, of length, ou would successivel add 6 to get 6, 0, 6,, and as the -values for the ke points on the graph. 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. Eample Sketch the graph of sin. Scaling: Horizontal Stretching The amplitude is. Moreover, because b, the period is b. 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. Intercept Maimum Intercept Minimum Intercept 0, 0,,,, 0,,, and, 0 The graph is shown in Figure.5. FIGURE.5 = sin = sin Now tr Eercise 9. Period:

45 Section.5 Graphs of Sine and Cosine Functions 5 Translations of Sine and Cosine Curves The constant c in the general equations a sinb c and 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 a cosb c 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. FIGURE.5 ( ) = sin 5 8 Period: Eample Horizontal Translation Sketch the graph of sin The amplitude is and the period is. B solving the equations and 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 Maimum Intercept Minimum Intercept, 0, 5 6,, 7 The graph is shown in Figure.5.., 0, Now tr Eercise 5. 6,, and 7, 0.

46 6 Chapter Trigonometr = cos( + ) Eample 5 Horizontal Translation Sketch the graph of cos. The amplitude is and the period is 0. B solving the equations Period FIGURE.55 and 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.55. Now tr Eercise 7. FIGURE.56 = + cos 5 Period The final tpe of transformation is the vertical translation caused b the constant d in the equations 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. Eample 6 Sketch the graph of Vertical Translation cos. 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.56. Compared with the graph of f cos, the graph of cos is shifted upward two units. Now tr Eercise 5.,,

47 Depth (in feet) FIGURE.57 Changing Tides A.M. 8 A.M. Noon Time 0 0 = 5.6 cos(0.5t.09) FIGURE.58 Time, t (.7, 0) (7., 0) = 0 Depth, Midnight. A.M. 8.7 A.M.. 6 A.M A.M..8 0 A.M. 0. Noon. t 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. Eample 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.) 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? a. Begin b graphing the data, as shown in Figure.57. You can use either a sine or cosine model. Suppose ou 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 5.6 cos0.5t b. The depths at 9 A.M. and P.M. are as follows. 5.6 cos foot 5.6 cos feet Section.5 Graphs of Sine and Cosine Functions 7 9 A.M. P.M. 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.58. 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.. Now tr Eercise

48 8 Chapter Trigonometr.5 Eercises VOCABULARY CHECK: Fill in the blanks.. One period of a sine or cosine function function is called one of the sine curve or cosine curve.. The of a sine or cosine curve represents half the distance between the maimum and minimum values of the function.. The period of a sine or cosine function is given b. c. For the function given b a sinb c, represents the of the graph of the function. b 5. For the function given b d a cosb c, d represents a of the graph of the function. In Eercises, find the period and amplitude.. sin. cos 5. sin sin sin 0 0. sin 8. cos. 5 cos. 5 cos. sin cos cos.. sin cos 0 In Eercises 5, describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts. 5. f sin 6. f cos g sin g cos 7. f cos 8. f sin g cos g sin 9. f cos 0. f sin g cos g sin. f sin. f cos g sin g cos In Eercises 6, describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts... f f g g g f g f

49 Section.5 Graphs of Sine and Cosine Functions 9 In Eercises 7, graph f and g on the same set of coordinate aes. (Include two full periods.) 7.. f sin g sin g cos f sin g sin g cos In Eercises 5 56, sketch the graph of the function. (Include two full periods.) sin 6. sin 7. cos 8. cos. cos.. sin. 9. t sin cos 5. 0 cos cos cos cos cos 56. cos6 In Eercises 57 6, use a graphing utilit to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. 57. sin 58. sin 59. cos 60. cos f sin g sin 9. f cos 0. f cos g cos. f sin g sin. f cos. f cos g cos 9. cos 0. sin sin 0 cos 6 5. sin 6. sin 7. cos 8. cos sin 0 00 sin 0t Graphical Reasoning In Eercises 6 66, find a and d for the function f a cos d such that the graph of f matches the figure Graphical Reasoning In Eercises 67 70, find a, b, and c for the function f a sinb c such that the graph of f matches the figure f f f f In Eercises 7 and 7, use a graphing utilit to graph and in the interval [, ]. Use the graphs to find real numbers such that. 7. sin 7. cos 5 f f f f

50 0 Chapter Trigonometr 7. Respirator Ccle For a person at rest, the velocit v (in liters per second) of air flow during a respirator ccle (the time from the beginning of one breath to the beginning of t 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. 7. Respirator Ccle After eercising for a few minutes, a person has a respirator ccle for which the velocit of air t flow 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. 75. Data Analsis: Meteorolog The table shows the maimum dail high temperatures for Tallahassee T and Chicago C (in degrees Fahrenheit) for month t, with t corresponding to Januar. (Source: National Climatic Data Center) (c) Use a graphing utilit to graph the data points and the model for the temperatures in Chicago. 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. 76. 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. 77. 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? Month, t Tallahassee, T Chicago, C (a) A model for the temperature in Tallahassee is given b t Tt cos Find a trigonometric model for Chicago. (b) Use a graphing utilit to graph the data points and the model for the temperatures in Tallahassee. How well does the model fit the data? Model It 78. Data Analsis: Astronom The percent of the moon s face that is illuminated on da of the ear 007, where represents Januar, is 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,

51 Section.5 Graphs of Sine and Cosine Functions 79. Fuel Consumption The dail consumption C (in gallons) of diesel fuel on a farm is modeled b 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. 80. 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. Snthesis True or False? In Eercises 8 8, determine whether the statement is true or false. Justif our answer. 8. 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. 8. The function given b cos has an amplitude that is twice that of the function given b cos. 8. The graph of cos is a reflection of the graph of sin in the -ais. 8. Writing 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 85 and 86, graph f and g on the same set of coordinate aes. Include two full periods. Make a conjecture about the functions C 0..6 sin t f sin, g cos f sin, g cos 87. Eploration Using calculus, it can be shown that the sine and cosine functions can be approimated b the polnomials sin 5! 5! and 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 predict the net term in each. Then repeat parts (a) and (b). How did the accurac of the approimations change when an additional term was added? 88. Eploration Use the polnomial approimations for the sine and cosine functions in Eercise 87 to approimate the following function values. Compare the results with those given b a calculator. Is the error in the approimation the same in each case? Eplain. (a) sin (b) sin (c) sin 6 (d) cos0.5 (e) cos (f) cos Skills Review cos! In Eercises 89 9, use the properties of logarithms to write the epression as a sum, difference, and/or constant multiple of a logarithm. 89. log 90. log 0 t ln z ln t z In Eercises 9 96, write the epression as the logarithm of a single quantit. 9. log 0 log 0 9. log log 95. ln ln 96. ln ln ln 97. Make a Decision To work an etended application analzing the normal dail maimum temperature and normal precipitation in Honolulu, Hawaii, visit this tet s website at college.hmco.com. (Data Source: NOAA)!

52 Chapter Trigonometr.6 Graphs of Other Trigonometric Functions What ou should learn 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. Wh ou should learn it Trigonometric functions can be used to model real-life situations such as the distance from a television camera to a unit in a parade as in Eercise 76 on page. 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.59. 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 RANGE: n, VERTICAL ASYMPTOTES: n Photodisc/Gett Images FIGURE.59 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.

53 Section.6 Graphs of Other Trigonometric Functions Eample Sketching the Graph of a Tangent Function Sketch the graph of tan. = 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 FIGURE.60 tan Undef. 0 Undef. Now tr Eercise 7. Eample Sketching the Graph of a Tangent Function Sketch the graph of tan. B solving the equations and 6 = tan ou can see that two consecutive vertical asmptotes occur at and. Between these two asmptotes, plot a few points, including the -inter- cept, as shown in the table. Three ccles of the graph are shown in Figure tan Undef. 0 Undef. 8 FIGURE.6 6 Now tr Eercise 9. 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.

54 Chapter Trigonometr Technolog 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.6. 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 RANGE: n, VERTICAL ASYMPTOTES: n FIGURE.6 Eample Sketching the Graph of a Cotangent Function = cot 6 Sketch the graph of cot. B solving the equations 0 0 and ou can see that two consecutive vertical asmptotes occur at 0 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.6. Note that the period is, the distance between consecutive asmptotes. FIGURE.6 0 cot 9 Undef. 0 Undef. Now tr Eercise 9.

55 Section.6 Graphs of Other Trigonometric Functions 5 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.6. = csc = sec = sin = cos Sine: maimum Cosecant: relative maimum FIGURE.65 Cosecant: relative minimum Sine: minimum PERIOD: DOMAIN: ALL RANGE:, VERTICAL ASYMPTOTES: SYMMETRY: ORIGIN FIGURE.6 n, n PERIOD: DOMAIN: ALL RANGE:, VERTICAL ASYMPTOTES: SYMMETRY: -AXIS 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.65. Additionall, -intercepts of the sine and cosine functions become vertical asmptotes of the cosecant and secant functions, respectivel (see Figure.65). n, n

56 6 Chapter Trigonometr = csc + FIGURE.66 ( ) = sin + ( ) Eample Sketching the Graph of a Cosecant Function Sketch the graph of csc Begin b sketching the graph of sin For this function, the amplitude is and the period is. B solving the equations 0 and 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.66. 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.66. Now tr Eercise 5. 7 Eample 5 Sketching the Graph of a Secant Function Sketch the graph of sec. = sec = cos Begin b sketching the graph of cos, as indicated b the gra curve in Figure.67. Then, form the graph of sec as the black curve in the figure. Note that the -intercepts of cos, 0,, 0,, 0,... correspond to the vertical asmptotes,,,... of the graph of sec. Moreover, notice that the period of cos and sec is. FIGURE.67 Now tr Eercise 7.

57 Section.6 Graphs of Other Trigonometric Functions 7 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 Using properties of absolute value and the fact that ou have 0 sin. Consequentl, sin. sin, sin which means that the graph of f sin lies between the lines and. Furthermore, because and f sin ± at n f sin 0 at n FIGURE.68 f() = sin 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.68. In the function f sin, the factor is called the damping factor. Eample 6 Damped Sine Wave Do ou see wh the graph of f sin touches the lines ± at n and wh the graph has -intercepts at n? Recall that the sine function is equal to at,, 5,... odd multiples of and is equal to 0 at,,,... multiples of. f() = e sin 6 = e = e 6 FIGURE.69 Sketch the graph of f e sin. Consider f as the product of the two functions and each of which has the set of real numbers as its domain. For an real number, ou know that and sin So, e sin e 0. e, which means that Furthermore, because and e e e sin e. f e sin ±e f e sin 0 sin at at the graph of f touches the curves e and e at 6 n and has intercepts at n. A sketch is shown in Figure.69. Now tr Eercise n n

58 8 Chapter Trigonometr Figure.70 summarizes the characteristics of the si basic trigonometric functions. = tan = sin = cos 5 DOMAIN: ALL REALS RANGE:, PERIOD: DOMAIN: ALL REALS RANGE:, PERIOD:, DOMAIN: ALL RANGE: PERIOD: n = csc = sin = sec = cos = cot = tan DOMAIN: ALL RANGE: PERIOD: FIGURE.70 n,, n,, PERIOD: DOMAIN: ALL RANGE: DOMAIN: ALL RANGE: PERIOD: n, W RITING ABOUT MATHEMATICS Combining Trigonometric Functions Recall from Section.8 that functions can be combined arithmeticall. This also applies to trigonometric functions. For each of the functions h sin and (a) identif two simpler functions f and g that comprise the combination, (b) use a table to show how to obtain the numerical values of h from the numerical values of f and g, and (c) use graphs of f and g to show how h ma be formed. Can ou find functions f d a sinb c such that f g 0 for all? h cos sin and g d a cosb c

59 Section.6 Graphs of Other Trigonometric Functions 9.6 Eercises VOCABULARY CHECK: Fill in the blanks.. 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.. The period of tan is. 5. The domain of cot is all real numbers such that. 6. The range of sec is. 7. The period of csc is. In Eercises 6, 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 7 0, sketch the graph of the function. Include two full periods. 7. tan 8. tan 9. tan 0. tan. sec. sec. csc. csc 5. sec 6. sec 7. csc 8. csc (c) (e) (d) (f) 9. cot 0.. sec.. tan. 5. csc sec csc cot tan tan csc sec cot In Eercises 0, use a graphing utilit to graph the function. Include two full periods.. tan. tan. sec.. cot sec tan csc sec. sec. sec 5. tan 6. cot 7. csc 8. sec sec 0. tan

60 0 Chapter Trigonometr In Eercises 8, use a graph to solve the equation on the interval [, ].. tan. tan. cot. cot 5. sec 6. sec 7. csc 8. In Eercises 9 and 50, use the graph of the function to determine whether the function is even, odd, or neither. 5. Graphical Reasoning Consider the functions given b and 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? 5. Graphical Reasoning Consider the functions given b and 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. In Eercises 5 56, 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 csc 9. f sec 50. f tan f sin f tan 5. sin csc, 5. sin sec, cos sin, sec, cot g csc g sec tan tan In Eercises 57 60, 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 6 6, graph the functions f and g. Use the graphs to make a conjecture about the relationship between the functions f cos f sin g sin g cos f sin cos f sin cos,, In Eercises 65 68, 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. 65. g e sin 66. f e cos 67. f cos 68. h sin Eploration In Eercises 69 7, use a graphing utilit to graph the function. Describe the behavior of the function as approaches zero cos, > sin, > 0 (b) (d) g sin f sin, g cos f cos, g cos g 0

61 Section.6 Graphs of Other Trigonometric Functions 7. g sin 7. f cos 7. f sin 7. h sin 75. 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 < <. Model It (continued) R 5,000 5,000 cos t. (a) Use a graphing utilit to graph both models in the same viewing window. Use the window setting 0 t 00. (b) Use the graphs of the models in part (a) to eplain the oscillations in the size of each population. (c) The ccles of each population follow a periodic pattern. Find the period of each model and describe several factors that could be contributing to the cclical patterns. 76. 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 Camera Model It Not drawn to scale d 7 mi 77. Predator-Pre Model The population C of cootes (a predator) at time t (in months) in a region is estimated to be t C sin and the population R of rabbits (its pre) is estimated to be 78. Sales The projected monthl sales S (in thousands of units) of lawn mowers (a seasonal product) are modeled b S 7 t 0 cost6, where t is the time (in months), with t corresponding to Januar. Graph the sales function over ear. 79. Meterolog The normal monthl high temperatures H (in degrees Fahrenheit) for Erie, Pennslvania are approimated b and the normal monthl low temperatures L are approimated b where t is the time (in months), with t corresponding to Januar (see figure). (Source: National Oceanic and Atmospheric Administration) Te mperature (in degrees Fahrenheit) t t Ht cos 5.69 sin 6 6 t t Lt cos.6 sin H(t) L(t) Month of ear (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. t

62 Chapter Trigonometr 80. 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, where is the distance (in feet) and t is the time (in seconds). (a) Use a graphing utilit to graph the function. (b) Describe the behavior of the displacement function for increasing values of time t. Snthesis True or False? In Eercises 8 and 8, determine whether the statement is true or false. Justif our answer. 8. The graph of csc can be obtained on a calculator b graphing the reciprocal of sin. 8. The graph of sec can be obtained on a calculator b graphing a translation of the reciprocal of sin. f cos. t > 0 Equilibrium 8. Writing Describe the behavior of f tan as approaches from the left and from the right. 8. Writing Describe the behavior of f csc as approaches from the left and from the right. 85. Eploration Consider the function given b (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? 86. 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? 87. Approimation Using calculus, it can be shown that the secant function can be approimated b the polnomial sec 5!! 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? 88. Pattern Recognition (a) Use a graphing utilit to graph each function. (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. Skills Review 6 5 5! sin sin sin sin 5 sin 5 In Eercises 89 9, solve the eponential equation. Round our answer to three decimal places. 89. e e t 5 In Eercises 9 98, solve the logarithmic equation. Round our answer to three decimal places. 9. ln 7 9. ln ln. 96. ln log 8 log log 6 log 6 log 6 6

63 Section.7 Inverse Trigonometric Functions.7 Inverse Trigonometric Functions What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse trigonometric functions. Evaluate and graph the compositions of trigonometric functions. Wh ou should learn it You can use inverse trigonometric functions to model and solve real-life problems. For instance, in Eercise 9 on page 5, an inverse trigonometric function can be used to model the angle of elevation from a television camera to a space shuttle launch. NASA When evaluating the inverse sine function, it helps to remember the phrase the arcsine of is the angle (or number) whose sine is. Inverse Sine Function Recall from Section.9 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.7, ou can see that sin does not pass the test because different values of ield the same -value. FIGURE.7 However, if ou restrict the domain to the interval (corresponding to the black portion of the graph in Figure.7), 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 has an inverse function on this interval. sin. = 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 Inverse Sine Function The inverse sine function is defined b arcsin if and onl if where and. The domain of arcsin is,, and the range is,. sin

64 Chapter Trigonometr Eample Evaluating the Inverse Sine Function 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. If possible, find the eact value. a. b. sin c. sin arcsin a. Because sin for it follows that 6, 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,. 6. Now tr Eercise. Eample Graphing the Arcsine Function ( ), ( ), 6 ( ), FIGURE.7 (0, 0) (, ) ( ), 6 = arcsin (, ) Sketch a graph of arcsin. B definition, the equations arcsin and sin 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. 0 6 sin The resulting graph for arcsin is shown in Figure.7. Note that it is the reflection (in the line ) of the black portion of the graph in Figure.7. Be sure ou see that Figure.7 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,. Now tr Eercise

65 Section.7 Inverse Trigonometric Functions 5 Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval 0 shown in Figure.7., as = cos cos has an inverse function on this interval. FIGURE.7 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 0 0. Definitions of the Inverse Trigonometric Functions Function Domain Range arcsin if and onl if sin arccos if and onl if cos arctan if and onl if tan < < 0 < < The graphs of these three inverse trigonometric functions are shown in Figure.7. = arcsin = arccos = arctan, DOMAIN: RANGE:, FIGURE.7 DOMAIN:, RANGE: 0, DOMAIN: RANGE:,,

66 6 Chapter Trigonometr Eample Evaluating Inverse Trigonometric Functions Find the eact value. a. arccos b. cos c. arctan 0 d. tan a. Because cos, and lies in 0,, it follows that arccos. Angle whose cosine is b. Because cos, and cos. lies in 0,, it follows that 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. Now tr Eercise. Eample Calculators and Inverse Trigonometric Functions It is important to 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 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,. Now tr Eercise 5. 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.

67 Compositions of Functions Section.7 Inverse Trigonometric Functions 7 Recall from Section.9 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,.. Eample 5 Using Inverse Properties If possible, find the eact value. a. tanarctan5 b. arcsin sin 5 c. coscos 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,. Now tr Eercise..

68 8 Chapter Trigonometr Eample 6 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 Angle whose cosine is FIGURE.75 = 5 5 ( ) = u = arcsin ( 5 5 Angle whose sine is 5 FIGURE.76 ( Eample 6 Find the eact value. a. b. tan arccos Evaluating Compositions of Functions a. If ou let u arccos then cos u,. Because cos u is positive, u is a firstquadrant angle. You can sketch and label angle u as shown in Figure.75. Consequentl, tan arccos 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.76. Consequentl, cos arcsin 5 cos arcsin 5 tan u opp adj cos u adj hp 5. Now tr Eercise Eample 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.77 () 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.77. From this triangle, ou can easil convert each epression to algebraic form. a. b. sinarccos sin u opp hp 9, cotarccos cot u adj opp 9, Now tr Eercise < In Eample 7, similar arguments can be made for interval, 0. -values ling in the

69 Section.7 Inverse Trigonometric Functions 9.7 Eercises VOCABULARY CHECK: Fill in the blanks. Function Alternative Notation Domain Range. arcsin. cos. arctan In Eercises 6, evaluate the epression without using a calculator.. arcsin. arcsin 0. arccos. arccos 0 5. arctan 6. arctan 7. cos arctan 0. arctan arccos... sin. 5. tan 0 6. cos sin arcsin tan In Eercises 7 and 8, 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 9, use a calculator to evaluate the epression. Round our result to the nearest hundredth. 9. arccos arcsin 0.5. arcsin0.75. arccos0.7. arctan. arctan 5 5. sin cos arccos0. 8. arcsin arctan arctan.8. arcsin.. tan 7. arccos tan 95 7 In Eercises 5 and 6, determine the missing coordinates of the points on the graph of the function = arctan (, ) ( = arccos, ) (, ) In Eercises 7, use an inverse trigonometric function to write as a function of , 6 ( ) + (, ) (, ) In Eercises 8, use the properties of inverse trigonometric functions to evaluate the epression.. sinarcsin 0.. tanarctan 5 5. cosarccos0. 6. sinarcsin0. 7. arcsinsin 8. arccos cos

70 50 Chapter Trigonometr In Eercises 9 58, find the eact value of the epression. (Hint: Sketch a right triangle.) 9. sinarctan 50. secarcsin 5 5. costan 5. sin 5 cos 5 5. cosarcsin 5 5. cscarctan secarctan tanarcsin 57. sinarccos 58. cotarctan 5 8 In Eercises 59 68, write an algebraic epression that is equivalent to the epression. (Hint: Sketch a right triangle, as demonstrated in Eample 7.) 59. cotarctan 60. sinarctan 6. cosarcsin 6. secarctan 6. sinarccos 6. secarcsin csc arctan 68. In Eercises 69 and 70, 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 7 7, fill in the blank. 7. tan arccos cot arctan cos arcsin h r arctan 9 arcsin, f sinarctan, f tan arccos, g 0 g In Eercises 77 8, sketch a graph of the function. 77. arccos 78. gt arccost 79. f ) arctan 80. f arctan 8. hv tanarccos v 8. In Eercises 8 88, use a graphing utilit to graph the function. 8. f arccos 8. f arcsin 85. f arctan f arccos f arctan f sin f cos In Eercises 89 and 90, 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? 89. f t cos t sin t 90. f t cos t sin t 9. 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 arcsin arccos, 6 arccos 0 arcsin arccos arctan, ft s In Eercises 75 and 76, sketch a graph of the function and compare the graph of g with the graph of f arcsin. (a) Write as a function of s. (b) Find when s 0 feet and s 0 feet. 75. g arcsin 76. g arcsin

71 Section.7 Inverse Trigonometric Functions 5 9. 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. 9. Granular Angle of Repose Different tpes of granular substances naturall settle at different angles when stored in cone-shaped piles. This angle is called the angle of repose (see figure). When rock salt is stored in a coneshaped pile feet high, the diameter of the pile s base is about feet. (Source: Bulk-Store Structures, Inc.) (a) Write as a function of s. (b) Find when s 00 meters and s 00 meters. 750 m s Not drawn to scale 9. 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 arctan, Model It > 0. (a) Find the angle of repose for rock salt. (b) How tall is a pile of rock salt that has a base diameter of 0 feet? 95. Granular Angle of Repose When whole corn is stored in a cone-shaped pile 0 feet high, the diameter of the pile s base is about 8 feet. (a) Find the angle of repose for whole corn. (b) How tall is a pile of corn that has a base diameter of 00 feet? 96. Angle of Elevation An airplane flies at an altitude of 6 miles toward a point directl over an observer. Consider and as shown in the figure. 7 ft ft 6 mi Not drawn to scale ft ft α β (a) Write as a function of. (b) Find when 7 miles and mile. 97. Securit Patrol A securit car with its spotlight on is parked 0 meters from a warehouse. Consider and as shown in the figure. Not drawn to scale (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. (a) Write 0 m Not drawn to scale as a function of. (b) Find when 5 meters and meters.

72 5 Chapter Trigonometr Snthesis True or False? In Eercises 98 00, determine whether the statement is true or false. Justif our answer sin 5 6 tan 5 arctan arcsin arccos 0. Define the inverse cotangent function b restricting the domain of the cotangent function to the interval 0,, and sketch its graph. 0. Define the inverse secant function b restricting the domain of the secant function to the intervals 0, and,, and sketch its graph. 0. Define the inverse cosecant function b restricting the domain of the cosecant function to the intervals, 0 and 0,, and sketch its graph. 0. Use the results of Eercises 0 0 to evaluate each epression without using a calculator. (a) arcsec (b) arcsec (c) arccot (d) arccsc 05. Area In calculus, it is shown that the area of the region bounded b the graphs of 0,, a, and b is given b Area arctan b arctan a arcsin (see figure). Find the area for the following values of a and b. (a) a 0, b (b) a, b (c) a 0, b (d) a, b a b 06. Think About It Use a graphing utilit to graph the functions f and g 6 arctan. = 5 6 arctan 5 + For > 0, it appears that g > f. Eplain wh ou know that there eists a positive real number a such that g < f for > a. Approimate the number a. 07. Think About It Consider the functions given b f sin and (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? 08. Proof Prove each identit. (a) arcsin arcsin (b) arctan arctan (c) arctan arctan, (d) arcsin arccos (e) arcsin arctan Skills Review f arcsin. In Eercises 09, evaluate the epression. Round our result to three decimal places In Eercises 6, sketch a right triangle corresponding to the trigonometric function of the acute angle. Use the Pthagorean Theorem to determine the third side. Then find the other five trigonometric functions of.. sin. tan 5. cos sec 7. Partnership Costs A group of people agree to share equall in the cost of a $50,000 endowment to a college. If the could find two more people to join the group, each person s share of the cost would decrease b $650. How man people are presentl in the group? 8. Speed A boat travels at a speed of 8 miles per hour in still water. It travels 5 miles upstream and then returns to the starting point in a total of hours. Find the speed of the current. 9. Compound Interest A total of $5,000 is invested in an account that pas an annual interest rate of.5%. Find the balance in the account after 0 ears, if interest is compounded (a) quarterl, (b) monthl, (c) dail, and (d) continuousl. 0. Profit Because of a slump in the econom, a department store finds that its annual profits have dropped from $7,000 in 00 to $6,000 in 00. The profit follows an eponential pattern of decline. What is the epected profit for 008? (Let t represent 00.) > 0

73 Section.8 Applications and Models 5.8 Applications and Models What ou should learn Solve real-life problems involving right triangles. Solve real-life problems involving directional bearings. Solve real-life problems involving harmonic motion. Wh ou should learn it Right triangles often occur in real-life situations. For instance, in Eercise 6 on page 6, right triangles are used to determine the shortest grain elevator for a grain storage bin on a farm. Applications Involving Right Triangles In this section, the three angles of a right triangle are denoted b the letters A, B, and C (where C is the right angle), and the lengths of the sides opposite these angles b the letters a, b, and c (where c is the hpotenuse). Eample Solving a Right Triangle Solve the right triangle shown in Figure.78 for all unknown sides and angles. B c a FIGURE.78 Because C 90, it follows that A B 90 and B To solve for a, use the fact that So, a 9. tan..8. Similarl, to solve for c, use the fact that So, tan A opp adj a b adj cos A hp b c c cos. A. c Now tr Eercise. b = 9. a b tan A. b cos A. C B Eample Finding a Side of a Right Triangle c = 0 ft A 7 C b FIGURE.79 a A safet regulation states that the maimum angle of elevation for a rescue ladder is 7. A fire department s longest ladder is 0 feet. What is the maimum safe rescue height? A sketch is shown in Figure.79. From the equation sin A ac, it follows that a c sin A 0 sin So, the maimum safe rescue height is about 0.6 feet above the height of the fire truck. Now tr Eercise 5.

74 5 Chapter Trigonometr Eample Finding a Side of a Right Triangle FIGURE.80 s a 00 ft 5 5 At a point 00 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 5, whereas the angle of elevation to the top is 5, as shown in Figure.80. Find the height s of the smokestack alone. Note from Figure.80 that this problem involves two right triangles. For the smaller right triangle, use the fact that tan 5 a 00 to conclude that the height of the building is a 00 tan 5. For the larger right triangle, use the equation tan 5 a s 00 to conclude that a s 00 tan 5º. So, the height of the smokestack is s 00 tan 5 a 00 tan 5 00 tan 5 5. feet. Now tr Eercise 9. Eample Finding an Acute Angle of a Right Triangle A FIGURE.8 0 m Angle of depression. m.7 m A swimming pool is 0 meters long and meters wide. The bottom of the pool is slanted so that the water depth is. meters at the shallow end and meters at the deep end, as shown in Figure.8. Find the angle of depression of the bottom of the pool. Using the tangent function, ou can see that tan A opp adj So, the angle of depression is A arctan radian Now tr Eercise 5.

75 Trigonometr and Bearings Section.8 Applications and Models 55 In surveing and navigation, directions are generall given in terms of bearings. A bearing measures the acute angle that a path or line of sight makes with a fied north-south line, as shown in Figure.8. For instance, the bearing S 5 E in Figure.8 means 5 degrees east of south. N 80 N N 5 W E W E W E FIGURE.8 S 5 S 5 E S N 80 W S N 5 E Eample 5 Finding Directions in Terms of Bearings A ship leaves port at noon and heads due west at 0 knots, or 0 nautical miles (nm) per hour. At P.M. the ship changes course to N 5 W, as shown in Figure.8. Find the ship s bearing and distance from the port of departure at P.M. In air navigation, bearings are measured in degrees clockwise from north. Eamples of air navigation bearings are shown below. 0 N 70 W 70 W S 80 0 N S E 90 E 90 D b C FIGURE.8 For triangle BCD, ou have B The two sides of this triangle can be determined to be b 0 sin 6 and For triangle ACD, ou can find angle A as follows. tan A The angle with the north-south line is So, the bearing of the ship is N 78.8 W. Finall, from triangle ACD, ou have sin A bc, which ields c 0 nm d 5 b d 0 0 sin cos 6 0 A arctan radian.8 b 0 sin 6 sin A sin.8 B 57. nautical miles. Now tr Eercise. c d 0 cos 6. 0 nm = (0 nm) Distance from port W N S Not drawn to scale E A

76 56 Chapter Trigonometr Harmonic Motion The periodic nature of the trigonometric functions is useful for describing the motion of a point on an object that vibrates, oscillates, rotates, or is moved b wave motion. For eample, consider a ball that is bobbing up and down on the end of a spring, as shown in Figure.8. Suppose that 0 centimeters is the maimum distance the ball moves verticall upward or downward from its equilibrium (at rest) position. Suppose further that the time it takes for the ball to move from its maimum displacement above zero to its maimum displacement below zero and back again is t seconds. Assuming the ideal conditions of perfect elasticit and no friction or air resistance, the ball would continue to move up and down in a uniform and regular manner. 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm Equilibrium FIGURE.8 Maimum negative displacement Maimum positive displacement From this spring ou can conclude that the period (time for one complete ccle) of the motion is Period seconds its amplitude (maimum displacement from equilibrium) is Amplitude 0 centimeters and its frequenc (number of ccles per second) is Frequenc ccle per second. Motion of this nature can be described b a sine or cosine function, and is called simple harmonic motion.

77 Section.8 Applications and Models 57 Definition of Simple Harmonic Motion A point that moves on a coordinate line is said to be in simple harmonic motion if its distance d from the origin at time t is given b either d a sin t or d a cos t where a and are real numbers such that > 0. The motion has amplitude period, and frequenc. a, Eample 6 Simple Harmonic Motion Write the equation for the simple harmonic motion of the ball described in Figure.8, where the period is seconds. What is the frequenc of this harmonic motion? Because the spring is at equilibrium d 0 when t 0, ou use the equation d a sin t. Moreover, because the maimum displacement from zero is 0 and the period is, ou have Amplitude a Period Consequentl, the equation of motion is d 0 sin t. Note that the choice of a 0 or a 0 depends on whether the ball initiall moves up or down. The frequenc is Frequenc 0. FIGURE.85 ccle per second. FIGURE.86 Now tr Eercise 5. One illustration of the relationship between sine waves and harmonic motion can be seen in the wave motion resulting when a stone is dropped into a calm pool of water. The waves move outward in roughl the shape of sine (or cosine) waves, as shown in Figure.85. As an eample, suppose ou are fishing and our fishing bob is attached so that it does not move horizontall. As the waves move outward from the dropped stone, our fishing bob will move up and down in simple harmonic motion, as shown in Figure.86.