Chapter 4. Trigonometric Functions. Selected Applications

Transcription

1 Chapter Trigonometric Functions. Radian and Degree Measure. Trigonometric Functions: The Unit Circle. Right Triangle Trigonometr. Trigonometric Functions of An Angle.5 Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions.7 Inverse Trigonometric Functions.8 Applications and Models Selected Applications Trigonometric functions have man real-life applications. The applications listed below represent a small sample of the applications in this chapter. Sports, Eercise 95, page 7 Electrical Circuits, Eercise 7, page 75 Machine Shop Calculations, Eercise 8, page 87 Meteorolog, Eercise 09, page 9 Sales, Eercise 7, page 0 Predator-Pre Model, Eercise 59, page 7 Photograph, Eercise 8, page 9 Airplane Ascent, Eercises 9 and 0, page 9 Harmonic Motion, Eercises 55 58, page 5 The si trigonometric functions can be defined from a right triangle perspective and as functions of real numbers. In Chapter, ou will use both perspectives to graph trigonometric functions and solve application problems involving angles and triangles. You will also learn how to graph and evaluate inverse trigonometric functions. 5 5 Winfried Wisniewski/zefa/Corbis Trigonometric functions are often used to model repeating patterns that occur in real life. For instance, a trigonometric function can be used to model the populations of two species that interact, one of which (the predator) hunts the other (the pre). 57

2 58 Chapter Trigonometric Functions. Radian and Degree Measure 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, including 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. What ou should learn Describe angles. Use radian measure. Use degree measure and convert between degree and radian measure. Use angles to model and solve real-life problems. Wh ou should learn it Radian measures of angles are involved in numerous aspects of our dail lives. For instance, in Eercise 95 on page 7, ou are asked to determine the measure of the angle generated as a skater performs an ael jump. Terminal side Terminal side Verte Initial side Initial side Figure. Figure. 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 such as (alpha), (beta), and (theta), as well as uppercase letters such as A, B, and C. In Figure., note that angles and have the same initial and terminal sides. Such angles are coterminal. Stephen Jaffe/AFP/Gett Images Positive angle (counterclockwise) Negative angle (clockwise) α β β α Figure. Figure.

3 Section. Radian and Degree Measure 59 Radian Measure 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. r r s = r Definition of Radian One radian (rad) 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 Arc length radius when radian. Figure.5 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 s r. Moreover, because there are just over si radius lengths in a full circle, as shown in Figure.. 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.8, radians revolution radians These and other common angles are shown in Figure.7. radians r radians r radians Figure. r r radian r r 5 radians radians = Figure.7 Recall that the four quadrants in a coordinate sstem are numbered I, II, III, and IV. Figure.8 shows which angles between 0 and lie in each of the four quadrants. Note that angles between 0 and are acute and that angles between and are obtuse. Figure.8 Quadrant II < < Quadrant I 0 < < = = 0 Quadrant III Quadrant IV < < < < =

4 0 Chapter Trigonometric Functions 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 and. You can find an angle that is coterminal to a given angle b adding or subtracting (one revolution), as demonstrated in Eample. A given angle has infinitel man coterminal angles. For instance, is coterminal with n, where n is an integer. Eample Sketching and Finding Coterminal Angles a. For the positive angle subtract to obtain a coterminal angle, 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.. STUDY TIP The phrase the terminal side of lies in a quadrant is often abbreviated b simpl saing that lies in a quadrant. The terminal sides of the quadrant angles 0,,, and do not lie within quadrants. = 0 = 0 0 Figure.9 Figure.0 Figure. Now tr Eercise. 5 = 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

5 Eample Complementar and Supplementar Angles If possible, find the complement and the supplement of (a) Solution a. The complement of is 5 5 The supplement of 5 is Section. Radian and Degree Measure and (b) b. Because 5 is greater than, it has no complement. (Remember that complements are positive angles.) The supplement is Now tr Eercise Degree Measure A second wa to measure angles is in terms of degrees, denoted b the smbol. A measure of one degree is equivalent to a rotation of 0 of a complete revolution about the verte. To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure.. So, a full revolution (counterclockwise) corresponds to 0, a half revolution to 80, a quarter revolution to 90, and so on. Because radians corresponds to one complete revolution, degrees and radians are related b the equations 0 rad and 80 rad. From the second equation, ou obtain and rad rad which lead to the conversion rules at the top of the net page = (0 ) 0 = (0 ) 5 = (0 ) 8 0 = (0 ) Figure.

6 Chapter Trigonometric Functions 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 or ou impl that radians or radians. Eample, 80 0 Converting from Degrees to Radians rad a. 5 5 deg Multipl b 80 deg radians 80. rad b deg Multipl b deg radians rad c deg Multipl b 80 deg radians 80. Eample Now tr Eercise 9. Converting from Radians to Degrees 80 a. Multipl b. rad rad 80 deg 80 deg 0 80 b. rad rad.59 Multipl b. 9 rad rad c. Multipl b. rad 9 rad 80 deg 80 rad Now tr Eercise. TECHNOLOGY TIP 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 one second Consequentl, an angle of degrees, minutes, and 7 seconds was represented b Man calculators have special kes for converting angles in degrees, minutes, and seconds D M S to decimal degree form, and vice versa.

7 Linear and Angular Speed The radian measure formula circle. Arc Length sr For a circle of radius r, a central angle b s r Length of circular arc can be used to measure arc length along a intercepts an arc of length s given where is measured in radians. Note that if r, then s, and the radian measure of equals the arc length. Section. Radian and Degree Measure Eample 5 Finding Arc Length 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. Solution 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.7 inches r rad Note that the units for are determined b the units for r because is given in radian measure and therefore has no units. Now tr Eercise 8. radians s = 0 Figure.5 r = 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 and Angular Speed 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 of the particle is arc length Linear speed s time t. Moreover, if is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is central angle Angular speed time t. Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes.

8 Chapter Trigonometric Functions Eample Finding Linear Speed The second hand of a clock is 0. centimeters long, as shown in Figure.. Find the linear speed of the tip of this second hand. Solution 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 0 seconds. So, the linear speed of the tip of the second hand is Figure. 0. cm Linear speed s t 0. centimeters 0 seconds Now tr Eercise centimeters per second. Eample 7 Finding Angular and Linear Speed A 5-inch diameter tire on a car makes 9. revolutions per second (see Figure.7). a. Find the angular speed of the tire in radians per second. b. Find the linear speed of the car. Solution a. Because each revolution generates radians, it follows that the tire turns radians per second. In other words, the angular speed is Angular speed t b. The linear speed of the tire and car is Linear speed s t 8. radians second r t 58. inches second Now tr Eercise radians per second. 8.5 inches per second. Figure.7 5 in. Activities. Convert 0 from degrees to radians. Answer:. Find the supplement of an angle measuring 7. 5 Answer: 7. On a circle with a radius of 9 inches, find the length of the arc intercepted b a central angle of 0. Answer: inches 7

9 Section. Radian and Degree Measure 5. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. means measurement of triangles.. An is determined b rotating a ra about its endpoint.. An angle with its initial side coinciding with the positive -ais and the origin as its verte is said to be in.. Two angles that have the same initial and terminal sides are. 5. One is the measure of a central angle that intercepts an arc equal in length to the radius of the circle.. Two positive angles that have a sum of are angles. 7. 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. 9. The speed of a particle is the ratio of the arc length traveled to the time traveled. 0. The speed of a particle is the ratio of the change in the central angle to the time. In Eercises and, estimate the angle to the nearest onehalf radian.... (a) (b) 7 = 0 5 = 0 In Eercises, determine the quadrant in which each angle lies. (The angle measure is given in radians.) 7 In Eercises 7 0, sketch each angle in standard position. 7. (a) (b) 8. (a) 7 (b) 5. (a) (b). (a) 5 (b) 9 5. (a) (b). (a).5 (b).5 9. (a) (b) 0. (a) (b) In Eercises, determine two coterminal angles in radian measure (one positive and one negative) for each angle. (There are man correct answers).. (a) (b) = =. (a) 9 (b) In Eercises 5 0, find (if possible) the complement and supplement of the angle In Eercises and, estimate the number of degrees in the angle (a) (b)

10 Chapter Trigonometric Functions In Eercises, determine the quadrant in which each angle lies.. (a) 50 (b) 8. (a) 87.9 (b) (a) 50 (b) 0. (a) 5.5 (b).5 In Eercises 7 0, sketch each angle in standard position. 7. (a) 0 (b) (a) 70 (b) 0 9. (a) 05 (b) (a) 50 (b) 00 In Eercises, determine two coterminal angles in degree measure (one positive and one negative) for each angle. (There are man correct answers).. (a) (b) = 5. (a) (b) =. (a) 00 (b) 0. (a) 5 (b) 70 In Eercises 5 8, find (if possible) the complement and supplement of the angle In Eercises 9, rewrite each angle in radian measure as a multiple of. (Do not use a calculator.) 9. (a) 0 (b) (a) 5 (b) 0. (a) 0 (b) 0. (a) 70 (b) = = (a) (b) 0. (a) 5 (b) 5 In Eercises 7 5, convert the angle measure from degrees to radians. Round our answer to three decimal places In Eercises 5 58, convert the angle measure from radians to degrees. Round our answer to three decimal places In Eercises 59, use the angle-conversion capabilities of a graphing utilit to convert the angle measure to decimal degree form. Round our answer to three decimal places if necessar In Eercises 5 70, use the angle-conversion capabilities of a graphing utilit to convert the angle measure to D M S form In Eercises 7 7, find the angle in radians cm 5 cm 8 8. in. in. In Eercises, rewrite each angle in degree measure. (Do not use a calculator.) 7. m 7.. (a) (b). (a) (b) 7 7 m 75 ft 0 ft

11 Section. Radian and Degree Measure Find each angle (in radians) shown on the unit circle. 7. Find each angle (in degrees) shown on the unit circle. In Eercises 77 80, 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 inches 8 inches 78. feet 0 feet centimeters 5 centimeters kilometers 0 kilometers In Eercises 8 8, find the length of the arc on a circle of radius r intercepted b a central angle. Radius r 8. inches 8. 9 feet Central Angle 8. 7 meters radians 8. centimeters radians In Eercises 85 88, find the radius r of a circle with an arc length s and a central angle. Arc Length s Central Angle 85. feet radians 8. meters radians miles inches Distance In Eercises 89 and 90, find the distance between the cities. Assume that Earth is a sphere of radius 000 miles and the cities are on the same longitude (one cit is due north of the other). Cit 89. Miami Erie 90. Johannesburg, South Africa Jerusalem, Israel Latitude 9. Difference in Latitudes Assuming that Earth is a sphere of radius 78 kilometers, what is the difference in the latitudes of Sracuse, New York and Annapolis, Marland, where Sracuse is 50 kilometers due north of Annapolis? 9. Difference in Latitudes Assuming that Earth is a sphere of radius 78 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? 9. Instrumentation A voltmeter s pointer is centimeters in length (see figure). Find the number of degrees through which it rotates when it moves.5 centimeters on the scale. cm Figure for 9 Figure for 9 5 N 7 N 8 S N ft 0 in. Not drawn to scale 9. Electric Hoist An electric hoist is used to lift a piece of equipment feet (see figure). The diameter of the drum on the hoist is 0 inches. Find the number of degrees through which the drum must rotate. 95. Sports The number of revolutions made b a figure skater for each tpe of ael jump is given. Determine the measure of the angle generated as the skater performs each jump. Give the answer in both degrees and radians. (a) Single ael: (b) Double ael: (c) Triple ael:

12 8 Chapter Trigonometric Functions 9. Linear Speed A satellite in a circular orbit 50 kilometers above Earth makes one complete revolution ever 0 minutes. What is its linear speed? Assume that Earth is a sphere of radius 00 kilometers. 97. Construction The circular blade on a saw has a diameter of 7.5 inches (see figure) and rotates at 00 revolutions per minute. (a) Find the angular speed in radians per second. (b) Find the linear speed of the saw teeth (in feet per second) as the contact the wood being cut. 98. Construction The circular blade on a saw has a diameter of 7.5 inches and rotates at 800 revolutions per minute. (a) Find the angular speed of the blade in radians per second. (b) Find the linear speed of the saw teeth (in feet per second) as the contact the wood being cut. 99. Angular Speed A computerized spin balance machine rotates a 5-inch diameter tire at 80 revolutions per minute. (a) Find the road speed (in miles per hour) at which the tire is being balanced. (b) At what rate should the spin balance machine be set so that the tire is being tested for 70 miles per hour? 00. Angular Speed A DVD is approimatel centimeters in diameter. 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 disc as it rotates. (b) Find the linear speed of a point on the outermost track as the disc rotates. Snthesis 7.5 in. True or False? In Eercises 0 0, determine whether the statement is true or false. Justif our answer. 0. A degree is a larger unit of measure than a radian. 0. An angle that measures 0 lies in Quadrant III. 0. The angles of a triangle can have radian measures,, and. 0. 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. 05. 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. 0. Geometr Show that the area of a circular sector of radius r with central angle is A r, where is measured in radians. Geometr In Eercises 07 and 08, use the result of Eercise 0 to find the area of the sector Graphical Reasoning The formulas for the area of a circular sector and arc length are A r and s r, respectivel. (r is the radius and is the angle measured in radians.) (a) If write the area and arc length as functions of r. What is the domain of each function? Use a graphing utilit to graph the functions. Use the graphs to determine which function changes more rapidl as r increases. Eplain. (b) If r 0 centimeters, write the area and arc length as functions of. What is the domain of each function? Use a graphing utilit to graph and identif the functions. 0. Writing A fan motor turns at a given angular speed. How does the angular speed of the tips of the blades change if a fan of greater diameter is installed on the motor? Eplain.. Writing In our own words, write a definition for radian.. Writing In our own words, eplain the difference between radian and degree. 0.8, Skills Review 0 m Librar of Parent Functions In Eercises 8, sketch the graph of 5 and the specified transformation.. f 5. f 5 5. f 5. f 5 7. f 5 8. f 5 5 ft 5 ft

13 . Trigonometric Functions: The Unit Circle Section. Trigonometric Functions: The Unit Circle 9 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.8. Unit circle (0, ) What ou should learn Identif a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle. Use 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 75 on page 75, the displacement from equilibrium of an oscillating weight suspended b a spring is modeled as a function of time. (, 0) (, 0) Figure.8 (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.9. Photo credit (, ) t t > 0 Richard Megna/Fundamental Photographs (, 0) (, 0) t (, ) t < 0 Figure.9 As the real number line is wrapped around the unit circle, each real number t corresponds to a point, on the circle. For eample, the real number 0 corresponds to the point, 0. Moreover, because the unit circle has a circumference of, the real number also corresponds to the point, 0. 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.

14 70 Chapter Trigonometric Functions The Trigonometric Functions From the preceding discussion, it follows that the coordinates and are two functions of the real variable t. You can use these coordinates to define the si trigonometric functions of t. sine cosecant cosine secant tangent cotangent These si functions are normall abbreviated sin, csc, cos, sec, tan, and cot, respectivel. 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 tan t, 0 sec t, cot t, 0 0 (0, ),, ( ) ( ) Note that the functions in the second column are the reciprocals of the corresponding functions in the first column. 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.0, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, and.,,,,,,, Similarl, in Figure., the unit circle has been divided into equal arcs, corresponding to t-values of 0, and.,,,, 5,,,,,,, Using the, coordinates in Figures.0 and., ou can easil evaluate the eact values of trigonometric functions for common t-values. This procedure is demonstrated in Eamples and. You should stud and learn these eact values for common t-values because the will help ou in later sections to perform calculations quickl and easil , Figure.0 (, 0) ( ), (, ) (, ) Figure. ( ) (, 0) (0, ) (, ) (0, ) (0, ) (, 0) ( ), ( ), (, 0) (, ) (, ) (, )

15 Eample Evaluating Trigonometric Functions Evaluate the si trigonometric functions at each real number. a. t b. t 5 c. t 0 d. t Solution For each t-value, begin b finding the corresponding point, on the unit circle. Then use the definitions of trigonometric functions listed on page 70. a. t corresponds to the point,,. sin cos sec tan cot b. t 5 corresponds to the point,,. 5 sin 5 cos 5 tan c. t 0 corresponds to the point,, 0. Section. Trigonometric Functions: The Unit Circle 7 csc 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.

16 7 Chapter Trigonometric Functions Eample 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 sin t cos t Adding to each value of t in the interval 0, completes a second revolution around the unit circle, as shown in Figure.. 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) around the unit circle. This leads to the general result and Evaluating Trigonometric Functions Evaluate the si trigonometric functions at t. Solution Moving clockwise around the unit circle, it follows that t corresponds to the point,,. sin cos tan sint n sin t cost n cos t Now tr Eercise 5. csc sec cot for an integer n and real number t. Functions that behave in such a repetitive (or cclic) manner are called periodic. Definition of a Periodic Function A function f is periodic if there eists a positive real number c such that f t c f t for all t in the domain of f. The least number c for which f is periodic is called the period of f. Eploration With our graphing utilit in radian and parametric modes, enter XT cos T and YT sin Tand use the following settings. Tmin 0, Tma., 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 for and? (0, ) Unit circle (, 0) (, 0) t = t = (0, ) Figure., + t =,,... t =,,... 5, 5 +,... t = 7 t =, +, +,... +,... t = 0,,... 7, +,... t =, +, +,... Figure.

17 Section. Trigonometric Functions: The Unit Circle 7 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 Prerequisite Skills To review even and odd functions, see Section.. Eample Using the Period to Evaluate the Sine and Cosine a. Because ou have, b. Because 7 ou have, sin sin cos 7 cos sin. cos 0. STUDY TIP It follows from the definition of periodic function 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.. c. For sin t sint because the function is odd. 5, 5 Now tr Eercise 9. 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 (degrees or radians). 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. Eample SIN 8 ENTER Displa.59 Using a Calculator Function Mode Graphing Calculator Kestrokes Displa a. sin Radian SIN ENTER b. cot.5 Radian TAN.5 ENTER Now tr Eercise 55. TECHNOLOGY TIP 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 ENTER These kestrokes ield the correct value of 0.5.

18 7 Chapter Trigonometric Functions. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular 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 ft c ft for all t in the domain of f.. A function f is if ft ftand if ft ft. 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. ( ), t. t 7. t 7 8. t 5 9. t 0. t 5. t. t. t 7. t 5. t. t ( ) 5, ( ) 5, In Eercises 7 0, evaluate (if possible) the sine, cosine, and tangent of the real number. 7. t 8. t 9. t 7 0. t 5. t. t 5. t. 5. t. t 7. t 7 8. t 9. t 0. t In Eercises, evaluate (if possible) the si trigonometric functions of the real number.. t. 5. t. 5. t. t t 5 t t 7 In Eercises 7, evaluate the trigonometric function using its period as an aid. 7. sin 5 8. cos 7 9. cos 8 0. sin 9. cos.. sin 9. cos 8 sin 9

19 Section. Trigonometric Functions: The Unit Circle 75 In Eercises 5 50, use the value of the trigonometric function to evaluate the indicated functions. 5. sin t. cos t (a) sint (a) cost (b) csct (b) sect 7. cost 5 8. sint 8 (a) cos t (a) sin t (b) sect (b) csc t 9. sin t cos t 5 (a) (a) (b) sint (b) cost sin t In Eercises 5 8, use a calculator to evaluate the trigonometric epression. Round our answer to four decimal places. 5. sin 7 5. tan cos 5. sin csc. 5. cot cos cos csc sec.8. sec.8. sin.. cot.5. tan csc.5. tan.5 7. sec.5 8. csc5. Estimation In Eercises 9 and 70, use the figure and a straightedge to approimate the value of each trigonometric function. Check our approimation using a graphing utilit. To print an enlarged cop of the graph, go to the website 9. (a) sin 5 (b) cos 70. (a) sin 0.75 (b) cos cos t Estimation In Eercises 7 and 7, use the figure and a straightedge to approimate the solution of each equation, where 0 } t <. Check our approimation using a graphing utilit. To print an enlarged cop of the graph, go to the website 7. (a) sin t 0.5 (b) cos t (a) sin t 0.75 (b) cos t Electrical Circuits The initial current and charge in an electrical circuit are zero. The current when 00 volts is applied to the circuit is given b I 5e t sin t where the resistance, inductance, and capacitance are 80 ohms, 0 henrs, and 0.0 farad, respectivel. Approimate the current (in amperes) t 0.7 second after the voltage is applied. 7. Electrical Circuits Approimate the current (in amperes) in the electrical circuit in Eercise 7 t. seconds after the voltage is applied. 75. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos t where is the displacement (in feet) and t is the time (in seconds) (see figure). Find the displacement when (a) t 0, (b) t and (c) t,. ft 0 ft 0 ft 0 ft ft Equilibrium Maimum negative Maimum positive displacement displacement Simple Harmonic Motion ft ft ft ft Figure for 9 7

20 7 Chapter Trigonometric Functions 7. 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 where is the displacement (in feet) and t is the time (in seconds). (a) What is the initial displacement t 0? (b) Use a graphing utilit to complete the table. (c) The approimate times when the weight is at its maimum distance from equilibrium are shown in the table in part (b). Eplain wh the magnitude of the maimum displacement is decreasing. What causes this decrease in maimum displacement in the phsical sstem? What factor in the model measures this decrease? (d) Find the first two times that the weight is at the equilibrium point 0. Snthesis True or False? In Eercises 77 80, determine whether the statement is true or false. Justif our answer. 77. Because sint sin t, it can be said that the sine of a negative angle is a negative number. 78. sin a sina 79. The real number 0 corresponds to the point 0, on the unit circle. 80. t et cos t t cos 7 cos 8. 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 8. 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 sint. (c) Make a conjecture about an relationship between cos t and cost. sin t. cos t. 8. Verif that cos t cos t b approimating cos.5 and cos Verif that sint t sin t sin t b approimating sin 0.5, sin 0.75, and sin. 85. Use the unit circle to verif that the cosine and secant functions are even. 8. Use the unit circle to verif that the sine, cosecant, tangent, and cotangent functions are odd. 87. Think About It Because f t sin t is an odd function and gt cos t is an even function, what can be said about the function ht f tgt? 88. Think About It Because f t sin t and gt tan t are odd functions, what can be said about the function ht f tgt? Skills Review In Eercises 89 9, find the inverse function f of the one-to-one function f. Verif b using a graphing utilit to graph both f and in the same viewing window. 89. f 90. f 9. f, 9. In Eercises 9 9, sketch the graph of the rational function b hand. Show all asmptotes. Use a graphing utilit to verif our graph f f f f 0 8 f 5 8 In Eercises 97 00, identif the domain, an intercepts, and an asmptotes of the function ln 99. f 00. f, f 5 7 >

21 Section. Right Triangle Trigonometr 77. Right Triangle Trigonometr 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.. 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 ). Hpotenuse Side adjacent to Figure. Side opposite 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 You can use trigonometr to analze all aspects of a geometric figure. For instance, Eercise 8 on page 8 shows ou how trigonometric functions can be used to approimate the angle of elevation of a zip-line. 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. Right Triangle Definitions of Trigonometric Functions Let be an acute angle of a right triangle. Then 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 Jerr Driendl/Gett Images

22 78 Chapter Trigonometric Functions Eample Evaluating Trigonometric Functions Use the triangle in Figure.5 to find the eact values of the si trigonometric functions of. Solution B the Pthagorean Theorem, hp opp adj, it follows that hp 5 5. So, the si trigonometric functions of sin opp hp 5 cos adj hp 5 tan opp adj Now tr Eercise. are csc hp opp 5 sec hp adj 5 cot adj opp Hpotenuse Figure.5 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 for a given acute angle. To do this, ou can construct a right triangle having as one of its angles. Eample Evaluating Trigonometric Functions of 5 Find the eact values of sin 5, cos 5, and tan 5. Solution Construct a right triangle having 5 as one of its acute angles, as shown in Figure.. Choose as the length of the adjacent side. 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 can find the length of the hpotenuse to be. 5 sin 5 opp hp cos 5 adj hp tan 5 opp adj 5 Figure. Now tr Eercise 7. TECHNOLOGY TIP 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.

23 Section. Right Triangle Trigonometr 79 Eample Evaluating Trigonometric Functions of 0 and 0 Use the equilateral triangle shown in Figure.7 to find the eact values of sin 0, cos 0, sin 0, and cos 0. 0 Figure.7 0 Solution Use the Pthagorean Theorem and the equilateral triangle to verif the lengths of the sides given in Figure.7. For ou have adj, opp, and hp. So, sin 0 opp hp 0, and For adj, opp, and hp. So, sin 0 opp hp and Now tr Eercise 9. 0, cos 0 adj hp. cos 0 adj hp. STUDY TIP Because the angles 0, 5, and 0,, and occur frequentl in trigonometr, ou should learn to construct the triangles shown in Figures. and.7. Sines, Cosines, and Tangents of Special Angles sin 0 sin sin 5 sin sin 0 sin cos 0 cos cos 5 cos cos 0 cos tan 0 tan tan 5 tan tan 0 tan In the bo, note that sin 0 cos 0. This occurs because 0 and 0 are complementar angles, and, 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 80 Chapter Trigonometric Functions Trigonometric Identities In trigonometr, a great deal of time is spent studing relationships between trigonometric functions (identities). Fundamental Trigonometric Identities Reciprocal Identities sin csc csc sin Quotient Identities tan sin cos Pthagorean Identities sin cos tan sec cot csc cos sec sec cos cot cos sin tan cot cot tan Eploration Select a number t and use our graphing utilit to calculate sin t cos t. Repeat this eperiment for other values of t and eplain wh the answer is alwas the same. Is the result true in both radian and degree modes? Note that sin represents sin, cos represents cos, and so on. Eample Appling Trigonometric Identities Let be an acute angle such that cos 0.8. Find the values of (a) sin and (b) tan using trigonometric identities. Solution a. To find the value of sin, use the Pthagorean identit sin cos. So, ou have sin 0.8 Substitute 0.8 for cos. sin Subtract 0.8 from each side. sin Etract positive square root. b. Now, knowing the sine and cosine of, ou can find the tangent of to be tan sin cos 0.8 Use the definitions of cos and tan and the triangle shown in Figure.8 to check these results Now tr Eercise Figure.8 0.

25 Eample 5 Using Trigonometric Identities Use trigonometric identities to transform one side of the equation into the other 0 < <. a. cos sec b. sec tan sec tan Solution Simplif the epression on the left-hand side of the equation until ou obtain the right-hand side. a. cos sec Reciprocal identit sec sec Section. Right Triangle Trigonometr 8 Divide out common factor. b. sec tan sec tan sec sec tan sec tan tan sec tan Now tr Eercise 55. Distributive Propert Simplif. Pthagorean identit Evaluating Trigonometric Functions with a Calculator To use a calculator to evaluate trigonometric functions of angles measured in degrees, set the calculator to degree mode and then proceed as demonstrated in Section.. Eample Using a Calculator Use a calculator to evaluate sec5 0. Solution Begin b converting to decimal degree form Then use a calculator in degree mode to evaluate sec 5.7. Function Graphing Calculator Displa Kestrokes sec50 sec 5.7 COS 5.7 ENTER.009 Now tr Eercise 59. Prerequisite Skills To review evaluating trigonometric functions, see Section. STUDY TIP Remember that throughout this tet, it is assumed that angles are measured in radians unless noted otherwise. For eample, sin means the sine of radian and sin means the sine of degree. Applications Involving Right Triangles 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.

26 8 Chapter Trigonometric Functions In Eample 7, the angle ou are given is the angle of elevation, which represents the angle from the horizontal upward to the object. In other applications ou ma be given the angle of depression, which represents the angle from the horizontal downward to the object. Eample 7 Using Trigonometr to Solve a Right Triangle A surveor is standing 50 feet from the base of a large tree, as shown in Figure.9. The surveor measures the angle of elevation to the top of the tree as 7.5. How tall is the tree? Angle of elevation 7.5 = 50 ft Not drawn to scale Solution Figure.9 From Figure.9, ou can see that tan 7.5 opp adj where 50 and is the height of the tree. So, the height of the tree is tan tan feet. Now tr Eercise 77. Eample 8 Using Trigonometr to Solve a Right Triangle You are 00 ards from a river. Rather than walking directl to the river, ou walk 00 ards along a straight path to the river s edge. Find the acute angle between this path and the river s edge, as illustrated in Figure.0. Solution From Figure.0, ou can see that the sine of the angle sin opp 00 hp 00. Now ou should recognize that 0. Now tr Eercise 8. is 00 d Figure.0 00 d

27 Section. Right Triangle Trigonometr 8 In Eample 8, ou were able to recognize that is the acute angle that satisfies the equation sin. Suppose, however, that ou were given the equation sin 0. and were asked to find the acute angle. Because and sin sin , 0 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. TECHNOLOGY TIP Calculators and graphing utilities have both degree and radian modes. As ou progress through this chapter, be sure ou use the correct mode. Eample 9 Solving a Right Triangle Find the length c of the skateboard ramp shown in Figure.. 8. c ft Figure. Solution From Figure., ou can see that sin 8. opp hp c. So, the length of the ramp is c sin feet. Now tr Eercise 8.

28 8 Chapter Trigonometric Functions. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check. Match the trigonometric function with its right triangle definition. (a) sine (b) cosine (c) tangent (d) cosecant (e) secant (f) cotangent (i) hp opp opp (ii) (iii) adj adj hp (iv) adj hp adj (v) (vi) opp opp hp In Eercises and, fill in the blanks.. Relative to the acute angle, the three sides of a right triangle are the, the side, and the side.. 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 triangles. Eplain wh the function values are the same In Eercises 9, sketch a right triangle corresponding to the trigonometric function of the acute angle. Use the Pthagorean Theorem to determine the third side of the triangle and then find the other five trigonometric functions of. 9. sin 5 0. cot 5. sec. cos 7. tan. csc 7 5. cot 9. sin 8 In Eercises 7, construct an appropriate triangle to complete the table. 0 } } 90, 0 } } / Function 7. sin 8. cos deg 0 5 rad Function Value 9. tan 0. sec

29 Section. Right Triangle Trigonometr 85. cot. csc. cos Function Function Value. sin 5. cot. tan In Eercises 7, complete the identit. 7. sin 8. cos 9. tan 0. csc. sec sin90 8. cos90 9. tan90 0. cot90. sec90. csc90 In Eercises 8, use the given function value(s) and the trigonometric identities to find the indicated trigonometric functions.. sin 0, cos 0 (a) tan 0 (b) sin 0 (c) cos 0 (d) cot 0. sin 0 tan 0, (a) csc 0 (b) cot 0 (c) cos 0 (d) cot 0 5. csc, deg sin cos sec (a) sin (c) tan. sec 5, tan (a) cos (c) cot90 rad tan cot (b) cos (d) sec90 (b) cot (d) sin cot tan 7. cos (a) sec (c) cot 8. tan 5 (a) cot (c) tan90 In Eercises 9 5, use trigonometric identities to transform one side of the equation into the other 0 < < /. 9. tan cot 50. csc tan sec 5. tan cos sin 5. cot sin cos 5. cos cos sin 5. csc cot csc cot sin cos csc sec cos sin tan cot csc tan In Eercises 57, use a calculator to evaluate each function. Round our answers to four decimal places. (Be sure the calculator is in the correct angle mode.) 57. (a) sin (b) cos (a) tan 8.5 (b) cot (a) sec (b) csc (a) cos (b) sec (a) cot (b) tan 8. (a) sec.5 (b) cos.5 In Eercises 8, find each value of in degrees 0 < < 90 and radians 0 < < / without using a calculator.. (a) sin (b) csc. (a) cos (b) tan 5. (a) sec (b) cot. (a) tan (b) cos 7. (a) csc (b) sin 8. (a) cot (b) sec (b) sin (d) sin90 (b) cos (d) csc

30 8 Chapter Trigonometric Functions In Eercises 9 7, find the eact values of the indicated variables (selected from,, and r). 9. Find and r. 70. Find and. 0 r Find and. 7. Find and r. 5 0 (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the balloon? 79. Width A biologist wants to know the width w of a river (see figure) 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. From this sighting, it is determined that. How wide is the river? Verif our result numericall. C 58 r 8 w Find and r. 7. Find and r. r 0 r = 58 A 00 ft 80. Height of a Mountain In traveling across flat land ou notice a mountain directl in front of ou. Its angle of elevation (to the peak) is.5. After ou drive miles closer to the mountain, the angle of elevation is 9 (see figure). Approimate the height of the mountain Find and r. 7. Find and r. 5 r Height A si-foot person walks from the base of a streetlight directl toward the tip of the shadow cast b the streetlight. When the person is feet from the streetlight and 5 feet from the tip of the streetlight s shadow, the person s shadow starts to appear beond the streetlight s shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities and use a variable to indicate the height of the streetlight. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the streetlight? 78. Height A 0-meter line is used to tether a helium-filled balloon. Because of a breeze, the line makes an angle of approimatel 75 with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities and use a variable to indicate the height of the balloon. r mi 8. Angle of Elevation A zip-line steel cable is being constructed for a realit television competition show. The high end of the zip-line is attached to the top of a 50-foot pole while the lower end is anchored at ground level to a stake 50 feet from the base of the pole. 50 ft 50 ft Not drawn to scale (a) Find the angle of elevation of the zip-line. (b) Find the number of feet of steel cable needed for the zip-line. (c) A contestant takes seconds to reach the ground from the top of the zip-line. At what rate is the contestant moving down the line? At what rate is the contestant dropping verticall?

31 Section. Right Triangle Trigonometr Inclined Plane The Johnstown Inclined Plane in Pennslvania is one of the longest and steepest hoists in the world. The railwa cars travel a distance of 89.5 feet at an angle of approimatel 5., rising to a height of 9.5 feet above sea level. (a) Find the vertical rise of the inclined plane. (b) Find the elevation of the lower end of the inclined plane. (c) The cars move up the mountain at a rate of 00 feet per minute. Find the rate at which the rise verticall. 8. Machine Shop Calculations A steel plate has the form of one fourth of a circle with a radius of 0 centimeters. 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 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 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 ft (, ) (, ) 9.5 feet above sea level Not drawn to scale 5 0 Snthesis True or False? In Eercises 85 87, determine whether the statement is true or false. Justif our answer. 85. sin 0 csc 0 8. sin 5 cos cot 0 csc Think About It You are given the value tan. Is it possible to find the value of sec without finding the measure of? Eplain. 89. Eploration (a) Use a graphing utilit to complete the table. Round our results to four decimal places. (b) Classif each of the three trigonometric functions as increasing or decreasing for the table values. (c) From the values in the table, verif that the tangent function is the quotient of the sine and cosine functions. 90. Eploration Use a graphing utilit to complete the table and make a conjecture about the relationship between cos and sin90. What are the angles and 90 called? cos Skills Review sin cos tan sin 90 In Eercises 9 9, use a graphing utilit to graph the eponential function. 9. f e 9. f e 9. f e 9. f e In Eercises 95 98, use a graphing utilit to graph the logarithmic function cm 0 0 (, ) 95. f log 9. f log 97. f log 98. f log

32 88 Chapter Trigonometric Functions. Trigonometric Functions of An Angle 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, the definitions here 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 r tan, 0 sec r, 0 cos r cot, csc r, 0 0 r (, ) 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, Eercise 09 on page 9 shows ou how trigonometric functions can be used to model the monthl normal temperatures in Santa Fe, New Meico. 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 Let, be a point on the terminal side of. Find the sine, cosine, and tangent of. (, ) r Figure. Richard Elliott/Gett Images Prerequisite Skills For a review of the rectangular coordinate sstem (or the Cartesian plane), see Appendi B.. Solution Referring to Figure., ou can see that,, and r 5 5. So, ou have sin cos and tan r r 5, 5,. Now tr Eercise.

33 Section. Trigonometric Functions of An Angle 89 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.. Quadrant II sin : + cos : tan : Quadrant I sin : + cos : + tan : + Eample Evaluating Trigonometric Functions Given sin and tan > 0, find cos and cot. Solution Note that lies in Quadrant III because that is the onl quadrant in which the sine is negative and the tangent is positive. Moreover, using sin r and the fact that is negative in Quadrant III, ou can let and r. Because is negative in Quadrant III, 9 5, and ou have the following. cos r cot Eact value Approimate value Eact value Quadrant III sin : cos : tan : + < < < > 0 0 < < 0 0 < < Figure. Quadrant IV sin : cos : + tan : 0 < < > > 0 0 > < 0 0 < <. Approimate value Now tr Eercise 9. Eample Solution Trigonometric Functions of Quadrant Angles Evaluate the sine and cosine functions at the angles 0,, and,. To begin, choose a point on the terminal side of each angle, as shown in Figure.. For each of the four given points, r, and ou have the following. sin 0 r 0 0 sin r sin r 0 0 sin r Now tr Eercise 9. cos 0 r cos r 0 0 cos r cos r 0 0,, 0, 0,,, 0, 0, (, 0) (0, ) (, 0) 0 Figure. (0, )

34 90 Chapter Trigonometric Functions 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.5 shows the reference angles for in Quadrants II, III, and IV. Quadrant II Reference angle: Reference angle: Reference angle: Quadrant III Quadrant IV = (radians) = 80 (degrees) = (radians) = 80 (degrees) = (radians) = 0 (degrees) Figure.5 Eample Finding Reference Angles Find the reference angle. 00. a. b. c. 5 = 00 = 0 Solution a. Because 00 lies in Quadrant IV, the angle it makes with the -ais is Degrees b. Because. lies between.5708 and it follows that it is in Quadrant II and its reference angle is Radians c. First, determine that 5 is coterminal with 5, which lies in Quadrant III. So, the reference angle is Figure. shows each angle Now tr Eercise 5. Degrees., and its reference angle. (a) =. (b) = 5 5 =. 5 and 5 are coterminal. = 5 (c) Figure.

35 Section. Trigonometric Functions of An Angle 9 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.7. B definition, ou know that (, ) sin and tan r. For the right triangle with acute angle and sides of lengths and ou have sin opp hp r, opp r = hp adj 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. tan opp adj. Figure.7 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. B using reference angles and the special angles discussed in the previous 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 following table shows the eact values of the trigonometric functions of special angles and quadrant angles. Trigonometric Values of Common Angles degrees radians 0 STUDY TIP Learning the table of values at the left 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. sin 0 0 cos 0 0 tan 0 Undef. 0 Undef.

36 9 Chapter Trigonometric Functions Eample 5 Trigonometric Functions of Nonacute Angles Evaluate each trigonometric function. a. cos b. tan0 c. csc Solution a. Because lies in Quadrant III, the reference angle is as shown in Figure.8. 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. Therefore, the reference angle is as shown in Figure.9. Finall, because the tangent is negative in Quadrant II, ou have, , tan0 tan 0. c. Because, it follows that is coterminal with the second-quadrant angle. Therefore, the reference angle is as shown in Figure.0. Because the cosecant is positive in Quadrant II, ou have, csc csc Now tr Eercise. The fundamental trigonometric identities listed in the preceding section (for an acute angle ) are also valid when is an angle in the domain of the function.. sin = = 0 = Figure.8 Figure.9 = = 0 = Eample Using Trigonometric Identities Let be an angle in Quadrant II such that sin. Find cos b using trigonometric identities. Solution Using the Pthagorean identit sin cos, ou obtain cos cos Because cos < 0 in Quadrant II, ou can use the negative root to obtain cos 8 9. Now tr Eercise 5. Figure.0 Prerequisite Skills If ou have difficult with this eample, review the Fundamental Trigonometric Identities in Section..

37 You can use a calculator to evaluate trigonometric functions, as shown in the net eample. Section. Trigonometric Functions of An Angle 9 Eample 7 Using a Calculator Use a calculator to evaluate each trigonometric function. a. cot 0 b. sin7 c. sec 9 Solution Function Mode Graphing Calculator Kestrokes Displa a. cot 0 Degree TAN 0 ENTER b. sin7 Radian SIN 7 ENTER c. sec Radian COS ENTER Now tr Eercise 79. Librar of Parent Functions: Trigonometric Functions Trigonometric functions are transcendental functions. The si trigonometric functions, sine, cosine, tangent, cosecant, secant, and cotangent, have important uses in construction, surveing, and navigation. Their periodic behavior makes them useful for modeling phenomena such as business ccles, planetar orbits, pendulums, wave motion, and light ras. The si trigonometric functions can be defined in three different was.. As the ratio of two sides of a right triangle [see Figure.(a)].. As coordinates of a point, in the plane and its distance r from the origin [see Figure.(b)].. As functions of an real number, such as time t. Hpotenuse Opposite 9 (, ) Eploration Set our graphing utilit to degree mode and enter tan 90. What happens? Wh? Now set our graphing utilit to radian mode and enter tan. Eplain the graphing utilit s answer. STUDY TIP An algebraic function, such as a polnomial, can be epressed in terms of variables and constants. Transcendental functions are functions which are not algebraic, such as eponential functions, logarithmic functions, and trigonometric functions. (a) Figure. Adjacent To be efficient in the use of trigonometric functions, ou should learn the trigonometric function values of common angles, such as those listed on page 9. Because pairs of trigonometric functions are related to each other b a variet of identities, it is useful to know the fundamental identities presented in Section.. Finall, trigonometric functions and their identit relationships pla a prominent role in calculus. A review of trigonometric functions can be found in the Stud Capsules. r (b)

38 9 Chapter Trigonometric Functions. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check In Eercises, let be an angle in standard position with, a point on the terminal side of and r 0. Fill in the blanks. r. sin.. tan. sec 5.. 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. Librar of Parent Functions In Eercises, determine the eact values of the si trigonometric functions of the angle.. (a) (b). (a) (b). (a) (b) (, ). (a) (b) (, ) (, 5) (, ) ( 8, 5) (, ) (, ) (, ) In Eercises 5, 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,. 8, , 8., 0 9., ,. 0, 8., 9 In Eercises, state the quadrant in which. sin < 0 and. sec > 0 and 5. cot > 0 and. tan > 0 and lies. In Eercises 7, find the values of the si trigonometric functions of. Function Value Constraint 7. sin 5 lies in Quadrant II. 8. cos 5 lies in Quadrant III tan 5 8 csc sin < 0 cot < 0. sec 0. sin 0. cot is undefined.. tan is undefined. cos < 0 cot < 0 cos > 0 csc < 0

39 Section. Trigonometric Functions of An Angle 95 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. III 7. 0 III 8. 0 IV In Eercises 9, evaluate the trigonometric function of the quadrant angle. 9. sec 0. tan. cot. csc 0. sec cot. csc In Eercises 7, find the reference angle for the special angle. Then sketch and in standard position In Eercises 5 5, find the reference angle and in standard position and sketch In Eercises 5, evaluate the sine, cosine, and tangent of the angle without using a calculator csc In Eercises 5 70, find the indicated trigonometric value in the specified quadrant. Function Quadrant Trigonometric Value 5. sin 5 IV cos. cot II sin 7. csc IV cot 8. cos 5 8 I sec 9. sec 9 III tan 70. tan 5 IV csc In Eercises 7 7, use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. 7. sin cos < 0 7. cos 5, 7, 7. tan, cos < 0 7. cot 5, 75. csc tan < 0 7. sec,, In Eercises 77 88, 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.) 77. sin sec tan csc 0 8. cos0 8. cot0 8. sec80 8. csc tan 8. tan csc In Eercises 89 9, find two solutions of the equation. Give our answers in degrees 0 } < 0 and radians 0 } <. Do not use a calculator. 89. (a) sin (b) sin 90. (a) cos (b) cos 9. (a) csc (b) cot 9. (a) csc (b) csc 9. (a) sec (b) cos 9. (a) cot (b) sec 0 cos 5 sin < 0 sin > 0 cot > 0

40 9 Chapter Trigonometric Functions In Eercises 95 08, find the eact value of each function for the given angle for f sin and g cos. Do not use a calculator. (a) f g (b) g f (c) [g (d) fg (e) f (f) g Meteorolog The monthl normal temperatures T (in degrees Fahrenheit) for Santa Fe, New Meico are given b T cos t 7 where t is the time in months, with t corresponding to Januar. Find the monthl normal temperature for each month. (Source: National Climatic Data Center) (a) Januar (b) Jul (c) December 0. Sales A compan that produces water skis, which are seasonal products, forecasts monthl sales over a twoear period to be t S. 0.t. sin where S is measured in thousands of units and t is the time (in months), with t representing Januar 00. Estimate sales for each month. (a) Januar 00 (b) Februar 007 (c) Ma 00 (d) June 00. Distance An airplane fling at an altitude of miles is on a flight path that passes directl over an observer (see figure). If is the angle of elevation from the observer to the plane, find the distance from the observer to the plane when (a) (b) and (c) 0, d 90, mi. Writing Consider an angle in standard position with r centimeters, as shown in the figure. Write a short paragraph describing the changes in the magnitudes of,, sin, cos, and tan as increases continuall from 0 to 90. Snthesis True or False? In Eercises and, determine whether the statement is true or false. Justif our answer... cot sin 5 sin 9 5. Conjecture (a) Use a graphing utilit to complete the table. sin (b) Make a conjecture about the relationship between sin and sin80.. Writing Create a table of the si trigonometric functions comparing their domains, ranges, parit (evenness or oddness), periods, and zeros. Then identif, and write a short paragraph describing, an inherent patterns in the trigonometric functions. What can ou conclude? Skills Review sin80 cm (, ) cot In Eercises 7, solve the equation. Round our answer to three decimal places, if necessar e 5. ln. ln Not drawn to scale

41 .5 Graphs of Sine and Cosine Functions Section.5 Graphs of Sine and Cosine Functions 97 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., 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 wave repeats indefinitel to the right and left. The graph of the cosine function is shown in Figure.. To produce these graphs with a graphing utilit, make sure ou set the graphing utilit to radian mode. 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. and.? Range: Figure. = sin 5 Period: 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 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 77 on page 0, ou can use a trigonometric function to model the percent of the moon s face that is illuminated for an given da in 00. = cos Range: Figure. 5 Period: Jerr Lodriguss/Photo Researchers, Inc. The table below lists ke points on the graphs of sin and cos. 0 sin cos 0 0 Note in Figures. and. 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 whereas the cosine function is even.

42 98 Chapter Trigonometric Functions To sketch the graphs of the basic sine and cosine functions b hand, it helps to note five ke points in one period of each graph: the intercepts, the maimum points, and the minimum points. See Figure.. Figure. Maimum Intercept Minimum Intercept Intercept (, ) = sin (0, 0) Quarter period Half period Period: (, 0) (, ) Three-quarter period (, 0) Full period (0, ) Maimum Intercept Quarter period Period: = cos Minimum (, 0 ) (, 0) (, ) Half period Intercept Maimum (, ) Three-quarter period Full period Eample Using Ke Points to Sketch a Sine Curve Sketch the graph of sin b hand on the interval,. Solution Note that sin sin indicates that the -values of 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 Intercept Maimum Intercept Minimum Intercept 0, 0,,,, 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.5. Use a graphing utilit to confirm this graph. Be sure to set the graphing utilit to radian mode. and Eploration Enter the Graphing a Sine Function Program, found at this tetbook s Online Stud Center, into our graphing utilit. This program simultaneousl draws the unit circle and the corresponding points on the sine curve, as shown below. After the circle and sine curve are drawn, ou can connect the points on the unit circle with their corresponding points on the sine curve b pressing ENTER. Discuss the relationship that is illustrated Figure.5 Now tr Eercise 9.

43 Section.5 Graphs of Sine and Cosine Functions 99 Amplitude and Period of Sine and Cosine Curves In the rest 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. 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. Eample Scaling: Vertical Shrinking and Stretching On the same set of coordinate aes, sketch the graph of each function b hand. a. cos b. cos Solution a. Because the amplitude of is the maimum value is and the minimum value is. cos, Divide one ccle, 0, into four equal parts to get the ke points Maimum Intercept Minimum Intercept Maimum 0,,, and, 0,, b. A similar analsis shows that the amplitude of cos is, and the ke points are Maimum Intercept Minimum Intercept Maimum 0,,,, and,., 0, The graphs of these two functions are shown in Figure.. 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. Use a graphing utilit to confirm these graphs. Now tr Eercise 0., 0,, 0,,. Prerequisite Skills For a review of transformations of functions, see Section.. TECHNOLOGY TIP When using a graphing utilit to graph trigonometric functions, pa special attention to the viewing window ou use. For instance, tr graphing sin00 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. For instructions on how to use the zoom feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Figure.

44 00 Chapter Trigonometric Functions You know from Section. 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.7. Because a sin completes one ccle from 0 to, it follows that a sin b completes one ccle from 0 to b. = cos = cos 5 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 Period b. Figure.7 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 Scaling: Horizontal Stretching Sketch the graph of sin b hand. Solution 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,. The graph is shown in Figure.8. Use a graphing utilit to confirm this graph. Figure.8,,, 0, Now tr Eercise.,, and, 0.. STUDY TIP In general, to divide a periodinterval into four equal parts, successivel add period/, starting with the left endpoint of the interval. For instance, for the period-interval, of length, ou would successivel add to get, 0,,, and as the ke points on the graph.

45 Section.5 Graphs of Sine and Cosine Functions 0 Translations of Sine and Cosine Curves The constant c in the general equations a sinb c and a cosb c creates horizontal translations (shifts) 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 Prerequisite Skills To review horizontal and vertical shifts of graphs, see Section.. Left endpoint Right endpoint c b c b b. Period This implies that the period of a sinb c is b, and the graph of a sin b is shifted b an amount cb. The number cb is the phase shift. Graphs of Sine and Cosine Functions The graphs of a sinb c and a cosb c have the following characteristics. (Assume b > 0. ) Amplitude a Period b The left and right endpoints of a one-ccle interval can be determined b solving the equations b c 0 and b c. TECHNOLOGY SUPPORT For instructions on how to use the minimum feature, the maimum feature, and the zero or root feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Eample Horizontal Translation Analze the graph of sin Algebraic Solution. The amplitude is and the period is. B solving the equations 0 and ou see that the interval, 7 corresponds to one ccle of the graph. Dividing this interval into four equal parts produces the following ke points. Intercept Maimum Intercept Minimum Intercept, 0, 5,, 7, 0, Now tr Eercise.,, 7, 0 Graphical Solution Use a graphing utilit set in radian mode to graph sin, as shown in Figure.9. Use the minimum, maimum, and zero or root features of the graphing utilit to approimate the ke points.05, 0,., 0.5,.9, 0, 5.7, 0.5, and 7., 0. Figure.9 = sin ( 5 (

46 0 Chapter Trigonometric Functions Eample 5 Horizontal Translation Use a graphing utilit to analze the graph of cos. Solution The amplitude is and the period is. B solving the equations 0 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.50. Now tr Eercise 5. = cos( + ) Figure.50 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 Vertical Translation Use a graphing utilit to analze the graph of cos. Solution The amplitude is and the period is. The ke points over the interval 0, are 0, 5,,,,,,, and, 5. The graph is shown in Figure.5. Compared with the graph of f cos, the graph of cos is shifted upward two units. Now tr Eercise 9. Figure.5 = + cos Eample 7 Finding an Equation for a Graph Find the amplitude, period, and phase shift for the sine function whose graph is shown in Figure.5. Write an equation for this graph. Solution The amplitude of this sine curve is. The period is, and there is a right phase shift of. So, ou can write sin. Now tr Eercise 5. Figure.5

47 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 8 Finding a Trigonometric Model Throughout the da, the depth of the water at the end of a dock in Bangor, Washington varies with the tides. The table shows the depths (in feet) at various times during the morning. (Source: Nautical Software, Inc.) Section.5 Graphs of Sine and Cosine Functions 0 Time Depth, a. Use a trigonometric function to model the data. b. A boat needs at least 0 feet of water to moor at the dock. During what times in the evening can it safel dock? Solution a. Begin b graphing the data, as shown in Figure.5. You can use either a sine or cosine model. Suppose ou use a cosine model of the form a cosbt c d. Midnight. A.M. 7.8 A.M.. A.M A.M.. 0 A.M..7 Noon 0.9 The difference between the maimum height and 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 which implies that b p 0.9. Because high tide occurs hours after midnight, consider the left endpoint to be cb, so c.57. Moreover, because the average depth is. 0.9., it follows that d.. So, ou can model the depth with the function 5. cos0.9t.57.. b. Using a graphing utilit, graph the model with the line 0. Using the intersect feature, ou can determine that the depth is at least 0 feet between :0 P.M. t 8. and 9:8 P.M. t.8, as shown in Figure.5. Now tr Eercise 77. Depth (in feet) 0 8 Figure.5 = 0 (8., 0) (.8, 0) 0 0 = 5. cos(0.9t.57) +. Figure.5 Changing Tides A.M. 8 A.M. Noon Time TECHNOLOGY SUPPORT For instructions on how to use the intersect feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. t

48 0 Chapter Trigonometric Functions.5 Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. The of a sine or cosine curve represents half the distance between the maimum and minimum values of the function.. One period of a sine function is called of the sine curve.. The period of a sine or cosine function is given b.. For the equation a sinb c, c is the of the graph of the equation. b Librar of Parent Functions In Eercises and, use the graph of the function to answer the following. (a) Find the -intercepts of the graph of f. (b) Find the -intercepts of the graph of f. (c) Find the intervals on which the graph f is increasing and the intervals on which the graph f is decreasing. (d) Find the relative etrema of the graph of f.. f sin. f cos In Eercises, find the period and amplitude.. sin. cos sin cos 9. sin 0.. cos. 5 cos. sin. In Eercises 5, describe the relationship between the graphs of f and g. Consider amplitudes, periods, and shifts. 5. f ) sin. f cos 7. f cos 8. f sin 9. f cos 0. sin f g 5 cos. f sin. f cos In Eercises, describe the relationship between the graphs of f and g. Consider amplitudes, periods, and shifts... g sin g cos g 5 sin f g cos 5 cos g cos g( sin g sin g cos f g 5.. g f f g sin cos

49 Section.5 Graphs of Sine and Cosine Functions 05 In Eercises 7, sketch the graphs of f and g in the same coordinate plane. (Include two full periods.) 7. f sin 8. f sin g sin g cos g sin g cos Conjecture In Eercises 5 8, use a graphing utilit to graph f and g in the same viewing window. (Include two full periods.) Make a conjecture about the functions. g cos In Eercises 9, sketch the graph of the function b hand. Use a graphing utilit to verif our sketch. (Include two full periods.) 9. sin 0. cos In Eercises 7 0, use a graphing utilit to graph the function. (Include two full periods.) Identif the amplitude and period of the graph. 7. sin 8. t 9. 5 cos 50. g sin g sin 9. f cos 0. f cos g cos. f. f sin sin g sin. f cos. f cos g cos 5. f sin. f sin g cos 7. f cos 8. f cos. cos. sin. sin. sin 5. 8 cos. cos 0 cos sin g cos cos cos 5. sin 5. sin 55. cos 5. cos sin cos sin 0t 00 cos 50t Graphical Reasoning In Eercises, find a and d for the function fh a cos d such that the graph of f matches the figure Graphical Reasoning In Eercises 5 8, find a, b, and c for the function f a sinb c such that the graph of f matches the graph shown In Eercises 9 and 70, use a graphing utilit to graph and for all real numbers in the interval [,. Use the graphs to find the real numbers such that. 9. sin 70. cos 8

50 0 Chapter Trigonometric Functions 7. Health 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 the net) is given b v 0.85 sin t/, where t is the time (in seconds). Inhalation occurs when v > 0, and ehalation occurs when v < 0. (a) Use a graphing utilit to graph v. (b) Find the time for one full respirator ccle. (c) Find the number of ccles per minute. (d) The model is for a person at rest. How might the model change for a person who is eercising? Eplain. 7. Sales A compan that produces snowboards, which are seasonal products, forecasts monthl sales for ear to be t S cos where S is the sales in thousands of units and t is the time in months, with t corresponding to Januar. (a) Use a graphing utilit to graph the sales function over the one-ear period. (b) Use the graph in part (a) to determine the months of maimum and minimum sales. 7. Recreation You are riding a Ferris wheel. Your height h (in feet) above the ground at an time t (in seconds) can be modeled b h 5 sin t The Ferris wheel turns for 5 seconds before it stops to let the first passengers off. (a) Use a graphing utilit to graph the model. (b) What are the minimum and maimum heights above the ground? 7. Health The pressure P (in millimeters of mercur) against the walls of the blood vessels of a person is modeled b P 00 0 cos 8 t where t is the time (in seconds). Use a graphing utilit to graph the model. One ccle is equivalent to one heartbeat. What is the person s pulse rate in heartbeats per minute? 75. Fuel Consumption The dail consumption C (in gallons) of diesel fuel on a farm is modeled b C 0.. sin t where t is the time in das, with t corresponding to Januar. (a) What is the period of the model? Is it what ou epected? Eplain. (b) What is the average dail fuel consumption? Which term of the model did ou use? Eplain. (c) Use a graphing utilit to graph the model. Use the graph to approimate the time of the ear when consumption eceeds 0 gallons per da. 7. Data Analsis The motion of an oscillating weight suspended from a spring was measured b a motion detector. The data was collected, and the approimate maimum displacements from equilibrium are labeled in the figure. The distance from the motion detector is measured in centimeters and the time t is measured in seconds. 0 0 (0.5,.5) (0.75,.5) (a) Is a function of t? Eplain. (b) Approimate the amplitude and period. (c) Find a model for the data. (d) Use a graphing utilit to graph the model in part (c). Compare the result with the data in the figure. 77. Data Analsis The percent (in decimal form) of the moon s face that is illuminated on da of the ear 00, where represents Januar, is shown in the table. (Source: U.S. Naval Observator) Da, Percent, (a) Create a scatter plot of the data. (b) Find a trigonometric model for 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 percent illumination of the moon on June 9,

51 Section.5 Graphs of Sine and Cosine Functions Data Analsis The table shows the average dail high temperatures for Nantucket, Massachusetts N and Athens, Georgia A (in degrees Fahrenheit) for month t, with t corresponding to Januar. (Source: U.S. Weather Bureau and the National Weather Service) (a) A model for the temperature in Nantucket is given b Find a trigonometric model for Athens. (b) Use a graphing utilit to graph the data and the model for the temperatures in Nantucket in the same viewing window. How well does the model fit the data? (c) Use a graphing utilit to graph the data and the model for the temperatures in Athens in the same viewing window. How well does the model fit the data? (d) Use the models to estimate the average dail high 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. Snthesis Month, t Nantucket, N Athens, A Nt 58 9 sin t 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 d a sinb c for different values of a, b, c, and d. Write a paragraph describing the changes in the graph corresponding to changes in each constant. Librar of Parent Functions In Eercises 8 8, determine which function is represented b the graph. Do not use a calculator (a) (b) (c) (d) (e) f sin (a) (b) (c) (d) f sin f sin f cos f cos f sin f cos f sin f cos (a) f cos (b) f cos (c) f sin( (d) f cos (e) (a) f cos (b) (c) (d) f sin f sin( f cos (e) f sin (e) f sin 5 5 f sin True or False? In Eercises 79 8, determine whether the statement is true or false. Justif our answer. 79. The graph of sin has a period of The function cos has an amplitude that is twice that of the function cos.

52 08 Chapter Trigonometric Functions 87. Eploration In Section., it was shown that f cos is an even function and g sin is an odd function. Use a graphing utilit to graph h and use the graph to determine whether h is even, odd, or neither. (a) h cos (b) h sin (c) h( sin cos 88. Conjecture If f is an even function and g is an odd function, use the results of Eercise 87 to make a conjecture about each of the following. (a) h f (b) h g (c) h f g 89. Eploration Use a graphing utilit to eplore the ratio sin, which appears in calculus. (a) Complete the table. Round our results to four decimal places sin sin (b) Use a graphing utilit to graph the function f sin. Use the zoom and trace features to describe the behavior of the graph as approaches 0. (c) Write a brief statement regarding the value of the ratio based on our results in parts (a) and (b). 90. Eploration Use a graphing utilit to eplore the ratio cos /, which appears in calculus. (a) Complete the table. Round our results to four decimal places cos cos (b) Use a graphing utilit to graph the function f Use the zoom and trace features to describe the behavior of the graph as approaches 0. (c) Write a brief statement regarding the value of the ratio based on our results in parts (a) and (b). 9. 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? 9. Eploration Use the polnomial approimations found in Eercise 9(c) to approimate the following values. Round our answers to four decimal places. Compare the results with those given b a calculator. How does the error in the approimation change as approaches 0? (a) sin (b) sin (c) sin 8 (d) cos (e) cos (f) Skills Review cos. In Eercises 9 and 9, plot the points and find the slope of the line passing through the points. 9. 0,,, 7 9.,,, In Eercises 95 and 9, convert the angle measure from radians to degrees. Round our answer to three decimal places cos!! 0.8 cos 97. Make a Decision To work an etended application analzing the normal dail maimum temperature and normal precipitation in Honolulu, Hawaii, visit this tetbook s Online Stud Center. (Data Source: NOAA)

53 . Graphs of Other Trigonometric Functions Section. Graphs of Other Trigonometric Functions 09 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 function is undefined at values at which cos 0. Two such values are ± ± tan Undef Undef. tan approaches as tan approaches as approaches from the right. approaches from the left 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 You can use tangent, cotangent, secant, and cosecant functions to model real-life data. For instance, Eercise on page 8 shows ou how a tangent function can be used to model and analze the distance between a television camera and a parade unit. As indicated in the table, tan increases without bound as approaches from the left, and it decreases without bound as approaches from the right. So, the graph of tan has vertical asmptotes at and, as shown in Figure.55. Moreover, because the period of the tangent function is, vertical asmptotes also occur at 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 Figure.55 Period: Domain: all real numbers, ecept n Range:, Vertical asmptotes: n 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 asmptotes can be found b solving the equations b c and b c. The midpoint between two consecutive asmptotes is an -intercept of the graph. The period of the function a tanb c is the distance between two consecutive 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. A. Rame/PhotoEdit

54 0 Chapter Trigonometric Functions Eample Sketching the Graph of a Tangent Function Sketch the graph of tan b hand. Solution B solving the equations and, ou can see that two consecutive 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.5. Use a graphing utilit to confirm this graph. tan Undef. 0 Undef. 0 Figure.5 Now tr Eercise 5. Eample Sketching the Graph of a Tangent Function Sketch the graph of tan b hand. Solution B solving the equations and, ou can see that two consecutive asmptotes occur at and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three complete ccles of the graph are shown in Figure tan Undef. 0 Undef. TECHNOLOGY TIP Your graphing utilit ma connect parts of the graphs of tangent, cotangent, secant, and cosecant functions that are not supposed to be connected. So, in this tet, these functions are graphed on a graphing utilit using the dot mode. A blue curve is placed behind the graphing utilit s displa to indicate where the graph should appear. For instructions on how to use the dot mode, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Figure.57 Now tr Eercise 7.

55 Section. Graphs of Other Trigonometric Functions TECHNOLOGY TIP Graphing utilities are helpful in verifing sketches of trigonometric functions. You can use a graphing utilit set in radian and dot modes to graph the function tan from Eample, as shown in Figure.58. You can use the zero or root feature or the zoom and trace features to approimate the ke points of the graph. 5 = tan 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. 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.59. Eample Sketching the Graph of a Cotangent Function Sketch the graph of cot b hand. Solution To locate two consecutive vertical asmptotes of the graph, solve the equations 0 and to see that two consecutive asmptotes occur at 0 and. Then, 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.0. Use a graphing utilit to confirm this graph. Enter the function as tan. Note that the period is, the distance between consecutive asmptotes. Figure.58 5 = cot Period: Domain: all real numbers, ecept n Range:, Vertical asmptotes: n Figure cot Undef. 0 Undef. Now tr Eercise 5. Figure.0 Eploration Use a graphing utilit to graph the functions cos and sec cos in the same viewing window. How are the graphs related? What happens to the graph of the secant function as approaches the zeros of the cosine function?

56 Chapter Trigonometric Functions Graphs of the Reciprocal Functions The graphs of the two remaining trigonometric functions can be obtained from the graphs of the sine and cosine functions using the reciprocal identities csc and sec sin cos. For instance, at a given value of, the -coordinate for sec is the reciprocal of the -coordinate for 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 (i.e., the values at which the cosine is zero). Similarl, cot cos sin and 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 the reciprocals of the -coordinates to obtain points on the graph of csc. You can use this procedure to obtain the graphs shown in Figure.. Prerequisite Skills To review the reciprocal identities of trigonometric functions, see Section.. = sin = csc = cos = sec Period: Domain: all real numbers, ecept n Range:, ] [, Vertical asmptotes: Smmetr: origin Figure. n Period: Domain: all real numbers, ecept n Range:, ] [, Vertical asmptotes: n Smmetr: -ais 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 local minimum) on the cosecant curve, and a valle (or minimum point) on the

57 Section. Graphs of Other Trigonometric Functions sine curve corresponds to a hill (a local maimum) on the cosecant curve, as shown in Figure.. Additionall, -intercepts of the sine and cosine functions become vertical asmptotes of the cosecant and secant functions, respectivel (see Figure.). Eample Comparing Trigonometric Graphs Use a graphing utilit to compare the graphs of sin and Solution The two graphs are shown in Figure.. Note how the hills and valles of the graphs are related. For the function sin, the amplitude is and the period is. B solving the equations and 0 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 thick curve in Figure.. Because the sine function is zero at the endpoints of this interval, the corresponding cosecant function csc sin has vertical asmptotes at,, 7, and so on. Now tr Eercise 5. csc. Cosecant local minimum Figure. Figure. Sine maimum Cosecant local maimum 5 5 ( Sine minimum = sin + = csc ( + ( ( Eample 5 Comparing Trigonometric Graphs Use a graphing utilit to compare the graphs of cos and sec. Solution Begin b graphing cos and sec cos in the same viewing window, as shown in Figure.. Note that the -intercepts of cos = 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. = sec Now tr Eercise 7.

58 Chapter Trigonometric Functions 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 f sin ± at n and f sin 0 at n the graph of f touches the line or the line at n and has -intercepts at n. A sketch of f is shown in Figure.5. In the function f sin, the factor is called the damping factor. = Figure.5 = f() = sin Eample Analze the graph of Solution Analzing a Damped Sine Curve Consider f as the product of the two functions e and sin each of which has the set of real numbers as its domain. For an real number, ou know that and sin. So, e sin e 0 e, which means that Furthermore, because and f e sin. e e sin e. f e sin ±e at f e at n sin 0 the graph of f touches the curves e and e at n and has intercepts at n. The graph is shown in Figure.. Now tr Eercise 5. n STUDY TIP 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 = e = e Figure.

59 Figure.7 summarizes the si basic trigonometric functions. Section. Graphs of Other Trigonometric Functions 5 = sin = cos Domain: all real numbers Range: [, ] Period: Domain: all real numbers Range: [, ] Period: = tan = cot = tan Domain: all real numbers, Range: Period: ecept n, Domain: all real numbers, ecept n Range:, Period: = csc = sin = sec = cos Domain: all real numbers, ecept n Range:, ] [, Period: Figure.7 Domain: all real numbers, ecept n Range:, ] [, Period:

60 Chapter Trigonometric Functions. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. The graphs of the tangent, cotangent, secant, and cosecant functions have asmptotes.. To sketch the graph of a secant or cosecant function, first make a sketch of its function.. For the function f g sin, g is called the factor of the function. Librar of Parent Functions In Eercises, use the graph of the function to answer the following. (a) Find all -intercepts of the graph of f. (b) Find all -intercepts of the graph of f. (c) Find the intervals on which the graph f is increasing and the intervals on which the graph f is decreasing. (d) Find all relative etrema, if an, of the graph of f. (e) Find all vertical asmptotes, if an, of the graph of f.. f tan. f cot 7. tan 8. tan 9. sec 0. sec. csc. csc. cot. cot In Eercises 5 0, use a graphing utilit to graph the function (include two full periods). Graph the corresponding reciprocal function and compare the two graphs. Describe our viewing window. 5. csc. csc 7. sec 8. sec csc sec. f sec. f csc In Eercises, use a graph of the function to approimate the solution to the equation on the interval [,.. tan. cot. sec. csc In Eercises 5, sketch the graph of the function. (Include two full periods.) Use a graphing utilit to verif our result. 5. tan. tan 7. tan 8. tan 9. sec 0. sec. sec. sec. csc. csc 5.. cot cot In Eercises 5 8, use the graph of the function to determine whether the function is even, odd, or neither. 5. f sec. f tan 7. f csc 8. f cot In Eercises 9, 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. 9. sin csc, 0. sin sec, tan. cos sin, cot. sec, tan

61 Section. Graphs of Other Trigonometric Functions 7 In Eercises, 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) (b) (d). f cos. 5.. Conjecture In Eercises 7 50, use a graphing utilit to graph the functions f and g. Use the graphs to make a conjecture about the relationship between the functions g sin f sin, f cos, f sin cos f sin cos In Eercises 5 5, 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. 5. f e cos 5. f e sin 5. h e cos 5. g e sin,, Eploration In Eercises 55 and 5, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) as approaches from the right (b) as approaches from the left (c) as approaches from the right (d) as approaches from the left 55. f tan 5. f sec g 0 g cos g sin g cos f sin g cos Eploration In Eercises 57 and 58, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) As 0, the value of f. (b) As 0, the value of f. (c) As, the value of f. (d) As, the value of f. 57. f cot 58. f csc 59. Predator-Pre Model The population P of cootes (a predator) at time t (in months) in a region is estimated to be P 0, sin t and the population p of rabbits (its pre) is estimated to be p 5, cos t. Use the graph of the models to eplain the oscillations in the size of each population. Population 0,000 5,000 0,000 5, Time (in months) 0. Meteorolog The normal monthl high temperatures H (in degrees Fahrenheit) for Erie, Pennslvania are approimated b t t Ht cos 5.9 sin and the normal monthl low temperatures L are approimated b t t Lt cos. sin p P where t is the time (in months), with t corresponding to Januar. (Source: National Oceanic and Atmospheric Association) (a) Use a graphing utilit to graph each function. 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 the farthest north 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 8 Chapter Trigonometric Functions. Distance A plane fling at an altitude of 5 miles over level ground will pass directl over a 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 < <.. Numerical and Graphical Reasoning A crossed belt connects a 0-centimeter pulle on an electric motor with a 0-centimeter pulle on a saw arbor (see figure). The electric motor runs at 700 revolutions per minute. 0 cm 0 cm 5 mi φ d. Television Coverage A television camera is on a reviewing platform 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 Not drawn to scale (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let L be the total length of the belt. Write L as a function of, where is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of belt around each pulle.) (d) Use a graphing utilit to complete the table. m d L Camera. Harmonic Motion An object weighing W pounds is suspended from a 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 t > 0. Equilibrium (a) Use a graphing utilit to graph the function. (b) Describe the behavior of the displacement function for increasing values of time t. (e) As increases, do the lengths of the straight sections of the belt change faster or slower than the lengths of the belts around each pulle? (f) Use a graphing utilit to graph the function over the appropriate domain. Snthesis True or False? In Eercises 5 and, determine whether the statement is true or false. Justif our answer. 5. The graph of has an asmptote at. 8 tan. For the graph of sin, as approaches, approaches Graphical Reasoning Consider the functions f sin and g csc on the interval 0,. (a) Use a graphing utilit to graph f and g in the same viewing window.

63 Section. Graphs of Other Trigonometric Functions 9 (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? 8. Pattern Recognition (a) Use a graphing utilit to graph each function. sin sin sin sin 5 sin 5 (b) Identif the pattern 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. Eploration In Eercises 9 and 70, use a graphing utilit to eplore the ratio f, which appears in calculus. (a) Complete the table. Round our results to four decimal places f f (b) Use a graphing utilit to graph the function f. Use the zoom and trace features to describe the behavior of the graph as approaches 0. (c) Write a brief statement regarding the value of the ratio based on our results in parts (a) and (b). 9. f tan tan 70. f Librar of Parent Functions In Eercises 7 and 7, determine which function is represented b the graph. Do not use a calculator (a) f tan (b) f tan (b) f csc (c) f tan (c) f csc (d) f tan (d) f sec (e) f tan (e) f csc 7. 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? 7. 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? Skills Review 5 5! In Eercises 75 78, identif the rule of algebra illustrated b the statement a 9 5a a b 0 a b 0 In Eercises 79 8, determine whether the function is one-to-one. If it is, find its inverse function. 79. f f 7 8. f 8. f 5 (a) f sec

64 0 Chapter Trigonometric Functions.7 Inverse Trigonometric Functions Inverse Sine Function Recall from Section. that for a function to have an inverse function, it must be one-to-one that is, it must pass the Horizontal Line Test. In Figure.8 it is obvious that sin does not pass the test because different values of ield the same -value. Figure.8 However, if ou restrict the domain to the interval (corresponding to the black portion of the graph in Figure.8), 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. = sin Sin has an inverse function on this interval. 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. What ou should learn Evaluate and graph inverse sine functions. Evaluate other inverse trigonometric functions. Evaluate compositions of trigonometric functions. Wh ou should learn it Inverse trigonometric functions can be useful in eploring how aspects of a real-life problem relate to each other. Eercise 8 on page 9 investigates the relationship between the height of a cone-shaped pile of rock salt, the angle of the cone shape, and the diameter of its base. Francoise Sauze/Photo Researchers Inc. Definition of Inverse Sine Function The inverse sine function is defined b arcsin if and onl if sin where and. The domain of arcsin is, and the range is,. When evaluating the inverse sine function, it helps to remember the phrase the arcsine of is the angle (or number) whose sine is.

65 Section.7 Inverse Trigonometric Functions Eample Evaluating the Inverse Sine Function If possible, find the eact value. a. b. sin c. sin arcsin Solution a. Because sin and lies in it follows that,,, arcsin Angle whose sine is. b. Because sin and lies in, sin. it follows that,, Angle whose sine is c. It is not possible to evaluate sin at because there is no angle whose sine is. Remember that the domain of the inverse sine function is,. Now tr Eercise. STUDY TIP As with the trigonometric functions, much of the work with the inverse trigonometric functions can be done b eact calculations rather than b calculator approimations. Eact calculations help to increase our understanding of the inverse functions b relating them to the triangle definitions of the trigonometric functions. Eample Graphing the Arcsine Function Sketch a graph of arcsin b hand. Solution B definition, the equations arcsin and sin are equivalent for. So, their graphs are the same. For the interval,, ou can assign values to in the second equation to make a table of values. 0 sin 0 Then plot the points and connect them with a smooth curve. The resulting graph of arcsin is shown in Figure.9. Note that it is the reflection (in the line ) of the black portion of the graph in Figure.8. Use a graphing utilit to confirm this graph. Be sure ou see that Figure.9 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 0. Figure.9

66 Chapter Trigonometric Functions Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval 0 shown in Figure.70., as = cos = arcsin Cos has an inverse function on this interval. Figure.70 Consequentl, on this interval the cosine function has an inverse function the inverse cosine function denoted b arccos or cos. Because arccos and cos are equivalent for 0, their graphs are the same, and can be confirmed b the following table of values. Domain: [, ]; Range: [, ] 0 cos 0 5 = arccos 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 Domain: [, ]; Range: [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 < < = arctan The graphs of these three inverse trigonometric functions are shown in Figure.7. Domain:, ; Range: Figure.7,

67 Section.7 Inverse Trigonometric Functions Eample Evaluating Inverse Trigonometric Functions Find the eact value. a. arccos b. cos c. arctan 0 d. tan Solution a. Because cos, and lies in 0,, it follows that arccos Angle whose cosine is. b. Because cos, and lies in 0,, it follows that cos. Angle whose cosine is c. Because tan 0 0, and 0 lies in,, it follows that arctan 0 0. Angle whose tangent is 0 d. Because tan, and lies in,, it follows that tan. Now tr Eercise 5. Angle whose tangent is TECHNOLOGY TIP You can use the SIN, COS, and TAN kes on our calculator to approimate values of inverse trigonometric functions. To evaluate the inverse cosecant function, the inverse secant function, or the inverse cotangent function, ou can use the inverse sine, inverse cosine, and inverse tangent functions, respectivel. For instance, to evaluate sec., enter the epression as shown below. Eample Calculators and Inverse Trigonometric Functions Use a calculator to approimate the value (if possible). a. arctan8.5 b. sin 0.7 c. arccos Solution Function Mode Graphing Calculator Kestrokes a. arctan8.5 Radian TAN 8.5 ENTER From the displa, it follows that arctan b. sin 0.7 Radian SIN 0.7 ENTER From the displa, it follows that sin c. arccos Radian COS 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. TECHNOLOGY TIP In Eample, if ou had set the calculator to degree mode, the displa would have been in degrees rather than in radians. This convention is peculiar to calculators. B definition, the values of inverse trigonometric functions are alwas in radians.

68 Chapter Trigonometric Functions Compositions of Functions Recall from Section. that for all in the domains of f and f, inverse functions have the properties f f and f f. Inverse Properties If and, then sinarcsin and arcsinsin. If and 0, then cosarccos and arccoscos. If is a real number and < <, then tanarctan and arctantan. Eploration Use a graphing utilit to graph arcsinsin. What are the domain and range of this function? Eplain wh arcsinsin does not equal. Now graph sinarcsin and determine the domain and range. Eplain wh sinarcsin is not defined. Keep in mind that these inverse properties do not appl for arbitrar values of and. For instance, arcsin sin arcsin. In other words, the propert arcsinsin 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 Solution a. Because 5 lies in the domain of the arctangent 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..

69 Eample shows how to use right triangles to find eact values of compositions of inverse functions. Eample Find the eact value. Evaluating Compositions of Functions a. tan arccos b. cos arcsin 5 Section.7 Inverse Trigonometric Functions 5 Algebraic Solution a. If ou let u arccos then cos u,. Because cos u is positive, u is a first-quadrant angle. You can sketch and label angle u as shown in Figure.7. Consequentl, u Figure.7 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.7. tan arccos 5 = 5 5 ( ) = u tan u opp adj 5. Graphical Solution a. Use a graphing utilit set in radian mode to graph tanarccos, as shown in Figure.7. Use the value feature or the zoom and trace features of the graphing utilit to find that the value of the composition of functions when 0.7 is.8 5. Figure.7 b. Use a graphing utilit set in radian mode to graph cosarcsin, as shown in Figure.75. Use the value feature or the zoom and trace features of the graphing utilit to find that the value of the composition of functions when 5 0. is = tan(arccos ) Consequentl, Figure.7 cos arcsin 5 cos u adj hp 5. = cos(arcsin ) Now tr Eercise 9. Figure.75

70 Chapter Trigonometric Functions Librar of Parent Functions: Inverse Trigonometric Functions The inverse trigonometric functions are obtained from the trigonometric functions in much the same wa that the logarithmic function was developed from the eponential function. However, unlike the eponential function, the trigonometric functions are not one-to-one, and so it is necessar to restrict their domains to intervals on which the pass the Horizontal Line Test. Consequentl, the inverse trigonometric functions have restricted domains and ranges, and the are not periodic. A review of inverse trigonometric functions can be found in the Stud Capsules. One prominent role plaed b inverse trigonometric functions is in solving a trigonometric equation in which the argument (angle) of the trigonometric function is the unknown quantit in the equation. You will learn how to solve such equations in the net chapter. Inverse trigonometric functions pla a unique role in calculus. There are two basic operations of calculus. One operation (called differentiation) transforms an inverse trigonometric function (a transcendental function) into an algebraic function. The other operation (called integration) produces the opposite transformation from algebraic to transcendental. Eample 7 Some Problems from Calculus Write each of the following as an algebraic epression in. a. sinarccos, 0 b. cotarccos, 0 < Solution If ou let u arccos, then cos u, where. Because ou can sketch a right triangle with acute angle u, as shown in Figure.7. From this triangle, ou can easil convert each epression to algebraic form. a. b. cos u adjhp sinarccos sin u opp hp 9, cotarccos cot u adj opp 9, 0 0 < ( ) u = arccos Figure.7 A similar argument can be made here for -values ling in the interval, 0. Now tr Eercise 55.

71 Section.7 Inverse Trigonometric Functions 7.7 Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks. Function Alternative Notation Domain Range. arcsin. cos. arctan Librar of Parent Functions In Eercises 9, find the eact value of each epression without using a calculator.. (a) arcsin (b) arcsin 0. (a) arccos (b) arccos 0. (a) arcsin (b) arccos. (a) arctan (b) arctan 0 5. (a) arctan (b) arctan. (a) cos (b) sin 7. (a) arctan (b) arctan 8. (a) (b) arccos 9. (a) sin (b) 0. Numerical and Graphical Analsis Consider the function arcsin. (a) Use a graphing utilit to complete the table. arcsin tan (b) Plot the points from the table in part (a) and graph the function. (Do not use a graphing utilit.) (c) Use a graphing utilit to graph the inverse sine function and compare the result with our hand-drawn graph in part (b). (d) Determine an intercepts and smmetr of the graph.. Numerical and Graphical Analsis Consider the function arccos. (a) Use a graphing utilit to complete the table (b) Plot the points from the table in part (a) and graph the function. (Do not use a graphing utilit.) (c) Use a graphing utilit to graph the inverse cosine function and compare the result with our hand-drawn graph in part (b). (d) Determine an intercepts and smmetr of the graph.. Numerical and Graphical Analsis Consider the function arctan. (a) Use a graphing utilit to complete the table (b) Plot the points from the table in part (a) and graph the function. (Do not use a graphing utilit.) (c) Use a graphing utilit to graph the inverse tangent function and compare the result with our hand-drawn graph in part (b). (d) Determine the horizontal asmptotes of the graph.

72 8 Chapter Trigonometric Functions In Eercises and, determine the missing coordinates of the points on the graph of the function... = arctan = arccos (, ) (, ) (, ) In Eercises 5 0, use a calculator to approimate the value of the epression. Round our answer to the nearest hundredth. 5. cos sin arcsin arccos arctan 0. tan 5.9 In Eercises, use an inverse trigonometric function to write as a function of..... In Eercises 5 8, find the length of the third side of the triangle in terms of. Then find in terms for all three inverse trigonometric functions (, ) 0 (, ) ( ), In Eercises 9, use the properties of inverse functions to find the eact value of the epression. 9. sinarcsin tanarctan 5. cosarccos0.. sinarcsin0.. arcsinsin tan. sin sin tan 5 7. sin 8. cos cos sin 5 9. sin 0. cos tan tan. tanarcsin 0. cos[arctan. sinarctan. sinarctan 5.. In Eercises 7 5, find the eact value of the epression. Use a graphing utilit to verif our result. (Hint: Make a sketch of a right triangle.) 7. sinarctan 8. secarcsin 5 9. cosarcsin cscarctan 5 5. secarctan 5 5. tanarcsin 5. sinarccos 5. cotarctan 5 8 In Eercises 55, write an algebraic epression that is equivalent to the epression. (Hint: Sketch a right triangle, as demonstrated in Eample 7.) 55. cotarctan 5. sinarctan 57. sinarccos 58. secarcsin csc arctan. 7 In Eercises, complete the equation arcsin cos tan arccos 5 arctan arcsin, > 0 arcsin arccos, arccos 0 arcsin arccos cos 7 arccos sin cot arctan 0. arccos arctan, < < cos arcsin h r

73 Section.7 Inverse Trigonometric Functions 9 In Eercises 7 7, use a graphing utilit to graph the function. 7. arccos 8. arcsin 9. f arcsin 70. gt arccost 7. f arctan 7. f arccos In Eercises 7 and 7, write the function in terms of the sine function b using the identit (a) Draw a diagram that gives a visual representation of the problem. Label all known and unknown quantities. (b) Find the angle of repose for rock salt. (c) How tall is a pile of rock salt that has a base diameter of 0 feet? 8. Photograph A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let s be the height of the shuttle. A cos t B sin ta B sin t arctan A B. Use a graphing utilit to graph both forms of the function. What does the graph impl? 7. f t cos t sin t 7. f t cos t sin t Eploration In Eercises 75 80, find the value. If not possible, state the reason. 75. As, the value of arcsin. 7. As, the value of arccos. 77. As, the value of arctan. 78. As, the value of arcsin. 79. As, the value of arccos. 80. As, the value of arctan. 8. Docking a Boat A boat is pulled in b means of a winch located on a dock 0 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. (a) Write as a function of s. (b) Find when s 00 meters and when s 00 meters. 8. Photograph A photographer takes a picture of a threefoot painting hanging 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, 750 m > 0. s Not drawn to scale ft 0 ft s ft α β (a) Write as a function of s. (b) Find when s 5 feet and when s feet. 8. 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. When rock salt is stored in a cone-shaped pile feet high, the diameter of the pile s base is about feet. (Source: Bulk-Store Structures, Inc.) 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.

74 0 Chapter Trigonometric Functions 85. Angle of Elevation An airplane flies at an altitude of miles toward a point directl over an observer. Consider and as shown in the figure. (a) Write as a function of. (b) Find when 0 miles and miles. 8. Securit Patrol A securit car with its spotlight on is parked 0 meters from a warehouse. Consider and as shown in the figure. (a) Write as a function of. (b) Find when 5 meters and when meters. Snthesis True or False? In Eercises 87 and 88, determine whether the statement is true or false. Justif our answer sin 5 Not drawn to scale arctan arcsin arccos 0 m arcsin 89. Define the inverse cotangent function b restricting the domain of the cotangent function to the interval 0,, and sketch the inverse function s graph. 90. Define the inverse secant function b restricting the domain of the secant function to the intervals 0, and,, and sketch the inverse function s graph. 9. Define the inverse cosecant function b restricting the domain of the cosecant function to the intervals, 0 and 0,, and sketch the inverse function s graph. 5 mi Not drawn to scale 9. Use the results of Eercises 89 9 to eplain how to graph (a) the inverse cotangent function, (b) the inverse secant function, and (c) the inverse cosecant function on a graphing utilit. In Eercises 9 9, use the results of Eercises 89 9 to evaluate the epression without using a calculator. 9. arcsec 9. arcsec 95. arccot 9. arccsc Proof In Eercises 97 99, prove the identit. 97. arcsin arcsin 98. arctan arctan 99. arcsin arccos 00. 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 (see figure). Find the area for each value of a and b. (a) a 0, b (b) a, b (c) a 0, b (d) a, b Skills Review a b In Eercises 0 0, simplif the radical epression In Eercises 05 08, 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. 05. sin 5 0. tan 07. sin 08. sec = +

75 Section.8 Applications and Models.8 Applications and Models 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.77 for all unknown sides and angles. B c a 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 You can use trigonometric functions to model and solve real-life problems. For instance, Eercise 0 on page shows ou how a trigonometric function can be used to model the harmonic motion of a buo. A. b = 9. C Figure.77 Solution 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 9... cos. Now tr Eercise. a b tan A. c b cos A. Mar Kate Denn/PhotoEdit Recall from Section. that the term angle of elevation denotes the angle from the horizontal upward to an object and that the term angle of depression denotes the angle from the horizontal downward to an object. An angle of elevation and an angle of depression are shown in Figure.78. Observer Object Angle of elevation Horizontal Observer Horizontal Angle of depression Object Figure.78

76 Chapter Trigonometric Functions Eample Finding a Side of a Right Triangle 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? B Solution A sketch is shown in Figure.79. From the equation sin A ac, it follows that c = 0 ft a a c sin A 0 sin So, the maimum safe rescue height is about 0. feet above the height of the fire truck. A 7 C Now tr Eercise 7. b Figure.79 Eample Finding a Side of a Right Triangle At a point 00 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 5, and the angle of elevation to the top is 5, as shown in Figure.80. Find the height s of the smokestack alone. Solution This problem involves two right triangles. For the smaller right triangle, use the fact that tan 5 a00 to conclude that the height of the building is s a 00 tan 5. Now, for the larger right triangle, use the equation tan 5 a s 00 a 00 ft 5 5 to conclude that s 00 tan 5 a. So, the height of the smokestack is Figure.80 s 00 tan 5 a 00 tan 5 00 tan feet. Now tr Eercise. Eample Finding an Angle of Depression A swimming pool is 0 meters long and meters wide. The bottom of the pool is slanted such 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. Solution Using the tangent function, ou see that tan A opp adj So, the angle of depression is A arctan radian 7.9. Now tr Eercise 7. A Figure.8 0 m Angle of depression. m.7 m

77 Trigonometr and Bearings In surveing and navigation, directions are generall given in terms of bearings. A bearing measures the acute angle a path or line of sight makes with a fied north south line, as shown in Figure.8. For instance, the bearing of S 5 E in Figure.8(a) means 5 degrees east of south. Section.8 Applications and Models N 80 N N 5 W E W E W E S 5 S 5 E (a) (b) (c) Figure.8 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. D b C Figure.8 W E c 0 nm S 5 B A d 0 nm = (0 nm) N Not drawn to scale STUDY TIP In air navigation, bearings are measured in degrees clockwise from north. Eamples of air navigation bearings are shown below. 70 W 0 N 0 E 90 Solution For triangle BCD, ou have B The two sides of this triangle can be determined to be b 0 sin and d 0 cos. S 80 0 N In triangle ACD, ou can find angle A as follows. tan A b d 0 0 sin cos 0 70 W 5 E 90 A arctan radian.8 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 S 80 c b 0 sin 57.9 sin A sin.8 nautical miles. Distance from port Now tr Eercise.

78 Chapter Trigonometric Functions 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 Maimum negative Maimum positive displacement displacement Figure.8 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.

79 Section.8 Applications and Models 5 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 The motion has amplitude period, and frequenc. a, > 0. Eample Simple Harmonic Motion Write the equation for the simple harmonic motion of the ball illustrated in Figure.8, where the period is seconds. What is the frequenc of this motion? Solution 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 the following. Amplitude a 0 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 ccle per second. Figure.85 Now tr Eercise 5. One illustration of the relationship between sine waves and harmonic motion is the wave motion that results 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.8. Figure.8