292 Chapter 4 Trigonometry
4.2 TRIGONOMETRIC FUNCTIONS : T HE UNIT CIRCLE
What you should learn The Unit Circle • Identify a unit circle and describe its relationship to real numbers. The two historical perspectives of trigonometry incorporate different methods for • Evaluate trigonometric functions introducing the trigonometric functions. Our first introduction to these functions is using the unit circle. based on the unit circle. • Use the domain and period to Consider the unit circle given by evaluate sine and cosine functions. x2 ϩ y 2 ϭ 1 Unit circle • Use a calculator to evaluate trigonometric functions. as shown in Figure 4.20. Why you should learn it y Trigonometric functions are used to (0, 1) model the movement of an oscillating weight. For instance, in Exercise 60 on page 298, the displacement from equilibrium of an oscillating weight x suspended by a spring is modeled as (− 1, 0) (1, 0) a function of time.
(0,− 1)
FIGURE 4.20
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 4.21.
y y
t > 0 (x , y ) t t < 0 t
θ (1, 0)
Richard Richard Megna/Fundamental Photographs x x (1, 0) θ
t (x , y ) t
FIGURE 4.21
As the real number line is wrapped around the unit circle, each real number t corresponds to a point ͑x, y͒ on the circle. For example, the real number 0 corresponds to the point ͑1, 0 ͒. Moreover, because the unit circle has a circumference of 2, the real number 2 also corresponds to the point ͑1, 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 ϭ r (with r ϭ 1 ) indicates that the real number t is the (directional) length of the arc intercepted by the angle , given in radians. Section 4.2 Trigonometric Functions: The Unit Circle 293
The Trigonometric Functions From the preceding discussion, it follows that the coordinates x and y are two functions of the real variable t. You can use these coordinates to define the six trigonometric functions of t. sine cosecant cosine secant tangent cotangent These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively.
Definitions of Trigonometric Functions Let t be a real number and let ͑x, y͒ be the point on the unit circle corresponding Note in the definition at the to t. right that the functions in the second row are the reciprocals y sin t ϭ y cos t ϭ x tan t ϭ , x 0 of the corresponding functions x in the first row. 1 1 x csc t ϭ , y 0 sec t ϭ , x 0 cot t ϭ , y 0 y x y
In the definitions of the trigonometric functions, note that the tangent and secant are not defined when x ϭ 0. For instance, because t ϭ ͞2 corresponds to y ͑x, y͒ ϭ ͑0, 1 ͒, it follows that tan ͑͞2͒ and sec ͑͞2͒ are undefined . Similarly, the (0, 1) cotangent and cosecant are not defined when y ϭ 0. For instance, because t ϭ 0 2 2 2 , 2 corresponds to ͑x, y͒ ϭ ͑1, 0 ͒, cot 0 and csc 0 are undefined . (− , ) ( 2 2 ) 2 2 In Figure 4.22, the unit circle has been divided into eight equal arcs, corresponding to t -values of
x 3 5 3 7 (− 1, 0) (1, 0) 0, , , , , , , , and 2. 4 2 4 4 2 4 Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs, 2 2 corresponding to t -values of − , − 2, − 2 (2 2 ) − ( 2 2 ) (0, 1) 2 5 7 4 3 5 11 0, , , , , , , , , , , , and 2. FIGURE 4.22 6 3 2 3 6 6 3 2 3 6 Ί2 Ί2 To verify the points on the unit circle in Figure 4.22, note that , also lies 2 2 y 1 , 3 (2 2 ) on the line y ϭ x. So, substituting x for y in the equation of the unit circle produces − 1 , 3 (0, 1) (2 2 ) the following. 3 , 1 − 3 , 1 (2 2 ) 1 Ί2 (2 2 ) x2 ϩ x2 ϭ 1 2x2 ϭ 1 x2 ϭ x ϭ ± 2 2
x Ί2 (− 1, 0) (1, 0) Because the point is in the first quadrant,x ϭ and because y ϭ x, you also 2 Ί2 have y ϭ . You can use similar reasoning to verify the rest of the points in (− 3 , − 1 ) 2 2 2 Figure 4.22 and the points in Figure 4.23. 1 , − 3 (2 2 ) Using the ͑x, y͒ coordinates in Figures 4.22 and 4.23, you can evaluate the trigono- − 1 , − 3 (0,− 1) metric functions for common t -values. This procedure is demonstrated in Examples 1, (2 2 ) 3 , − 1 (2 2 ) 2, and 3. You should study and learn these exact function values for common t -values FIGURE 4.23 because they will help you in later sections to perform calculations. 294 Chapter 4 Trigonometry
Example 1 Evaluating Trigonometric Functions
You can review dividing Evaluate the six trigonometric functions at each real number. fractions and rationalizing 5 a.t ϭ b.t ϭ c.t ϭ 0 d. t ϭ denominators in Appendix A.1 6 4 and Appendix A.2, respectively. Solution For each t -value, begin by finding the corresponding point ͑x, y͒ on the unit circle. Then use the definitions of trigonometric functions listed on page 293. Ί3 1 a. t ϭ corresponds to the point ͑x, y͒ ϭ , . 6 2 2 1 1 1 sin ϭ y ϭ csc ϭ ϭ ϭ 2 6 2 6 y 1͞2 Ί3 1 2 2Ί3 cos ϭ x ϭ sec ϭ ϭ ϭ 6 2 6 x Ί3 3 y 1͞2 1 Ί3 x Ί3͞2 tan ϭ ϭ ϭ ϭ cot ϭ ϭ ϭ Ί3 6 x Ί3͞2 Ί3 3 6 y 1͞2 5 Ί2 Ί2 b. t ϭ corresponds to the point ͑x, y͒ ϭ Ϫ , Ϫ . 4 2 2 5 Ί2 5 1 2 sin ϭ y ϭ Ϫ csc ϭ ϭ Ϫ ϭ Ϫ Ί2 4 2 4 y Ί2 5 Ί2 5 1 2 cos ϭ x ϭ Ϫ sec ϭ ϭ Ϫ ϭ Ϫ Ί2 4 2 4 x Ί2 5 y ϪΊ2͞2 5 x ϪΊ2͞2 tan ϭ ϭ ϭ 1 cot ϭ ϭ ϭ 1 4 x ϪΊ2͞2 4 y ϪΊ2͞2 c. t ϭ 0 corresponds to the point ͑x, y͒ ϭ ͑1, 0 ͒. 1 sin 0 ϭ y ϭ 0 csc 0 ϭ is undefined. y 1 1 cos 0 ϭ x ϭ 1 sec 0 ϭ ϭ ϭ 1 x 1 y 0 x tan 0 ϭ ϭ ϭ 0 cot 0 ϭ is undefined. x 1 y d. t ϭ corresponds to the point ͑x, y͒ ϭ ͑Ϫ1, 0 ͒. 1 sin ϭ y ϭ 0 csc ϭ is undefined. y 1 1 cos ϭ x ϭ Ϫ 1 sec ϭ ϭ ϭ Ϫ 1 x Ϫ1 y 0 x tan ϭ ϭ ϭ 0 cot ϭ is undefined. x Ϫ1 y Now try Exercise 23. Section 4.2 Trigonometric Functions: The Unit Circle 295
Example 2 Evaluating Trigonometric Functions
Evaluate the six trigonometric functions at t ϭ Ϫ . 3 Solution Moving clockwise around the unit circle, it follows that t ϭ Ϫ ͞3 corresponds to the point ͑x, y͒ ϭ ͑1͞2, ϪΊ3͞2͒. Ί3 2 2Ί3 sin Ϫ ϭ Ϫ csc Ϫ ϭ Ϫ ϭ Ϫ 3 2 3 Ί3 3 1 cos Ϫ ϭ sec Ϫ ϭ 2 3 2 3 ϪΊ3͞2 1͞2 1 Ί3 tan Ϫ ϭ ϭ Ϫ Ί3 cot Ϫ ϭ ϭ Ϫ ϭ Ϫ 3 1͞2 3 ϪΊ3͞2 Ί3 3
Now try Exercise 33.
Domain and Period of Sine and Cosine y The domain of the sine and cosine functions is the set of all real numbers. To determine (0, 1) the range of these two functions, consider the unit circle shown in Figure 4.24. By definition,sin t ϭ y and cos t ϭ x. Because ͑x, y͒ is on the unit circle, you know that Ϫ1 y 1 and Ϫ1 x 1. So, the values of sine and cosine also range between (−1, 0) (1, 0) Յ Յ Յ Յ x −1 ≤ y ≤ 1 Ϫ1 and 1. Ϫ1 Յ y Յ 1 Ϫ1 Յ x Յ 1 and (0, −1) Ϫ1 Յ sin t Յ 1 Ϫ1 Յ cos t Յ 1 Adding 2 to each value of t in the interval ͓0, 2 ͔ completes a second revolution −1 ≤ x ≤ 1 around the unit circle, as shown in Figure 4.25. The values of sin ͑t ϩ 2͒ and FIGURE 4.24 cos ͑t ϩ 2͒ correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result sin ͑t ϩ 2n͒ ϭ sin t and π π π t = , + 2 π, + 4π , ... 2 2 y 2 cos ͑t ϩ 2 n͒ ϭ cos t π π π π t = 3 3 π t π 4 , 4 + 2 , ... = 4 , 4 + 2 , ... for any integer n and real number t. Functions that behave in such a repetitive (or cyclic) manner are called periodic. t = π , 3 π , ... x Definition of Periodic Function t = 0, 2π , ... A function f is periodic if there exists a positive real number c such that π π t = 5 5 π f t ϩ c ϭ f t 4 , 4 + 2 , ... π π ͑ ͒ ͑ ͒ t 7 7 π = 4, 4 + 2 , ... π π π for all t in the domain of f. The smallest number c for which f is periodic is t = 3, 3+ 2π , 3 + 4 π , ... 2 2 2 called the period of f. FIGURE 4.25 296 Chapter 4 Trigonometry
Recall from Section 1.5 that a function f is even if f ͑Ϫt͒ ϭ f ͑t͒, and is odd if f ͑Ϫt͒ ϭ Ϫ f ͑t͒.
Even and Odd Trigonometric Functions The cosine and secant functions are even . cos ͑Ϫt͒ ϭ cos t sec ͑Ϫt͒ ϭ sec t The sine, cosecant, tangent, and cotangent functions are odd . sin ͑Ϫt͒ ϭ Ϫ sin t csc ͑Ϫt͒ ϭ Ϫ csc t tan ͑Ϫt͒ ϭ Ϫ tan t cot ͑Ϫt͒ ϭ Ϫ cot t
Example 3 Using the Period to Evaluate the Sine and Cosine
13 13 1 a. Because ϭ 2 ϩ , you have sin ϭ sin 2 ϩ ϭ sin ϭ . 6 6 6 6 6 2 7 From the definition of periodic b. Because Ϫ ϭ Ϫ 4 ϩ , you have 2 2 function, it follows that the sine and cosine functions are 7 cos Ϫ ϭ cos Ϫ4 ϩ ϭ cos ϭ 0. periodic and have a period of 2 2 2 2. The other four trigonometric 4 4 functions are also periodic, and c. For sin t ϭ , sin ͑Ϫt͒ ϭ Ϫ because the sine function is odd. will be discussed further in 5 5 Section 4.6. Now try Exercise 37.
Evaluating Trigonometric Functions with a Calculator When evaluating a trigonometric function with a calculator, you need to set the TECHNOLOGY calculator to the desired mode of measurement ( degree or radian ). Most calculators do not have keys for the cosecant, secant, and cotangent When evaluating trigonometric functions. To evaluate these functions, you can use the x Ϫ1 key with their respective functions with a calculator, reciprocal functions sine, cosine, and tangent. For instance, to evaluate csc ͑͞8͒, use remember to enclose all fractional the fact that angle measures in parentheses. 1 For instance, if you want to csc ϭ 8 sin ͑͞8͒ /6, you ؍ evaluate sin t for t should enter and enter the following keystroke sequence in radian mode.
1 SIN ͑ Ϭ 6͒ ENTER . ͑ SIN ͑ Ϭ 8 ͒ ͒ x Ϫ ENTER Display 2.6131259 These keystrokes yield the correct value of 0.5. Note that some Example 4 Using a Calculator calculators automatically place a left parenthesis after trigonometric Function Mode Calculator Keystrokes Display functions. Check the user’s guide 2 a. sin RadianSIN ͑ 2 Ϭ 3͒ ENTER 0.8660254 for your calculator for specific 3 keystrokes on how to evaluate b. cot 1.5 Radian͑ TAN ͑ 1.5͒ ͒ x Ϫ1 ENTER 0.0709148 trigonometric functions. Now try Exercise 55. Section 4.2 Trigonometric Functions: The Unit Circle 297
4.2 EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. Each real number t corresponds to a point ͑x, y͒ on the ______. 2. A function f is ______if there exists a positive real number c such that f ͑t ϩ c͒ ϭ f ͑t͒ for all t in the domain of f. 3. The smallest number c for which a function f is periodic is called the ______of f. 4. A function f is ______if f ͑Ϫt͒ ϭ Ϫ f ͑t͒ and ______if f ͑Ϫt͒ ϭ f ͑t͒. SKILLS AND APPLICATIONS
In Exercises 5–8, determine the exact values of the six 11 5 23.t ϭ 24. t ϭ trigonometric functions of the real number t. 6 3 5.y 6. y 3 − 8, 15 25.t ϭ Ϫ 26. t ϭ Ϫ 2 12, 5 ( 17 17 ( 2 (13 13 ( t θ θ t In Exercises 27–34, evaluate (if possible) the six trigonometric x x functions of the real number. 2 5 27.t ϭ 28. t ϭ 3 6 4 7 7.y 8. y 29.t ϭ 30. t ϭ t 3 4 3 3 θ t θ 31.t ϭ 32. t ϭ 4 2 x x 33.t ϭ Ϫ 34. t ϭ Ϫ 12, − 5 2 (13 13 ( (− 4 , − 3 ( 5 5 In Exercises 35– 42, evaluate the trigonometric function using its period as an aid. In Exercises 9–16, find the point ͑x, y͒ on the unit circle that corresponds to the real number t. 35.sin 4 36. cos 3 7 9 37.cos 38. sin 9.t ϭ 10. t ϭ 3 4 2 17 19 39.cos 40. sin 11.t ϭ 12. t ϭ 4 6 4 3 8 9 5 3 41.sin Ϫ 42. cos Ϫ 13.t ϭ 14. t ϭ 3 4 6 4 4 5 15.t ϭ 16. t ϭ In Exercises 43–48, use the value of the trigonometric 3 3 function to evaluate the indicated functions. 1 3 In Exercises 17–26, evaluate (if possible) the sine, cosine, and 43.sin t ϭ 2 44. sin ͑Ϫt͒ ϭ 8 tangent of the real number. (a)sin ͑Ϫt͒ (a) sin t (b)csc ͑Ϫt͒ (b) csc t 17.t ϭ 18. t ϭ 1 3 4 3 45.cos ͑Ϫt͒ ϭ Ϫ 5 46. cos t ϭ Ϫ 4 (a)cos t (a) cos ͑Ϫt͒ 19.t ϭ Ϫ 20. t ϭ Ϫ 6 4 (b)sec ͑Ϫt͒ (b) sec ͑Ϫt͒ 4 4 7 4 47.sin t ϭ 5 48. cos t ϭ 5 21.t ϭ Ϫ 22. t ϭ Ϫ 4 3 (a)sin ͑ Ϫ t͒ (a) cos ͑ Ϫ t͒ (b)sin ͑t ϩ ͒ (b) cos ͑t ϩ ͒ 298 Chapter 4 Trigonometry
In Exercises 49–58, use a calculator to evaluate the trigono- (b) Make a conjecture about any relationship between metric function. Round your answer to four decimal places. sin t1 and sin ͑ Ϫ t1͒. (Be sure the calculator is set in the correct angle mode.) (c) Make a conjecture about any relationship between cos t1 and cos ͑ Ϫ t1͒. 49.sin 50. tan 4 3 66. Use the unit circle to verify that the cosine and secant functions are even and that the sine, cosecant, tangent, 2 51.cot 52. csc and cotangent functions are odd. 4 3 67. Verify that cos 2 t 2 cos t by approximating cos 1.5 53.cos ͑Ϫ1.7 ͒ 54. cos ͑Ϫ2.5 ͒ and 2 cos 0.75. 55.csc 0.8 56. sec 1.8 68. Verify that sin ͑t1 ϩ t2͒ sin t1 ϩ sin t2 by approximating 57.sec ͑Ϫ22.8 ͒ 58. cot ͑Ϫ0.9 ͒ sin 0.25, sin 0.75, and sin 1. THINK ABOUT IT 59. HARMONIC MOTION The displacement from 69. Because f͑t͒ ϭ sin t is an odd equilibrium of an oscillating weight suspended by a function and g͑t͒ ϭ cos t is an even function, what can 1 be said about the function h͑t͒ ϭ f͑t͒g͑t͒? spring is given by y͑t͒ ϭ 4 cos 6 t, where y is the displacement (in feet) and t is the time (in seconds). Find 70. THINK ABOUT IT Because f͑t͒ ϭ sin t and 1 1 the displacements when (a) t ϭ 0, (b) t ϭ 4, and (c) t ϭ 2. g͑t͒ ϭ tan t are odd functions, what can be said about 60. HARMONIC MOTION The displacement from the function h͑t͒ ϭ f͑t͒g͑t͒? equilibrium of an oscillating weight suspended by a 71. GRAPHICAL ANALYSIS With your graphing utility in spring and subject to the damping effect of friction is radian and parametric modes, enter the equations given by y͑t͒ ϭ 1eϪt cos 6 t, where y is the displacement 4 X1T ϭ cos T and Y 1T ϭ sin T (in feet) and t is the time (in seconds). (a) Complete the table. and use the following settings. Tminϭ 0, Tmaxϭ 6.3, Tstepϭ 0.1 t 0 1 1 3 1 4 2 4 Xminϭ Ϫ 1.5, Xmaxϭ 1.5, Xsclϭ 1 y Yminϭ Ϫ 1, Ymaxϭ 1, Ysclϭ 1 (a) Graph the entered equations and describe the graph. (b) Use the table feature of a graphing utility to approxi- (b) Use the trace feature to move the cursor around the mate the time when the weight reaches equilibrium. graph. What do the t -values represent? What do the (c) What appears to happen to the displacement as t x- and y- values represent? increases? (c) What are the least and greatest values of x and y? EXPLORATION 72. CAPSTONE A student you are tutoring has used a TRUE OR FALSE? In Exercises 61– 64, determine whether unit circle divided into 8 equal parts to complete the the statement is true or false. Justify your answer. table for selected values of t. What is wrong? 61. Because sin Ϫt ϭ Ϫ sin t, it can be said that the sine of ͑ ͒ 3 a negative angle is a negative number. t 0 4 2 4 62. tan a ϭ tan ͑a Ϫ 6͒ Ί2 Ί2 63. The real number 0 corresponds to the point ͑0, 1 ͒ on the x 1 0 Ϫ Ϫ1 unit circle. 2 2 7 y 0 Ί2 1 Ί2 0 64. cos Ϫ ϭ cos ϩ 2 2 2 2 Ί2 Ί2 65. Let x , y and x , y be points on the unit circle sin t 1 0 Ϫ Ϫ1 ͑ 1 1͒ ͑ 2 2͒ 2 2 corresponding to t ϭ t1 and t ϭ Ϫ t1, respectively. (a) Identify the symmetry of the points ͑x , y ͒ and cos t 0 Ί2 1 Ί2 0 1 1 2 2 ͑x2, y2͒. tan t Undef. 1 0 Ϫ1 Undef. Section 4.3 Right Triangle Trigonometry 299
4.3 RIGHT TRIANGLE TRIGONOMETRY
What you should learn The Six Trigonometric Functions • Evaluate trigonometric functions of acute angles. Our second look at the trigonometric functions is from a right triangle perspective. • Use fundamental trigonometric Consider a right triangle, with one acute angle labeled , as shown in Figure 4.26. identities. Relative to the angle , the three sides of the triangle are the hypotenuse, the • Use a calculator to evaluate opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to trigonometric functions. the angle ). • Use trigonometric functions to model and solve real-life problems.
Why you should learn it θ Trigonometric functions are often used to analyze real-life situations. For
instance, in Exercise 76 on page 309, Hypotenuse you can use trigonometric functions
Side opposite to find the height of a helium-filled balloon. θ Side adjacent to θ
FIGURE 4.26
Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle . sine cosecant cosine secant tangent cotangent In the following definitions, it is important to see that 0Њ < < 90 Њ ͑ 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. The six 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.)
Joseph Joseph Sohm/Visions of America/Corbis opp adj opp sin ϭ cos ϭ tan ϭ hyp hyp adj hyp hyp adj csc ϭ sec ϭ cot ϭ opp adj opp The abbreviations opp, adj, and hyp represent the lengths of the three sides of a right triangle. opp ϭ the length of the side opposite adj ϭ the length of the side adjacent to hyp ϭ the length of the hypotenuse 300 Chapter 4 Trigonometry
Example 1 Evaluating Trigonometric Functions
Use the triangle in Figure 4.27 to find the values of the six trigonometric functions of . se u 4 Solution 2 2 2 Hypoten By the Pythagorean Theorem,͑hyp ͒ ϭ ͑opp ͒ ϩ ͑adj ͒ , it follows that hyp ϭ Ί42 ϩ 32 θ ϭ Ί25 3 ϭ 5. FIGURE 4.27 So, the six trigonometric functions of are opp 4 hyp 5 sin ϭ ϭ csc ϭ ϭ hyp 5 opp 4 You can review the Pythagorean adj 3 hyp 5 cos ϭ ϭ sec ϭ ϭ Theorem in Section 1.1. hyp 5 adj 3 opp 4 adj 3 tan ϭ ϭ cot ϭ ϭ . adj 3 opp 4 HISTORICAL NOTE Now try Exercise 7. Georg Joachim Rhaeticus In Example 1, you were given the lengths of two sides of the right triangle, but not (1514–1574) was the leading the angle . Often, you will be asked to find the trigonometric functions of a given acute Teutonic mathematical angle . To do this, construct a right triangle having as one of its angles. astronomer of the 16th century. He was the first to define the trigonometric functions as ratios Example 2 Evaluating Trigonometric Functions of 45 ؇ of the sides of a right triangle. Find the values of sin 45 Њ, cos 45 Њ, and tan 45 Њ. Solution Construct a right triangle having 45 Њ as one of its acute angles, as shown in Figure 4.28. Choose the length of the adjacent side to be 1. From geometry, you know that the other 45 ° acute angle is also 45 Њ. So, the triangle is isosceles and the length of the opposite side is also 1. Using the Pythagorean Theorem, you find the length of the hypotenuse to 2 1 be Ί2. opp 1 Ί2 sin 45 Њ ϭ ϭ ϭ hyp Ί2 2 45 ° adj 1 Ί2 1 cos 45 Њ ϭ ϭ ϭ hyp Ί2 2 FIGURE 4.28 opp 1 tan 45 Њ ϭ ϭ ϭ 1 adj 1 Now try Exercise 23. Section 4.3 Right Triangle Trigonometry 301
Example 3 Evaluating Trigonometric Functions of 30؇ and 60 ؇
Because the angles 30 Њ, 45 Њ, and Use the equilateral triangle shown in Figure 4.29 to find the values of sin 60 Њ, 60 Њ ͑͞6, ͞4, and ͞3͒ occur cos 60 Њ, sin 30 Њ, and cos 30 Њ. frequently in trigonometry, you should learn to construct the triangles shown in Figures 4.28 and 4.29. 30 ° 2 3 2
60 ° 1 1 FIGURE 4.29 Solution Use the Pythagorean Theorem and the equilateral triangle in Figure 4.29 to verify the lengths of the sides shown in the figure. For ϭ 60 Њ, you have adj ϭ 1, opp ϭ Ί3, and TECHNOLOGY hyp ϭ 2. So, opp Ί3 adj 1 sin 60 Њ ϭ ϭ and cos 60 Њ ϭ ϭ . You can use a calculator to hyp 2 hyp 2 convert the answers in Example 3 to decimals. However, the For ϭ 30 Њ, adj ϭ Ί3, opp ϭ 1, and hyp ϭ 2. So, radical form is the exact value opp 1 adj Ί3 sin 30 Њ ϭ ϭ and cos 30 Њ ϭ ϭ . and in most cases, the exact hyp 2 hyp 2 value is preferred. Now try Exercise 27.
Sines, Cosines, and Tangents of Special Angles 1 Ί3 Ί3 sin 30 Њ ϭ sin ϭ cos 30 Њ ϭ cos ϭ tan 30 Њ ϭ tan ϭ 6 2 6 2 6 3 Ί2 Ί2 sin 45 Њ ϭ sin ϭ cos 45 Њ ϭ cos ϭ tan 45 Њ ϭ tan ϭ 1 4 2 4 2 4 Ί3 1 sin 60 Њ ϭ sin ϭ cos 60 Њ ϭ cos ϭ tan 60 Њ ϭ tan ϭ Ί3 3 2 3 2 3
1 In the box, note that sin 30 Њ ϭ 2 ϭ cos 60 Њ. This occurs because 30 Њ and 60 Њ are complementary angles. In general, it can be shown from the right triangle definitions that cofunctions of complementary angles are equal . That is, if is an acute angle, the following relationships are true. sin ͑90 Њ Ϫ ͒ ϭ cos cos ͑90 Њ Ϫ ͒ ϭ sin tan ͑90 Њ Ϫ ͒ ϭ cot cot ͑90 Њ Ϫ ͒ ϭ tan sec ͑90 Њ Ϫ ͒ ϭ csc csc ͑90 Њ Ϫ ͒ ϭ sec 302 Chapter 4 Trigonometry
Trigonometric Identities In trigonometry, a great deal of time is spent studying relationships between trigono- metric functions (identities).
Fundamental Trigonometric Identities Reciprocal Identities 1 1 1 sin ϭ cos ϭ tan ϭ csc sec cot 1 1 1 csc ϭ sec ϭ cot ϭ sin cos tan Quotient Identities sin cos tan ϭ cot ϭ cos sin Pythagorean Identities sin 2 ϩ cos 2 ϭ 1 1 ϩ tan 2 ϭ sec 2 1 ϩ cot 2 ϭ csc 2
Note that sin 2 represents ͑sin ͒2, cos 2 represents ͑cos ͒2, and so on.
Example 4 Applying Trigonometric Identities
Let be an acute angle such that sin ϭ 0.6. Find the values of (a) cos and (b) tan using trigonometric identities. Solution a. To find the value of cos , use the Pythagorean identity sin 2 ϩ cos 2 ϭ 1. So, you have
͑0.6 ͒2 ϩ cos 2 ϭ 1 Substitute 0.6 for sin .
cos 2 ϭ 1 Ϫ ͑0.6 ͒ 2 ϭ 0.64 Subtract ͑0.6 ͒2 from each side.
cos ϭ Ί0.64 ϭ 0.8. Extract the positive square root. b. Now, knowing the sine and cosine of , you can find the tangent of to be sin tan ϭ cos 0.6 1 ϭ 0.6 0.8 ϭ 0.75. θ Use the definitions of cos and tan , and the triangle shown in Figure 4.30, to check 0.8 these results.
FIGURE 4.30 Now try Exercise 33. Section 4.3 Right Triangle Trigonometry 303
Example 5 Applying Trigonometric Identities
Let be an acute angle such that tan ϭ 3. Find the values of (a) cot and (b) sec using trigonometric identities. Solution 1 a. cot ϭ Reciprocal identity tan 1 cot ϭ 3
b. sec 2 ϭ 1 ϩ tan 2 Pythagorean identity 10 3 sec 2 ϭ 1 ϩ 32 sec 2 ϭ 10 sec ϭ Ί10 θ Use the definitions of cot and sec , and the triangle shown in Figure 4.31, to check these results. 1 FIGURE 4.31 Now try Exercise 35.
Evaluating Trigonometric Functions with a Calculator To use a calculator to evaluate trigonometric functions of angles measured in degrees, You can also use the reciprocal first set the calculator to degree mode and then proceed as demonstrated in Section 4.2. identities for sine, cosine, and For instance, you can find values of cos 28 Њ and sec 28 Њ as follows. tangent to evaluate the cosecant, secant, and cotangent functions Function Mode Calculator Keystrokes Display with a calculator. For instance, a. cos 28 Њ DegreeCOS 28ENTER 0.8829476 you could use the following b. sec 28 Њ Degree͑ COS ͑ 28͒ ͒ x Ϫ1 ENTER 1.1325701 keystroke sequence to evaluate sec 28 Њ. Throughout this text, angles are assumed to be measured in radians unless noted otherwise. For example, sin 1 means the sine of 1 radian and sin 1 Њ means the sine of 1Ϭ COS 28 ENTER 1 degree. The calculator should display 1.1325701. Example 6 Using a Calculator
Use a calculator to evaluate sec ͑5Њ 40 Ј 12 Љ͒. Solution 1 1 Begin by converting to decimal degree form. [Recall that 1Ј ϭ 60 ͑1Њ͒ and 1Љ ϭ 3600 ͑1Њ͔͒ . 40 Њ 12 Њ 5Њ 40 Ј 12 Љ ϭ 5Њ ϩ ϩ ϭ 5.67 Њ 60 3600 Then, use a calculator to evaluate sec 5.67 Њ. Function Calculator Keystrokes Display
sec ͑5Њ 40 Ј 12 Љ ͒ ϭ sec 5.67 Њ ͑ COS ͑ 5.67͒ ͒ x Ϫ1 ENTER 1.0049166 Now try Exercise 51. 304 Chapter 4 Trigonometry
Object Applications Involving Right Triangles Many applications of trigonometry involve a process called solving right triangles. In An gle of this type of application, you are usually given one side of a right triangle and one of the elevation Observer acute angles and are asked to find one of the other sides, or you are given two sides and Horizontal are asked to find one of the acute angles. In Example 7, the angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the Horizontal Observer horizontal, it is common to use the term angle of depression, as shown in Figure 4.32. An gle of depression Example 7 Using Trigonometry to Solve a Right Triangle Object A surveyor is standing 115 feet from the base of the Washington Monument, as shown FIGURE 4.32 in Figure 4.33. The surveyor measures the angle of elevation to the top of the monument as 78.3 Њ. How tall is the Washington Monument? Solution From Figure 4.33, you can see that y opp y An gle of tan 78.3 Њ ϭ ϭ elevation adj x 78.3 ° where x ϭ 115 and y is the height of the monument. So, the height of the Washington Monument is y ϭ x tan 78.3 Њ Ϸ 115 ͑4.82882 ͒ Ϸ 555 feet. x = 115 ft Not drawn to scale Now try Exercise 67. FIGURE 4.33
Example 8 Using Trigonometry to Solve a Right Triangle
A historic lighthouse is 200 yards from a bike path along the edge of a lake. A walk- way to the lighthouse is 400 yards long. Find the acute angle between the bike path and the walkway, as illustrated in Figure 4.34.
θ
200 yd 400 yd
FIGURE 4.34 Solution From Figure 4.34, you can see that the sine of the angle is opp 200 1 sin ϭ ϭ ϭ . hyp 400 2 Now you should recognize that ϭ 30 Њ. Now try Exercise 69. Section 4.3 Right Triangle Trigonometry 305
By now you are able to recognize that ϭ 30 Њ is the acute angle that satisfies the 1 equation sin ϭ 2. Suppose, however, that you were given the equation sin ϭ 0.6 and were asked to find the acute angle . Because 1 sin 30 Њ ϭ 2 ϭ 0.5000 and 1 sin 45 Њ ϭ Ί2 Ϸ 0.7071 you might guess that lies somewhere between 30 Њ and 45 Њ. In a later section, you will study a method by which a more precise value of can be determined.
Example 9 Solving a Right Triangle
Find the length c of the skateboard ramp shown in Figure 4.35.
c 4 ft 18.4 °
FIGURE 4.35 Solution From Figure 4.35, you can see that opp sin 18.4 Њ ϭ hyp 4 ϭ . c So, the length of the skateboard ramp is 4 c ϭ sin 18.4 Њ 4 Ϸ 0.3156 Ϸ 12.7 feet. Now try Exercise 71. 306 Chapter 4 Trigonometry
4.3 EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY 1. Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent hypotenuse adjacent hypotenuse adjacent opposite opposite (i) (ii) (iii) (iv) (v) (vi) adjacent opposite opposite hypotenuse hypotenuse adjacent
In Exercises 2–4, fill in the blanks. 2. Relative to the angle , the three sides of a right triangle are the ______side, the ______side, and the ______. 3. Cofunctions of ______angles are equal. 4. 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 ______.
SKILLS AND APPLICATIONS
In Exercises 5–8, find the exact values of the six trigono- In Exercises 13–20, sketch a right triangle corresponding to metric functions of the angle shown in the figure. (Use the the trigonometric function of the acute angle . Use the Pythagorean Theorem to find the third side of the triangle.) Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of . 5. 6. 13 3 5 5 13.tan ϭ 4 14. cos ϭ 6 6 θ 3 4 15.sec ϭ 2 16. tan ϭ 5 θ 1 17 8 17.sin ϭ 5 18. sec ϭ 7 19.cot ϭ 3 20. csc ϭ 9 7.41 8. θ 9 In Exercises 21–30, construct an appropriate triangle to complete the table. ͧ0؇ Յ Յ 90 ؇, 0 Յ Յ /2ͨ 4 θ Function (deg) (rad) Function Value 4 21. sin 30 Њ 22. cos 45 Њ In Exercises 9–12, find the exact values of the six trigonometric functions of the angle for each of the two 23. sec triangles. Explain why the function values are the same. 4 9. 10. 1.25 24. tan 3 8 θ θ 1 Ί3 5 25. cot 15 3 26. csc Ί2 θ 4 θ 7.5 4 27. csc 6 11.3 12. θ 1 1 28. sin θ 2 4 6 29. cot 1 2 3 Ί3 θ θ 30. tan 6 3 Section 4.3 Right Triangle Trigonometry 307
In Exercises 31–36, use the given function value(s), and 48. (a)tan 23.5 Њ (b) cot 66.5 Њ trigonometric identities (including the cofunction identities), 49. (a)sin 16.35 Њ (b) csc 16.35 Њ to find the indicated trigonometric functions. 50. (a)cot 79.56 Њ (b) sec 79.56 Њ Ί3 1 31. sin 60 Њ ϭ , cos 60 Њ ϭ 51. (a)cos 4 Њ 50 Ј 15 Љ (b) sec 4 Њ 50 Ј 15 Љ 2 2 52. (a)sec 42 Њ 12 Ј (b) csc 48 Њ 7 Ј (a)sin 30 Њ (b) cos 30 Њ 53. (a)cot 11 Њ 15 Ј (b) tan 11 Њ 15 Ј (c)tan 60 Њ (d) cot 60 Њ 54. (a)sec 56 Њ 8 Ј 10 Љ (b) cos 56 Њ 8 Ј 10 Љ 1 Ί3 32. sin 30 Њ ϭ , tan 30 Њ ϭ 55. (a)csc 32 Њ 40 Ј 3Љ (b) tan 44 Њ 28 Ј 16 Љ 2 3 9 9 56. (a)sec ͑5 и 20 ϩ 32 ͒Њ (b) cot ͑5 и 30 ϩ 32 ͒Њ (a)csc 30 Њ (b) cot 60 Њ (c)cos 30 Њ (d) cot 30 Њ In Exercises 57–62, find the values of in degrees 0؇ < < 90 ؇ͨ and radians ͧ0 < < /2ͨ without the aidͧ 1 33. cos ϭ 3 of a calculator. (a)sin (b) tan 1 (c)sec (d) csc ͑90 Њ Ϫ ͒ 57. (a)sin ϭ 2 (b) csc ϭ 2 Ί2 34. sec ϭ 5 58. (a)cos ϭ (b) tan ϭ 1 (a)cos (b) cot 2 (c)cot ͑90 Њ Ϫ ͒ (d) sin 59. (a)sec ϭ 2 (b) cot ϭ 1 1 35. cot ␣ ϭ 5 60. (a)tan ϭ Ί3 (b) cos ϭ 2 2Ί3 Ί2 (a)tan ␣ (b) csc ␣ 61. (a)csc ϭ (b) sin ϭ 3 2 (c)cot ͑90 Њ Ϫ ␣͒ (d) cos ␣ Ί3 Ί7 62. (a)cot ϭ (b) sec ϭ Ί2 36. cos  ϭ 3 4 (a)sec  (b) sin  In Exercises 63–66, solve for x, y , or r as indicated. (c)cot  (d) sin ͑90 Њ Ϫ ͒ 63. Solve for y. 64. Solve for x. In Exercises 37– 46, use trigonometric identities to transform the left side of the equation into the right side ͧ0 < < /2ͨ. 30 18 y 30 ° 37. tan cot ϭ 1 x 38. cos sec ϭ 1 60 ° 39. tan ␣ cos ␣ ϭ sin ␣ 65. Solve for x. 66. Solve for r. 40. cot ␣ sin ␣ ϭ cos ␣ 41. ͑1 ϩ sin ͒͑ 1 Ϫ sin ͒ ϭ cos 2 42. ͑1 ϩ cos ͒͑ 1 Ϫ cos ͒ ϭ sin 2 32 r 43. ͑sec ϩ tan ͒͑ sec Ϫ tan ͒ ϭ 1 20 44. sin 2 Ϫ cos 2 ϭ 2 sin 2 Ϫ 1 60 ° 45 ° sin cos x 45. ϩ ϭ csc sec cos sin 67. EMPIRE STATE BUILDING You are standing 45 meters tan  ϩ cot  from the base of the Empire State Building. You 46. ϭ csc 2  tan  estimate that the angle of elevation to the top of the 86th floor (the observatory) is 82 Њ. If the total height of the In Exercises 47–56, use a calculator to evaluate each function. building is another 123 meters above the 86th floor, Round your answers to four decimal places. (Be sure the what is the approximate height of the building? One of calculator is in the correct angle mode.) your friends is on the 86th floor. What is the distance between you and your friend? 47. (a)sin 10 Њ (b) cos 80 Њ 308 Chapter 4 Trigonometry
68. HEIGHT A six-foot person walks from the base of a 72. HEIGHT OF A MOUNTAIN In traveling across flat broadcasting tower directly toward the tip of the land, you notice a mountain directly in front of you. Its shadow cast by the tower. When the person is 132 feet angle of elevation (to the peak) is 3.5 Њ. After you drive from the tower and 3 feet from the tip of the shadow, the 13 miles closer to the mountain, the angle of elevation person’s shadow starts to appear beyond the tower’s is 9Њ. Approximate the height of the mountain. shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. 3.5 ° 9° 13 mi (b) Use a trigonometric function to write an equation Not drawn to scale involving the unknown quantity. MACHINE SHOP CALCULATIONS (c) What is the height of the tower? 73. A steel plate has the form of one-fourth of a circle with a radius of 69. ANGLE OF ELEVATION You are skiing down a 60 centimeters. Two two-centimeter holes are to be mountain with a vertical height of 1500 feet. The drilled in the plate positioned as shown in the figure. distance from the top of the mountain to the base is Find the coordinates of the center of each hole. 3000 feet. What is the angle of elevation from the base to the top of the mountain? d 70. WIDTH OF A RIVER A biologist wants to know the y width w of a river so that instruments for studying the 60 3° pollutants in the water can be set properly. From point 56 A, the biologist walks downstream 100 feet and sights to (x2, y2) point C (see figure). From this sighting, it is determined 15 cm that ϭ 54 Њ. How wide is the river? (x1, y1) 30 ° C 30 ° 30 ° 5 cm x 56 60 w FIGURE FOR 73 FIGURE FOR 74 74. MACHINE SHOP CALCULATIONS A tapered shaft θ = 54 ° has a diameter of 5 centimeters at the small end and is A 100 ft 15 centimeters long (see figure). The taper is 3Њ. Find the diameter d of the large end of the shaft. 71. LENGTH A guy wire runs from the ground to a cell 75. GEOMETRY Use a compass to sketch a quarter of a tower. The wire is attached to the cell tower 150 feet circle of radius 10 centimeters. Using a protractor, above the ground. The angle formed between the wire construct an angle of 20 Њ in standard position (see and the ground is 43 Њ (see figure). figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. By actual measurement, calculate the coordinates ͑x, y͒ of the point of intersection and use these measurements to approximate the six trigonometric functions of a 20 Њ angle. y 150 ft 10 (x, y) θ = 43 °
10 cm ° (a) How long is the guy wire? 20 x (b) How far from the base of the tower is the guy wire 10 anchored to the ground? Section 4.3 Right Triangle Trigonometry 309
76. HEIGHT A 20-meter line is used to tether a 84. THINK ABOUT IT helium-filled balloon. Because of a breeze, the line (a) Complete the table. makes an angle of approximately 85 Њ with the ground. (a) Draw a right triangle that gives a visual representa- 0Њ 18 Њ 36 Њ 54 Њ 72 Њ 90 Њ tion of the problem. Show the known quantities of sin the triangle and use a variable to indicate the height of the balloon. cos (b) Use a trigonometric function to write an equation involving the unknown quantity. (b) Discuss the behavior of the sine function for in the (c) What is the height of the balloon? range from 0 Њ to 90 Њ. (d) The breeze becomes stronger and the angle the (c) Discuss the behavior of the cosine function for in balloon makes with the ground decreases. How the range from 0 Њ to 90 Њ. does this affect the triangle you drew in part (a)? (d) Use the definitions of the sine and cosine functions (e) Complete the table, which shows the heights to explain the results of parts (b) and (c). (in meters) of the balloon for decreasing angle 85. WRITING measures . In right triangle trigonometry, explain why Њ ϭ 1 sin 30 2 regardless of the size of the triangle. Angle, 80 Њ 70 Њ 60 Њ 50 Њ 86. GEOMETRY Use the equilateral triangle shown in Figure 4.29 and similar triangles to verify the points in Height Figure 4.23 (in Section 4.2) that do not lie on the axes. 87. THINK ABOUT IT You are given only the value tan . Angle, 40 Њ 30 Њ 20 Њ 10 Њ Is it possible to find the value of sec without finding Height the measure of ? Explain.
(f) As the angle the balloon makes with the ground 88. CAPSTONE The Johnstown Inclined Plane in Њ approaches 0 , how does this affect the height of the Pennsylvania is one of the longest and steepest hoists balloon? Draw a right triangle to explain your in the world. The railway cars travel a distance of reasoning. 896.5 feet at an angle of approximately 35.4 Њ, rising to a height of 1693.5 feet above sea level. EXPLORATION
TRUE OR FALSE? In Exercises 77–82, determine whether the statement is true or false. Justify your answer. 77.sin 60 Њ csc 60 Њ ϭ 1 78. sec 30 Њ ϭ csc 60 Њ 896.5 ft 79.sin 45 Њ ϩ cos 45 Њ ϭ 1 80. cot 2 10 Њ Ϫ csc 2 10 Њ ϭ Ϫ 1 1693.5 feet above sea level sin 60 Њ 81.ϭ sin 2 Њ 82. tan 5Њ 2 ϭ tan 2 5Њ Њ ͓͑ ͒ ͔ 35.4 ° sin 30 Not drawn to scale
83. THINK ABOUT IT (a) Find the vertical rise of the inclined plane. (a) Complete the table. (b) Find the elevation of the lower end of the inclined plane. 0.1 0.2 0.3 0.4 0.5 (c) The cars move up the mountain at a rate of sin 300 feet per minute. Find the rate at which they rise vertically. (b) Is or sin greater for in the interval ͑0, 0.5 ͔? (c) As approaches 0, how do and sin compare? Explain. 310 Chapter 4 Trigonometry
4.4 TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
What you should learn Introduction • Evaluate trigonometric functions of any angle. In Section 4.3, the definitions of trigonometric functions were restricted to acute angles. • Find reference angles. In this section, the definitions are extended to cover any angle. If is an acute angle, these definitions coincide with those given in the preceding section. • Evaluate trigonometric functions of real numbers. Why you should learn it Definitions of Trigonometric Functions of Any Angle Let be an angle in standard position with ͑x, y͒ a point on the terminal side of You can use trigonometric functions and r ϭ Ίx2 ϩ y2 0. to model and solve real-life problems. For instance, in Exercise 99 on page y x y sin ϭ cos ϭ 318, you can use trigonometric r r functions to model the monthly (x , y ) normal temperatures in New York City y x tan ϭ , x 0 cot ϭ , y 0 and Fairbanks, Alaska. x y r r r sec ϭ , x 0 csc ϭ , y 0 θ x y x
Because r ϭ Ίx 2 ϩ y 2 cannot be zero, it follows that the sine and cosine functions are defined for any real value of . However, if x ϭ 0, the tangent and secant of are undefined. For example, the tangent of 90 Њ is undefined. Similarly, if y ϭ 0, the cotangent and cosecant of are undefined.
Example 1 Evaluating Trigonometric Functions
Let ͑Ϫ3, 4 ͒ be a point on the terminal side of . Find the sine, cosine, and tangent of . Solution Referring to Figure 4.36, you can see that x ϭ Ϫ 3, y ϭ 4, and James James Urbach/SuperStock r ϭ Ίx 2 ϩ y 2 ϭ Ί͑Ϫ3͒2 ϩ 42 ϭ Ί25 ϭ 5. So, you have the following. y 4 x 3 y 4 sin ϭ ϭ cos ϭ ϭ Ϫ tan ϭ ϭ Ϫ r 5 r 5 x 3
y
(− 3, 4) 4
3 r 2
1 The formula r ϭ Ίx2 ϩ y2 is θ x a result of the Distance Formula. −3−2 −1 1 You can review the Distance FIGURE 4.36 Formula in Section 1.1. Now try Exercise 9. Section 4.4 Trigonometric Functions of Any Angle 311
y The signs of the trigonometric functions in the four quadrants can be determined π π ϭ < θ < π 0 < θ < from the definitions of the functions. For instance, because cos x͞r, it follows that 2 2 cos is positive wherever x > 0, which is in Quadrants I and IV. (Remember,r is x < 0 x > 0 always positive.) In a similar manner, you can verify the results shown in Figure 4.37. y > 0 y > 0
x Example 2 Evaluating Trigonometric Functions
x < 0 x > 0 Given tan ϭ Ϫ 5 and cos > 0, find sin and sec . y < 0 y < 0 4 π π π < θ < 3 3 < θ < 2π Solution 2 2 Note that lies in Quadrant IV because that is the only quadrant in which the tangent is negative and the cosine is positive. Moreover, using y y tan ϭ Quadrant II Quadrant I x sin θ : + sin θ : + cos θ : − cos θ : + 5 tan θ : − tan θ : + ϭ Ϫ x 4 ϭ Ϫ ϭ Quadrant III Q uadrant IV and the fact that y is negative in Quadrant IV, you can let y 5 and x 4. So, ϭ ϩ ϭ sin θ : − sin θ : − r Ί16 25 Ί41 and you have cos θ : − cos θ : + tan θ : + tan θ : − y Ϫ5 sin ϭ ϭ r Ί41 FIGURE 4.37 Ϸ Ϫ0.7809 r Ί41 sec ϭ ϭ x 4 Ϸ 1.6008. Now try Exercise 23.
Example 3 Trigonometric Functions of Quadrant Angles
3 Evaluate the cosine and tangent functions at the four quadrant angles 0,, , and . 2 2 Solution To begin, choose a point on the terminal side of each angle, as shown in Figure 4.38. For each of the four points,r ϭ 1, and you have the following. y x 1 y 0 π cos 0 ϭ ϭ ϭ 1 tan 0 ϭ ϭ ϭ 0 ͑x, y͒ ϭ ͑1, 0 ͒ (0, 1) 2 r 1 x 1 x 0 y 1 cos ϭ ϭ ϭ 0 tan ϭ ϭ ⇒ undefined ͑x, y͒ ϭ ͑0, 1 ͒ (−1, 0) (1, 0) 2 r 1 2 x 0 x π 0 x Ϫ1 y 0 cos ϭ ϭ ϭ Ϫ 1 tan ϭ ϭ ϭ 0 ͑x, y͒ ϭ ͑Ϫ1, 0 ͒ r 1 x Ϫ1
3π (0, −1) 3 x 0 3 y Ϫ1 2 cos ϭ ϭ ϭ 0 tan ϭ ϭ ⇒ undefined ͑x, y͒ ϭ ͑0, Ϫ1͒ 2 r 1 2 x 0
FIGURE 4.38 Now try Exercise 37. 312 Chapter 4 Trigonometry
Reference Angles The values of the trigonometric functions of angles greater than 90 Њ (or less than 0Њ ) 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 Ј formed by the terminal side of and the horizontal axis.
Figure 4.39 shows the reference angles for in Quadrants II, III, and IV.
Quadrant II Reference Reference an gle: θ ′ θ an gle: θ ′ θ θ
Reference Quadrant an gle: θ ′ Quadrant III IV θ ′ = π − θ (radians) θ ′ = θ − π (radians) θ ′ = 2 π − θ (radians) θ′ = 180 ° − θ (de grees) θ′ = θ − 180 ° (de grees) θ′ = 360 ° − θ (de grees)
FIGURE 4.39 y Example 4 Finding Reference Angles
θ = 300 ° Ј x Find the reference angle . θ′ = 60 ° a. ϭ 300 Њ b. ϭ 2.3 c. ϭ Ϫ 135 Њ Solution FIGURE 4.40 a. Because 300 Њ lies in Quadrant IV, the angle it makes with the x -axis is Ј ϭ Њ Ϫ Њ y 360 300 ϭ 60 Њ. Degrees ϭ Њ Ј ϭ Њ θ′ = π − 2.3 θ = 2.3 Figure 4.40 shows the angle 300 and its reference angle 60 . x b. Because 2.3 lies between ͞2 Ϸ 1.5708 and Ϸ 3.1416, it follows that it is in Quadrant II and its reference angle is Ј ϭ Ϫ 2.3
FIGURE 4.41 Ϸ 0.8416. Radians Figure 4.41 shows the angle ϭ 2.3 and its reference angle Ј ϭ Ϫ 2.3. y c. First, determine that Ϫ135 Њ is coterminal with 225 Њ, which lies in Quadrant III. So, 225 ° and −135 ° the reference angle is 225 ° are coterminal. Ј ϭ 225 Њ Ϫ 180 Њ x θ′ = 45 ° θ = −135 ° ϭ 45 Њ. Degrees Figure 4.42 shows the angle ϭ Ϫ 135 Њ and its reference angle Ј ϭ 45 Њ.
FIGURE 4.42 Now try Exercise 45. Section 4.4 Trigonometric Functions of Any Angle 313
y Trigonometric Functions of Real Numbers (x, y) To see how a reference angle is used to evaluate a trigonometric function, consider the point ͑x, y͒ on the terminal side of , as shown in Figure 4.43. By definition, you know that r = hyp y y opp sin ϭ and tan ϭ . r x For the right triangle with acute angle Ј and sides of lengths ԽxԽ and ԽyԽ, you have θ′ θ opp ԽyԽ x sin Ј ϭ ϭ adj hyp r opp ϭ ԽyԽ, adj ϭ ԽxԽ and FIGURE 4.43 opp y tan Ј ϭ ϭ Խ Խ. adj ԽxԽ So, it follows that sin and sin Ј are equal, except poss ibly in s ign . The same is true for tan and tan Ј and for the other four trigonometric functions. In all cases, the sign of the function value can be determined by the quadrant in which lies.
Evaluating Trigonometric Functions of Any Angle To find the value of a trigonometric function of any angle : 1. Determine the function value for the associated reference angle Ј. 2. Depending on the quadrant in which lies, affix the appropriate sign to the function value.
By using reference angles and the special angles discussed in the preceding section, you can greatly extend the scope of exact trigonometric values. For instance, knowing the function values of 30 Њ means that you know the function values of all angles for which 30 Њ is a reference angle. For convenience, the table below shows the Learning the table of values exact values of the trigonometric functions of special angles and quadrant angles. at the right is worth the effort because doing so will increase Trigonometric Values of Common Angles both your efficiency and your confidence. Here is a pattern (degrees) 0Њ 30 Њ 45 Њ 60 Њ 90 Њ 180 Њ 270 Њ for the sine function that may 3 help you remember the values. (radians) 0 6 4 3 2 2 Њ Њ Њ Њ 90 Њ 1 0 30 45 60 sin 0 Ί2 Ί3 1 0 Ϫ1 2 2 2 Ί0 Ί1 Ί2 Ί3 Ί4 sin 1 2 2 2 2 2 cos 1 Ί3 Ί2 0 Ϫ1 0 2 2 2 Reverse the order to get cosine Ί3 values of the same angles. tan 0 1 Ί3 Undef. 0 Undef. 3 314 Chapter 4 Trigonometry
Example 5 Using Reference Angles
Evaluate each trigonometric function. 4 11 a.cos b.tan ͑Ϫ210 Њ͒ c. csc 3 4 Solution a. Because ϭ 4͞3 lies in Quadrant III, the reference angle is 4 Ј ϭ Ϫ ϭ 3 3 as shown in Figure 6.41. Moreover, the cosine is negative in Quadrant III, so 4 cos ϭ ͑Ϫ͒ cos 3 3 1 ϭ Ϫ . 2 b. Because Ϫ210 Њ ϩ 360 Њ ϭ 150 Њ, it follows that Ϫ210 Њ is coterminal with the second-quadrant angle 150 Њ. So, the reference angle is Ј ϭ180 Њ Ϫ 150 Њ ϭ 30 Њ, as shown in Figure 4.45. Finally, because the tangent is negative in Quadrant II, you have tan ͑Ϫ210 Њ͒ ϭ ͑Ϫ͒ tan 30 Њ Ί3 ϭ Ϫ . 3 c. Because ͑11 ͞4͒ Ϫ 2 ϭ 3͞4, it follows that 11 ͞4 is coterminal with the second-quadrant angle 3͞4. So, the reference angle is Ј ϭ Ϫ ͑3͞4͒ ϭ ͞4, as shown in Figure 4.46. Because the cosecant is positive in Quadrant II, you have 11 csc ϭ ͑ϩ͒ csc 4 4
ϭ 1 sin ͑͞4͒ ϭ Ί2.
y y y
θ′ ° π = 30 11 π θ = 4 θ = 3 θ′ = π 4 x x 4 x θ′ = π 3 θ = −210 °
FIGURE 4.44 FIGURE 4.45 FIGURE 4.46
Now try Exercise 59. Section 4.4 Trigonometric Functions of Any Angle 315
Example 6 Using Trigonometric Identities
ϭ 1 Let be an angle in Quadrant II such that sin 3. Find (a) cos and (b) tan by using trigonometric identities. Solution a. Using the Pythagorean identity sin 2 ϩ cos 2 ϭ 1, you obtain 1 2 ϩ cos 2 ϭ 1 Substitute 1 for sin . 3 3 1 8 cos 2 ϭ 1 Ϫ ϭ . 9 9 Because cos < 0 in Quadrant II, you can use the negative root to obtain Ί8 cos ϭ Ϫ Ί9 2Ί2 ϭ Ϫ . 3 sin b. Using the trigonometric identity tan ϭ , you obtain cos 1͞3 tan ϭ Substitute for sin and cos . Ϫ2Ί2͞3
ϭ Ϫ 1 2Ί2 Ί2 ϭ Ϫ . 4 Now try Exercise 69.
You can use a calculator to evaluate trigonometric functions, as shown in the next example.
Example 7 Using a Calculator
Use a calculator to evaluate each trigonometric function. a.cot 410 Њ b.sin ͑Ϫ7͒ c. sec 9
Solution Funct ion Mode Calculator Keystrokes Display a. cot 410 Њ Degree͑ TAN ͑ 410͒ ͒ x Ϫ1 ENTER 0.8390996 b. sin ͑Ϫ7͒ RadianSIN ͑ ͑Ϫ͒ 7 ͒ ENTER Ϫ0.6569866 c. sec Radian͑ COS ͑ ، 9͒ ͒ x Ϫ1 ENTER 1.0641778 9 Now try Exercise 79. 316 Chapter 4 Trigonometry
4.4 EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks.
.Ίx2 1 y2 # 0 ؍ In Exercises 1–6, let be an angle in standard position, with ͧx, yͨ a point on the terminal side of and r r 1.sin ϭ ______2. ϭ ______y 3.tan ϭ ______4. sec ϭ ______x x 5.ϭ ______6. ϭ ______r y 7. Because r ϭ Ίx2 ϩ y2 cannot be ______, the sine and cosine functions are ______for any real value of . 8. The acute positive angle that is formed by the terminal side of the angle and the horizontal axis is called the ______angle of and is denoted by Ј.
SKILLS AND APPLICATIONS
In Exercises 9–12, determine the exact values of the six In Exercises 19–22, state the quadrant in which lies. trigonometric functions of the angle . 19. sin > 0 and cos > 0 y y 9. (a) (b) 20. sin < 0 and cos < 0 (4, 3) θ 21. sin > 0 and cos < 0 θ x x 22. sec > 0 and cot < 0 (−8, 15) In Exercises 23–32, find the values of the six trigonometric functions of with the given constraint. 10. (a)y (b) y Funct ion Value Constra int ϭ Ϫ 15 > θ θ 23. tan 8 sin 0 x x ϭ 8 < 24. cos 17 tan 0 25. sin ϭ 3 lies in Quadrant II. (− 12, −5) (1, −1) 5 ϭ Ϫ 4 26. cos 5 lies in Quadrant III. y y 11. (a) (b) 27. cot ϭ Ϫ 3 cos > 0 θ 28. csc ϭ 4 cot < 0 θ x x 29. sec ϭ Ϫ 2 sin < 0 30. sin ϭ 0 sec ϭ Ϫ 1 (− 3, −1 ) (4, −1) 31. cot is undefined. ͞2 Յ Յ 3͞2 12. (a)y (b) y 32. tan is undefined. Յ Յ 2 θ (3, 1) In Exercises 33–36, the terminal side of lies on the given x x line in the specified quadrant. Find the values of the six θ (−4, 4) trigonometric functions of by finding a point on the line. Line Quadrant ϭ Ϫ In Exercises 13–18, the point is on the terminal side of an 33. y x II ϭ 1 angle in standard position. Determine the exact values of the 34. y 3x III six trigonometric functions of the angle. 35. 2x Ϫ y ϭ 0 III 13.͑5, 12 ͒ 14. ͑8, 15 ͒ 36. 4x ϩ 3y ϭ 0 IV 15.͑Ϫ5, Ϫ2͒ 16. ͑Ϫ4, 10 ͒ Ϫ 1 Ϫ 3 17.͑ 5.4, 7.2 ͒ 18. ͑32, 74͒ Section 4.4 Trigonometric Functions of Any Angle 317
In Exercises 37–44, evaluate the trigonometric function of In Exercises 75–90, use a calculator to evaluate the the quadrant angle. trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) 3 37.sin 38. csc 2 75.sin 10 Њ 76. sec 225 Њ 3 77.cos ͑Ϫ110 Њ͒ 78. csc ͑Ϫ330 Њ͒ 39.sec 40. sec 2 79.tan 304 Њ 80. cot 178 Њ 81.sec 72 Њ 82. tan ͑Ϫ188 Њ͒ 41.sin 42. cot 2 83.tan 4.5 84. cot 1.35 43.csc 44. cot 85.tan 86. tan Ϫ 2 9 9 87.sin ͑Ϫ0.65 ͒ 88. sec 0.29 In Exercises 45–52, find the reference angle , and sketch 11 15 and in standard position. 89.cot Ϫ 90. csc Ϫ 8 14 45. ϭ 160 Њ 46. ϭ 309 Њ 47. ϭ Ϫ 125 Њ 48. ϭ Ϫ 215 Њ In Exercises 91–96, find two solutions of the equation. Give ؇ Յ < ؇ 2 7 your answers in degrees ͧ0 360 ͨ and in radians 49. ϭ 50. ϭ ͧ0 Յ < 2ͨ. Do not use a calculator. 3 6 ϭ 1 ϭ Ϫ 1 51. ϭ 4.8 52. ϭ 11.6 91. (a)sin 2 (b) sin 2 Ί2 Ί2 92. (a)cos ϭ (b) cos ϭ Ϫ In Exercises 53–68, evaluate the sine, cosine, and tangent of 2 2 the angle without using a calculator. 2Ί3 93. (a)csc ϭ (b) cot ϭ Ϫ 1 53.225 Њ 54. 300 Њ 3 55.750 Њ 56. Ϫ405 Њ 94. (a)sec ϭ 2 (b) sec ϭ Ϫ 2 57.Ϫ150 Њ 58. Ϫ840 Њ 95. (a)tan ϭ 1 (b) cot ϭ Ϫ Ί3 2 3 Ί3 Ί3 59. 60. 96. (a)sin ϭ (b) sin ϭ Ϫ 3 4 2 2 5 7 61. 62. 97. DISTANCE An airplane, flying at an altitude of 6 miles, 4 6 is on a flight path that passes directly over an observer 63.Ϫ 64. Ϫ (see figure). If is the angle of elevation from the 6 2 observer to the plane, find the distance d from the ϭ Њ ϭ Њ 9 10 observer to the plane when (a) 30 , (b) 90 , 65. 66. and (c) ϭ 120 Њ. 4 3 3 23 67.Ϫ 68. Ϫ 2 4
In Exercises 69–74, find the indicated trigonometric value in d 6 mi the specified quadrant. θ Funct ion Quadrant Tr igonometr ic Value ϭ Ϫ 3 69. sin 5 IV cos Not drawn to scale 70. cot ϭ Ϫ 3 II sin 98. HARMONIC MOTION The displacement from 71. ϭ 3 III tan 2 sec equilibrium of an oscillating weight suspended by a 72. csc ϭ Ϫ 2 IV cot spring is given by y͑t͒ ϭ 2 cos 6 t, where y is the ϭ 5 73. cos 8 I sec displacement (in centimeters) and t is the time (in ϭ 74. sec ϭ Ϫ 9 III tan seconds). Find the displacement when (a) t 0, 4 ϭ 1 ϭ 1 (b) t 4, and (c) t 2.