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348 5 Trigonometric Functions SECTION 5-2 Circular Functions • • • • • • Definition of the Circular Functions DEFINITION 1 Definition of the Circular Functions Exact Values for Particular Real Numbers Sign Properties Basic Identities Calculator Evaluation In Section 5-1 we saw that the wrapping function W pairs each real number x with an ordered pair of real numbers (a, b), the coordinates of the circular point W(x). We use this association to construct the six circular functions, also called trigonometric functions*: sine, cosine, tangent, cotangent, secant, and cosecant. The values of these functions for a real number x are denoted by sin x, cos x, tan x, cot x, sec x, and csc x, respectively. These values are expressed in terms of the coordinates of the circular point W(x) (a, b) as indicated in Definition 1. Circular Functions If x is a real number and (a, b) are the coordinates of the circular point W(x), then v sin x b cos x a tan x b a 1 b b 0 sec x 1 a a 0 cot x a b b 0 csc x a 0 (a, b) W(x) (1, 0) u The domain of both the sine and cosine functions is the set of real numbers R. The range of both the sine and cosine functions is [1, 1]. This is the set of numbers assumed by b, for sine, and a, for cosine, as the circular point (a, b) moves around the unit circle. The domain of cosecant is the set of real numbers x such that b in W(x) (a, b) is not 0. Similar restrictions are made on the domains of the other three circular functions. We will have more to say about the domains and ranges of all six circular functions in subsequent sections. • Exact Values for Particular Real Numbers Using the results in Section 5-1, we can evaluate any one of the six circular functions exactly, when it exists, for integer multiples of the real numbers /6, /4, /3, and /2. Figure 8 in Section 5-1, which you should have memorized, and symmetry properties of the unit circle are central to the process. Later in this section we will show *Strictly speaking, the term trigonometric is used when we are dealing with angle domains and circular is used when we are dealing with real number domains. We will not insist on this distinction and often, as is the convention, use trigonometric for both. 5-2 Circular Functions 349 how a calculator can be used to evaluate the circular functions to 8 or more significant digits for arbitrary real numbers. You might ask why we don’t go directly to the calculator. The answer is that there are many situations in which it is more desirable to work with exact forms, if available, than the corresponding decimal approximations that are produced by a calculator. EXAMPLE 1 Exact Evaluation of Circular Functions Evaluate each circular function exactly for x /3. Solution From Section 5-1 we know that 3 12, 32 v a ( 3 W b 1 , 3 2 2 ) [See Fig. 1] Thus, (1, 0) u sin 3 b 3 2 csc 1 1 2 3 b 3/2 3 cos 1 a 3 2 sec 1 1 1 2 3 a 2 tan b 3/2 1 3 3 a 2 cot 1 a 1 2 3 b 3/2 3 FIGURE 1 Matched Problem 1 EXAMPLE 2 Evaluate each circular function exactly for x /6. Exact Evaluation of Circular Functions Evaluate exactly: (A) sin (5/6) Solutions (B) cot () (C) sec (2/3) (D) tan (7/4) Sketch a figure for each part, then use Figure 8 in Section 5-1 and symmetry properties of the unit circle. (A) v a b 3 1 , 2 2 ( ) 5 6 6 ( 32 , 12 ) (1, 0) u sin 5 1 b 6 2 350 5 Trigonometric Functions (B) v (1, 0) a b (1, 0) u cot () a 1 b 0 Not defined (C) v ( 12 , 32 ) 3 u (1, 0) 2 3 23 1a 1 sec 1 2 2 ( 12 , 32 ) a b (D) v ( 21 , 21 ) 4 7 4 (1, 0) a b 1 , 1 ( 2 Matched Problem 2 tan 7 b 1/2 1 4 a 1/2 ) Evaluate exactly: (A) cos (5/6) • Sign Properties 2 u (B) sin (3/4) (C) csc 3 (D) tan (/3) As a circular point W(x) moves from quadrant to quadrant, its coordinates (a, b) undergo sign changes. Hence, the circular functions also undergo sign changes. It is important to know the sign of each circular function in each quadrant. Table 1 shows the sign behavior for each function. It is not necessary to memorize Table 1, since the sign of each function for each quadrant is easily determined from its definition (which should be memorized). 5-2 TABLE 1 EXPLORE-DISCUSS 1 351 Circular Functions Sign Properties Sign in quadrant Circular function I II III IV sin x b II I a b (, ) a b (, ) csc x 1/b cos x a sec x 1/a tan x b/a cot x a/b v u a b (, ) a b (, ) III IV (A) Determine the quadrant in which both tan x 0 and sin x 0. Draw diagrams, and explain your reasoning. (B) Determine the quadrant in which both cos x 0 and cot x 0. Draw diagrams, and explain your reasoning. • Basic Identities Returning to the definitions of the circular functions and noting that sin x b and cos x a we can obtain the following useful relationships among the six circular functions: csc x 1 1 b sin x (1) sec x 1 1 a cos x (2) cot x a 1 1 b b/a tan x (3) tan x b sin x a cos x (4) cot x a cos x b sin x (5) v W(x) Because the circular points W(x) and W(x) are symmetric with respect to the horizontal axis (Fig. 2), we have the following sign properties: (a, b) u W(x) (a, b) FIGURE 2 Symmetry property. sin (x) b sin x (6) cos (x) a cos x (7) tan (x) b b tan x a a (8) 352 5 Trigonometric Functions Finally, because (a, b) (cos x, sin x) is on the unit circle u2 v2 1, it follows that (cos x)2 (sin x)2 1 which is usually written as sin2 x cos2 x 1 (9) where sin2 x and cos2 x are concise ways of writing (sin x)2 and (cos x)2, respectively. CAUTION (sin x)2 sin x2 (cos x)2 cos x2 Equations (1)–(9) are called basic identities. They hold true for all replacements of x by real numbers for which both sides of an equation are defined. These basic identities must be memorized along with the definitions of the six circular functions, since the material is used extensively in developments that follow. Note that most of Chapter 6 is devoted to trigonometric identities. We summarize the basic identities for convenient reference in Theorem 1. Theorem 1 Basic Trigonometric Identities For x any real number (in all cases restricted so that both sides of an equation are defined): Reciprocal Identities (1) (2) 1 csc x sin x 1 sec x cos x (3) cot x 1 tan x Quotient Identities (4) (5) sin x tan x cos x cot x cos x sin x Identities for Negatives (6) sin (x) sin x (7) cos (x) cos x (8) tan (x) tan x Pythagorean Identity (9) sin2 x cos2 x 1 EXAMPLE 3 Using Basic Identities Use the basic identities to find the values of the other five circular functions given sin x 12 and tan x 0. 5-2 Solution Circular Functions 353 We first note that the circular point W(x) is in quadrant III, since that is the only quadrant in which sin x 0 and tan x 0. We next find cos x using identity (9): sin2 x cos2 x 1 Pythagorean identity (9) ( 12)2 cos2 x 1 cos2 x 34 cos x 3 2 Since W(x) is in quadrant III Now, since we have values for sin x and cos x, we can find values for the other four circular functions using identities (1), (2), (4), and (5): csc x 1 1 1 2 sin x 2 Reciprocal identity (1) sec x 1 1 2 cos x 3/2 3 Reciprocal identity (2) tan x sin x 12 1 cos x 3/2 3 Quotient identity (4) cot x cos x 3/2 3 sin x 12 Quotient identity (5) [Note: We could also use identity (3).] In Example 3 it is important to note that we were able to find the values of the other five circular functions without finding x. Matched Problem 3 EXPLORE-DISCUSS 2 Use the basic identities to find the values of the other five circular functions given cos x 1/2 and cot x 0. Given the conditions on x in Example 3: sin x 12 and tan x 0. Find, using basic identities and the results in Example 3, each of the following: (A) sin (x) (B) sec (x) (C) tan (x) Verbally justify each step in your solution process. • Calculator Evaluation Evaluating circular functions for real numbers other than integer multiples of /6, /4, /3, and /2 is difficult without the use of a calculator. Using advanced mathematics, calculators are internally programmed to evaluate these functions automatically to an accuracy of 8 or more significant digits. If you look at the function keys on your calculator, you will find three keys labeled SIN COS TAN 354 5 Trigonometric Functions These keys are used to evaluate the sine, cosine, and tangent functions directly. A careful look at the function keys on your calculator also will reveal that there are no keys for cosecant, secant, and cotangent. Why is it not necessary to have these additional keys? Because of the reciprocal identities (1)–(3), we can use the function keys for sine, cosine, and tangent, along with the reciprocal function key x 1 or 1/x to obtain csc x, sec x, and cot x. Don’t use the keys marked sin1, cos1, and tan1 to evaluate csc, sec, and cot, respectively. You will see why in Section 5-9. Some examples should make the process of calculator evaluation of the circular functions clear. CAUTION EXAMPLE 4 Setting Calculator Mode: Before commencing with the examples and exercises, read the instruction book accompanying your calculator to determine how to put it in radian (rad) mode. It is in this mode that we can evaluate the circular functions for real numbers. (This process is justified in Section 5-4 when we discuss trigonometric functions with angle domains.) Forgetting to set the calculator in the correct mode before starting calculations involving circular or trigonometric functions is a frequent cause of error when using a calculator. Calculator Evaluation Evaluate to 4 significant digits using a calculator: (A) sin 2 Solutions (B) tan (1.612) (C) csc 3.2 (A) sin 2 0.9093 (B) tan (1.612) 24.26 (C) csc 3.2 17.13 Matched Problem 4 Evaluate to 4 significant digits using a calculator: (A) cos 4 EXPLORE-DISCUSS 3 (B) sec 1.605 (C) cot (3.133) Use a calculator to evaluate each of the following, and explain the results shown on the calculator: (A) tan (/2) (B) cot 0 (C) sec (/2) 5-2 355 Circular Functions Answers to Matched Problems 1. sin (/6) 12 , cos (/6) 3/2, tan (/6) 1/3, csc (/6) 2, sec (/6) 2/3, cot (/6) 3 2. (A) 3/2 (B) 1/2 (C) Not defined (D) 3 3. sin x 1/2, csc x 2, sec x 2, tan x 1, cot x 1 4. (A) 0.6536 (B) 29.24 (C) 116.4 EXERCISE 5-2 Figure 8 in Section 5-1, the definition of the circular functions, and the basic identities should now be memorized. Work the problems in this exercise without looking back in the text. Draw lots of pictures, if necessary. A 1. Write the value of each circular function in terms of the coordinates (a, b) of the circular point W(x). (A) cos x (B) csc x (C) cot x (D) sec x (E) tan x (F) sin x 2. Given W(x) (a, b), identify each quantity using one of the circular function values sin x, cos x, and so on. (A) b (B) 1/a (C) b/a (D) 1/b (E) a (F) a/b In Problems 3–20, find the exact value of each expression (if it exists) without the use of a calculator. 33. tan (/2) 34. sec (3/4) 35. csc (3/2) 36. sin (/3) 37. cos (5/3) 38. cot (3) 39. sec (/4) 40. tan (7/4) 41. sin (7/6) 42. cos (5/3) 43. cot (3/4) 44. csc (/3) 45. tan (2/3) 46. sec (5/2) 47. csc 5 48. cot (13/6) In Problems 49–52, find the value of each to one significant digit. Use only the accompanying figure, Definition 1, and a calculator as necessary for multiplication and division. Check your results by evaluating each directly on a calculator. 49. (A) sin 0.4 (B) cos 0.4 (C) tan 0.4 50. (A) sin 0.8 (B) cos 0.8 (C) cot 0.8 3. cos 0 4. sin 0 5. sin (/6) 51. (A) sec 2.2 (B) tan 5.9 (C) cot 3.8 6. cos (/6) 7. sin (/2) 8. cos (/2) 52. (A) csc 2.5 (B) cot 5.6 (C) tan 4.3 9. tan (/3) 10. cos (/3) 11. tan (/2) 12. cot 0 13. sec 0 14. cot (/4) 15. sec (/4) 16. csc (/3) 17. tan (/4) 18. tan 0 19. csc 0 20. cot (/6) b 2 Unit circle In Problems 21–26, in which quadrants must W(x) lie so that: 21. cos x 0 22. tan x 0 23. sin x 0 24. sec x 0 25. cot x 0 26. csc x 0 0.5 3 Evaluate Problems 27–32 to 4 significant digits using a calculator. 27. tan 4.728 28. cos 3.167 29. sec (1.489) 30. csc (13.25) 31. sin (9.841) 32. cot 6.386 B In Problems 33–48, find the exact value of each expression (if it exists) without the use of a calculator. 1 0.5 a 0.5 6 0.5 4 5 356 5 Trigonometric Functions In Problems 53–56, in which quadrants are the statements true and why? 53. sin x 0 and cot x 0 78. sec x 2 and sin x 0 79. tan x 3 and sin x 0 80. cot x 1 and sin x 0 54. cos x 0 and tan x 0 In Problems 81–86, find the smallest positive x (in terms of ) for which: 55. cos x 0 and sec x 0 56. sin x 0 and csc x 0 For which values of x, 0 x 2, is each of Problems 57–62 not defined? 57. cos x 58. sin x 59. tan x 60. cot x 61. sec x 62. csc x How does the functional value indicated in Problems 63 and 64 vary as x varies over the indicated intervals? [Hint: Draw a unit circle and note that W(x) (a, b) (cos x, sin x).] 81. cos x 1 82. sin x 3 2 83. cot x 3 84. tan x 1 85. sec x 2 3 86. csc x 2 63. sin x: (A) [0, /2] (C) [, 3/2] (B) [/2, ] (D) [3/2, 2] 64. cos x: (A) [0, /2] (C) [, 3/2] (B) [/2, ] (D) [3/2, 2] Complete Problems 65–68 to 4 significant digits using a calculator. 65. sin (cos 0.3157) 66. cos (tan 5.183) 67. cos [csc (1.408)] 68. sec [cot (3.566)] In Problems 87 and 88, fill in the blanks citing the appropriate identity (1)–(9). 87. Statement Reason cot2 x 1 Use appropriate identities to solve Problems 69–74. 69. Find sin (x) if sin x 13. 72. Find sec (x) if sec x 50. 73. Find cot (x) if cot x 25. 74. Find csc (x) if csc x 10. C Use the basic identities to find the values of the other five circular functions given the indicated information in Problems 75–80. 1 75. cos x and tan x 0 2 3 76. sin x and cot x 0 2 1 77. sin x and cos x 0 2 cos x (A) ________ cos2 x 1 sin2 x Algebra cos2 x sin2 x sin2 x Algebra 1 sin2 x sin1 x 70. Find cos (x) if cos x 13. 71. Find tan (x) if tan x 5. sin x 1 2 (B) ________ 2 Algebra csc2 x (C) ________ 88. Statement Reason tan2 x 1 cos x 1 sin x 2 (A) ________ sin2 x 1 cos2 x Algebra sin2 x cos2 x cos2 x Algebra 1 cos2 x 1 cos x sec2 x (B) ________ 2 Algebra (C) ________ 5-3 Angles and Their Measure 357 Approximating . Problems 93 and 94 refer to a sequence of numbers generated as follows: APPLICATIONS a1 If an n-sided regular polygon is inscribed in a circle of radius r, then it can be shown that the area of the polygon is given by A a2 a1 cos a1 1 1 2 2 nr sin 2 n cos an Compute each area exactly and then to 4 significant digits using a calculator if the area is not an integer. 0 89. n 12, r 5 meters an a3 a2 cos a2 . . . an1 an cos an 1 91. n 3, r 4 inches 93. Let a1 0.5, and compute the first five terms of the sequence to six decimal places and compare the fifth term with /2 computed to six decimal places. 92. n 8, r 10 centimeters 94. Repeat Problem 93, starting with a1 1. 90. n 4, r 3 inches SECTION 5-3 Angles and Their Measure • Angles • Degree and Radian Measure • From Degrees to Radians, and Vice Versa In this section, we will introduce the idea of angle and two measures of angles, degree and radian. • Angles The study of trigonometry depends on the concept of angle. An angle is formed by rotating (in a plane) a ray m, called the initial side of the angle, around its endpoint until it coincides with a ray n, called the terminal side of the angle. The common endpoint V of m and n is called the vertex (see Fig. 1). A counterclockwise rotation produces a positive angle, and a clockwise rotation produces a negative angle, as shown in Figure 2(a) and (b). The amount of rotation in either direction is not restricted. FIGURE 1 Angle or angle PVQ or V. Q n Terminal side V Initial side P m