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340 5 Trigonometric Functions Trigonometric functions seem to have had their origins with the Greeks’ investigation of the indirect measurement of distances and angles in the “celestial sphere.” (The ancient Egyptians had used some elementary geometry to build the pyramids and remeasure lands flooded by the Nile, but neither they nor the ancient Babylonians had developed the concept of angle measure.) The word “trigonometry,” based on the Greek words for triangle measurement, was first used as the title for a text by the German mathematician Pitiscus in A.D. 1600. Originally, the trigonometric functions were restricted to angles and their applications to the indirect measurement of angles and distances. These functions gradually broke free of these restrictions, and modern applications range over many types of problems that have little to do with angles—applications involving sound, light, and electrical waves; business cycles; and planetary motion. Our approach to the subject does not follow historical lines in that we first introduce trigonometric functions (circular functions) with real number domains; then we define trigonometric functions with angle domains. SECTION 5-1 The Wrapping Function • Definition of the Wrapping Function • Exact Values for Particular Real Numbers • The Wrapping Function Is Not One-to-One • Definition of the Wrapping Function v 1 1 u2 v 2 1 1 u 1 FIGURE 1 Unit circle. The important circular function definitions presented in the next section are based on a function W, called the wrapping function, whose domain is the set of real numbers and whose range is the set of points on the unit circle. By the unit circle we mean the circle of radius 1 with center at the origin of a rectangular coordinate system. Its equation is u2 v 2 1 (see Fig. 1).* In defining the wrapping function, we “wrap” a real number line with origin at (1, 0) around the unit circle—the positive real axis is wrapped counterclockwise, and the negative real axis is wrapped clockwise. In this way, each real number on the real line is paired with a unique point, called a circular point, on the unit circle, as shown in Figure 2. v x v x 2 0 x 2 2 1 (1, 0) v 1 3 u 2 1 (1, 0) 0 1 1 2 2 1 (1, 0) u 0 3 2 1 u 1 2 FIGURE 2 The wrapping function. *We use the variables u and v instead of x and y so that x can be used without ambiguity as an independent variable in defining the wrapping and circular functions. 341 5-1 The Wrapping Function To locate the circular point associated with a number such as 37 or 105, the number line is wrapped many times around the circle. An equivalent way of pairing real numbers with points on the unit circle is to think in terms of arc length. To find the circular point P associated with the real number x, we start at A(1, 0) and move x units along the unit circle, counterclockwise if x is positive and clockwise if x is negative. The length of arc AP is x (see Fig. 3). v FIGURE 3 The wrapping function and arc length. v x P 0 A(1, 0) u 0 A(1, 0) P x0 (a) u x x0 (b) It is important to be able to find the coordinates (a, b) of the circular point P associated with a given real number x so that we can write W(x) (a, b). In general, this is difficult and requires the use of a calculator. However, for certain real numbers, integer multiples of /6, /4, /3, and /2, we can find the exact coordinates of the corresponding circular points using simple geometric properties of a circle. • Exact Values for Particular Real Numbers We start our investigation by finding the circumference of the unit circle. Since radius r 1, the circumference is 2r 2(1) 2 Circumference of the unit circle One-fourth, one-half, and three-fourths of the circumference are, respectively, /2, , and 3/2. The circular points corresponding to these real numbers are on the coordinate axes, and hence, their coordinates are easily determined (see Fig. 4). W(0) (1, 0) W v W() (1, 0) (0, 1) 2 (1, 0) 2 (0, 1) 0, 2 (1, 0) 3 2 (0, 1) FIGURE 4 Circular points on the coordinate axes. u W 32 (0, 1) W(2) (1, 0) Following the same procedure, we can find the coordinates of any circular point on a coordinate axis—that is, for any circular point corresponding to a real number that is an integer multiple of /2. 342 5 Trigonometric Functions EXAMPLE 1 Finding the Coordinates of Circular Points Find the coordinates of the circular points: (A) W(/2) Solution (B) W(5/2) (A) Starting at (1, 0), we go one-fourth the way around the unit circle in a clockwise direction (see Fig. 4). Thus, W 2 (0, 1) (B) Starting at (1, 0) and proceeding counterclockwise, we count quarter-circle steps, /2, 2/2, 3/2, 4/2, and ending at 5/2. Thus, the circular point is on the positive vertical axis, and we have W Matched Problem 1 Find the coordinates of the circular points: (A) W() (0, 1) 4 vu (a, b) (1, 0) (B) W(3) We now find the coordinates of the circular point W(/4). Since /4 is one-half the arc joining (1, 0) and (0, 1), the circular point W(/4) must lie on the line v u, as shown in Figure 5. Since W(/4) is on the line v u and on the circle u2 v2 1, its coordinates (a, b) must satisfy both equations. That is, v 2 52 (0, 1) u ab and a2 b2 1 Substituting a for b in the second equation, we have FIGURE 5 Circular point W(/4). a2 a2 1 2a2 1 a2 1 2 a a 1 2 1 2 a 1/ 2 must be discarded, since W(/4) is in the first quadrant. Using the first equation, we see that ba 1 2 5-1 The Wrapping Function 343 Therefore, W 4 21 , 21 Using symmetry properties of a circle—the unit circle is symmetric with respect to both axes and the origin—we can easily find the coordinates of any circular point that is reflected across the vertical axis, horizontal axis, or origin from W(/4). EXAMPLE 2 Finding the Coordinates of Circular Points Find the coordinates of the circular points: (A) W(5/4) Solutions (B) W(/4) (A) Starting at (1, 0) and counting in one-eighth circle steps counterclockwise (/4, 2/4, 3/4, 4/4, 5/4), we find ourselves in the third quadrant on the circle halfway between (1, 0) and (0, 1), as indicated in Figure 6. Using symmetry with respect to the origin, we have W 54 21 , 21 v FIGURE 6 (0, 1) 2 3 4 4 (1, 0) ( 21 , 21 ) 4 4 5 4 4 4 (0, 1) ( 21 , 21 ) (1, 0) u ( 21 , 21 ) (B) Starting at (1, 0), we proceed one-eighth the way around the unit circle in a clockwise direction and end up in the fourth quadrant on the circle halfway between (0, 1) and (1, 0), as indicated in Figure 6. Using symmetry with respect to the horizontal axis, we see that 4 21 , 21 W Matched Problem 2 Find the coordinates of the circular points: (A) W(3/4) (B) W(7/4) 344 5 Trigonometric Functions v (a, b) (a, b) 2 3 (1, 0) 3 (1, 0) u We continue our investigation by finding the coordinates of the circular point W(/3). Referring to Figure 7, we divide the upper semicircle from (1, 0) to (1, 0) into thirds. The circular points W(/3) and W(2/3) are symmetric with respect to the v axis; hence, if W(/3) is given coordinates (a, b), then W(2/3) must have coordinates (a, b). The chord joining W(2/3) and W(/3) is thus 2a units long. Using the distance formula (see Section 2-1), we find the length of the chord joining W(0) and W(/3) to be given by (a 1)2 b2. The two chords are equal in length, since congruent arcs are opposite congruent chords on the same circle. Thus, (a 1)2 b2 2a FIGURE 7 Circular point W(/3). Squaring both sides, we obtain (a 1)2 b2 4a2 a2 2a 1 b2 4a2 a2 b2 2a 1 4a2 1 2a 1 4a2 a2 b2 1 (Why?) 4a2 2a 2 0 2a2 a 1 0 (2a 1)(a 1) 0 a 12 a 1 or a 12 Substitute a 1 2 a 1 must be discarded. (Why?) into a2 b2 1 and solve for b: b 1 1 2 2 2 b2 3 4 b b 3 2 3 2 b 3/2 must be discarded. (Why?) Thus, W 3 12, 32 Proceeding in a similar manner, or using symmetry with respect to the line v u, we can obtain W 6 32, 12 5-1 The Wrapping Function 345 The key results from the above discussion for the first quadrant are summarized in Figure 8. FIGURE 8 Coordinates of Key Circular Points v (0, 1) 2 ( 12 , 32 ) ( 21 , 21 ) 3 ( 32 , 12 ) 4 6 (1, 0) u It is important that you memorize these first-quadrant relationships. EXPLORE-DISCUSS 1 An effective memory aid for recalling the coordinates of the key circular points in Figure 8 can be created by writing the coordinates of the circular points W(0), W(/6), W(/4), W(/3), and W(/2), keeping this order, in a form where each numerator is the square root of an appropriate number and each denominator is 2. For example, W(0) (1, 0) (4/2, 0/2). Describe the pattern that results. The reason for memorizing the coordinates of key circular points in the first quadrant is that by using these, along with the symmetry of the unit circle, we can find the coordinates of any circular point that corresponds to any integer multiple of /6, /4, /3, and /2. EXAMPLE 3 Finding Coordinates of Circular Points Find the coordinates of the circular points: (A) W(5/6) Solutions (B) W(2/3) (A) Note that 5/6 is /6 less than 6/6. Locate 5/6 in the second quadrant and use Figure 8 and symmetry with respect to the vertical axis to find W(5/6). See Figure 9. W v FIGURE 9 (32 , 12 ) 5 6 6 ( 32 , 12 ) u 56 32, 12 346 5 Trigonometric Functions v (B) Locate 2/3 in the third quadrant and use Figure 8 and symmetry with respect to the origin to find W(2/3). See Figure 10. ( 12 , 32 ) 23 12, 32 3 2 3 3 (1, 0) W u ( 12 , 32 ) FIGURE 10 Matched Problem 3 Find the coordinates of the circular points: (A) W(5/3) • The Wrapping Function Is Not One-to-One (B) W(7/6) It is easy to see that the wrapping function is not a one-to-one function. Each domain value, a real number, corresponds to exactly one range value, a point on the unit circle. However, each range value, a point on the unit circle, corresponds to infinitely many domain values, real numbers. For example, we see that W 2 (0, 1) That is, exactly one range value corresponds to the domain value /2. But how many domain values correspond to the range value (0, 1)? Every time we go around the unit circle 2 units in either direction from (0, 1), we return to the same point. Thus, if we are asked to solve W(x) (0, 1) we have to write x 2k 2 k any integer and there are infinitely many domain values of W that correspond to the range value (0, 1). In general: Theorem 1 A Wrapping Function Property For all real numbers x, W(x) W(x 2k) v (a, b) W(x) k any integer* (1, 0) u *Think of a point P moving around the unit circle in either direction. Every time P covers a distance of 2, the circumference of the circle, it is back at the point where it started. 5-1 The Wrapping Function 347 We will have more to say about the implications of this important property of the wrapping function in subsequent sections. EXPLORE-DISCUSS 2 (A) Solve the circular point equation W(x) (0, 1), 2 x 2. (B) Describe an expression that would represent all solutions to W(x) (0, 1). Answers to Matched Problems 1. (A) (1, 0) (B) (1, 0) 2. (A) (1/2, 1/2) (B) (1/2, 1/2) 3. (A) (1/2, 3/2) (B) (3/2, 1/2) EXERCISE 5-1 A 0 x 2, then write an expression that represents all solutions for the equation without any restrictions on x. In Problems 1–12, find the coordinates for each circular point. Sketch your own figures—don’t look back in the text. 35. W(x) (1, 0) 36. W(x) (1, 0) 1. W() 2. W(0) 3. W(6) 37. W(x) (1/2, 1/2) 38. W(x) (1/2, 1/2) 4. W(3) 5. W() 6. W(5) 39. Describe in words why W(x) W(x 4) for every real number x. 7. W(3/2) 8. W(/2) 9. W(/2) 10. W(3/2) 11. W(11/2) 12. W(15/2) 40. Describe in words why W(x) W(x 6) for every real number x. B C In Problems 13–24, find the coordinates for each circular point. Sketch your own figures—don’t look back in the text. If W(x) (a, b), indicate whether the statements in Problems 41–46 are true (T) or false (F). Sketching figures should help you decide. 13. W(/3) 14. W(2/3) 15. W(/4) 16. W(5/4) 17. W(5/6) 18. W(/4) 41. W(x ) (a, b) 42. W(x ) (a, b) 19. W(13/4) 20. W(7/6) 21. W(19/2) 43. W(x) (a, b) 44. W(x) (a, b) 22. W(7) 23. W(13/6) 24. W(11/4) 45. W(x 2) (a, b) 46. W(x 2) (a, b) Determine the signs of a and b for the coordinates (a, b) of each circular point indicated in Problems 25–34. First determine the quadrant in which each circular point lies. [Note: /2 1.57, 3.14, 3/2 4.71, and 2 6.28.] 25. W(3.2) 26. W(0.9) 27. W(5.7) 28. W(1.3) 29. W(2) 30. W(3) 31. W(12.5) 32. W(37) 33. W(27) In Problems 47–52, find all solutions x, 2 x 2, such that: 21 , 21 1 3 49. W(x) , 2 2 3 1 51. W(x) , 2 2 47. W(x) 32, 12 1 3 50. W(x) , 2 2 1 1 52. W(x) , 2 2 48. W(x) 34. W(7.8) Find all solutions to each equation in Problems 53 and 54. In Problems 35–38, for each equation find all solutions for 53. W(x) W(/4) 54. W(x) W(2/3)