Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
6965_CH04_pp319-402.qxd 1/14/10 1:50 PM Page 361 SECTION 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant 361 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant What you’ll learn about • The Tangent Function • The Cotangent Function • The Secant Function The Tangent Function The graph of the tangent function is shown below. As with the sine and cosine graphs, this graph tells us quite a bit about the function’s properties. Here is a summary of tangent facts: • The Cosecant Function ... and why This will give us functions for the remaining trigonometric ratios. THE TANGENT FUNCTION [–3π /2, 3π /2] by [–4, 4] FIGURE 4.44A ƒ1x2 = tan x Domain: All reals except odd multiples of p/2 Range: All reals Continuous (i.e., continuous on its domain) Increasing on each interval in its domain Symmetric with respect to the origin (odd) Not bounded above or below No local extrema No horizontal asymptotes Vertical asymptotes: x = k # 1p/22 for all odd integers k End behavior: lim tan x and lim tan x do not exist. (The function values x: -q x: q continually oscillate between - q and q and approach no limit.) y 3 2 –2π π x tan x = –3 FIGURE 4.45 The tangent function has asymptotes at the zeros of cosine. y 3 2 1 –2π We now analyze why the graph of ƒ1x2 = tan x behaves the way it does. It follows from the definitions of the trigonometric functions (Section 4.2) that π x –3 FIGURE 4.46 The tangent function has zeros at the zeros of sine. sin x . cos x Unlike the sinusoids, the tangent function has a denominator that might be zero, which makes the function undefined. Not only does this actually happen, but it happens an infinite number of times: at all the values of x for which cos x = 0. That is why the tangent function has vertical asymptotes at those values (Figure 4.45). The zeros of the tangent function are the same as the zeros of the sine function: all the integer multiples of p (Figure 4.46). Because sin x and cos x are both periodic with period 2p, you might expect the period of the tangent function to be 2p also. The graph shows, however, that it is p. The constants a, b, h, and k influence the behavior of y = a tan 1b1x - h22 + k in much the same way that they do for the graph of y = a sin 1b1x - h22 + k. The constant a yields a vertical stretch or shrink, b affects the period, h causes a horizontal translation, and k causes a vertical translation. The terms amplitude and phase shift, however, are not used, as they apply only to sinusoids. 6965_CH04_pp319-402.qxd 362 1/14/10 1:50 PM Page 362 CHAPTER 4 Trigonometric Functions EXAMPLE 1 Graphing a Tangent Function Describe the graph of the function y = - tan 2x in terms of a basic trigonometric function. Locate the vertical asymptotes and graph four periods of the function. SOLUTION The effect of the 2 is a horizontal shrink of the graph of y = tan x by a factor of 1/2, while the effect of the -1 is a reflection across the x-axis. Since the vertical asymptotes of y = tan x are all odd multiples of p/2, the shrink factor causes the vertical asymptotes of y = tan 2x to be all odd multiples of p/4 (Figure 4.47a). The reflection across the x-axis (Figure 4.47b) does not change the asymptotes. Since the period of the function y = tan x is p, the period of the function y = - tan 2x is (thanks again to the shrink factor) p/2. Thus, any window of horizontal length 2p will show four periods of the graph. Figure 4.47b uses the window 3- p, p4 by 3-4, 44. Now try Exercise 5. [– π , π ] by [–4, 4] (a) The other three trigonometric functions (cotangent, secant, and cosecant) are reciprocals of tangent, cosine, and sine, respectively. (This is the reason that you probably do not have buttons for them on your calculators.) As functions they are certainly interesting, but as basic functions they are unnecessary—we can do our trigonometric modeling and equation solving with the other three. Nonetheless, we give each of them a brief section of its own in this book. The Cotangent Function The cotangent function is the reciprocal of the tangent function. Thus, cot x = [– π , π ] by [–4, 4] (b) cos x . sin x The graph of y = cot x will have asymptotes at the zeros of the sine function (Figure 4.48) and zeros at the zeros of the cosine function (Figure 4.49). FIGURE 4.47 The graph of (a) y = tan 2x is reflected across the x-axis to produce the graph of (b) y = - tan 2x. (Example 1) y y 3 2 1 3 2 x –2π –3 FIGURE 4.48 The cotangent has asymptotes at the zeros of the sine function. Cotangent on the Calculator If your calculator does not have a “cotan” button, you can use the fact that cotangent and tangent are reciprocals. For example, the function in Example 2 can be entered in a calculator as y = 3/tan 1x/22 + 1 or as y = 31tan1x/222-1 + 1. Remember that it cannot be entered as y = 3 tan-1 1x/22 + 1. (The - 1 exponent in that position represents a function inverse, not a reciprocal.) –2π 2π x –3 FIGURE 4.49 The cotangent has zeros at the zeros of the cosine function. EXAMPLE 2 Graphing a Cotangent Function Describe the graph of ƒ1x2 = 3 cot 1x/22 + 1 in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods. SOLUTION The graph is obtained from the graph of y = cot x by effecting a hori- zontal stretch by a factor of 2, a vertical stretch by a factor of 3, and a vertical translation up 1 unit. The horizontal stretch makes the period of the function 2p (twice the period of y = cot x), and the asymptotes are at the even multiples of p. Figure 4.50 shows two periods of the graph of ƒ. Now try Exercise 9. 6965_CH04_pp319-402.qxd 1/14/10 1:50 PM Page 363 SECTION 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant 363 The Secant Function Important characteristics of the secant function can be inferred from the fact that it is the reciprocal of the cosine function. Whenever cos x = 1, its reciprocal sec x is also 1. The graph of the secant function has asymptotes at the zeros of the cosine function. The period of the secant function is 2p, the same as its reciprocal, the cosine function. The graph of y = sec x is shown with the graph of y = cos x in Figure 4.51. A local maximum of y = cos x corresponds to a local minimum of y = sec x, while a local minimum of y = cos x corresponds to a local maximum of y = sec x. [–2π , 2π ] by [–10, 10] FIGURE 4.50 Two periods of ƒ1x2 = 3 cot 1x/22 + 1. (Example 2) y –1 –2 –3 Proving a Graphical Hunch Figure 4.52 shows that the graphs of y = sec x and y = - 2 cos x never seem to intersect. 3 2 –2π EXPLORATION 1 π 2π x If we stretch the reflected cosine graph vertically by a large enough number, will it continue to miss the secant graph? Or is there a large enough (positive) value of k so that the graph of y = sec x does intersect the graph of y = - k cos x? 1. Try a few other values of k in your calculator. Do the graphs intersect? FIGURE 4.51 Characteristics of the secant function are inferred from the fact that it is the reciprocal of the cosine function. 2. Your exploration should lead you to conjecture that the graphs of y = sec x and y = - k cos x will never intersect for any positive value of k. Verify this conjecture by proving algebraically that the equation - k cos x = sec x has no real solutions when k is a positive number. EXAMPLE 3 Solving a Trigonometric Equation Algebraically Find the value of x between p and 3p/2 that solves sec x = - 2. SOLUTION We construct a reference triangle in the third quadrant that has the ap- propriate ratio, hyp/adj, equal to -2. This is most easily accomplished by choosing an x-coordinate of -1 and a hypotenuse of 2 (Figure 4.53a). We recognize this as a 30°–60°–90° triangle that determines an angle of 240°, which converts to 4p/3 radians (Figure 4.53b). Therefore the answer is 4p/3. y y 240° –1 x x [–6.5, 6.5] by [–3, 3] 2 FIGURE 4.52 The graphs of y = sec x and y = - 2 cos x. (Exploration 1) (a) (b) FIGURE 4.53 A reference triangle in the third quadrant (a) with hyp/adj = - 2 determines an angle (b) of 240 degrees, which converts to 4p/3 radians. (Example 3) Now try Exercise 29. 6965_CH04_pp319-402.qxd 364 1/14/10 1:50 PM Page 364 CHAPTER 4 Trigonometric Functions The Cosecant Function Important characteristics of the cosecant function can be inferred from the fact that it is the reciprocal of the sine function. Whenever sin x = 1, its reciprocal csc x is also 1. The graph of the cosecant function has asymptotes at the zeros of the sine function. The period of the cosecant function is 2p, the same as its reciprocal, the sine function. The graph of y = csc x is shown with the graph of y = sin x in Figure 4.54. A local maximum of y = sin x corresponds to a local minimum of y = csc x, while a local minimum of y = sin x corresponds to a local maximum of y = csc x. y 3 2 1 x –2π –3 FIGURE 4.54 Characteristics of the cosecant function are inferred from the fact that it is the reciprocal of the sine function. EXAMPLE 4 Solving a Trigonometric Equation Graphically Are Cosecant Curves Parabolas? Find the smallest positive number x such that x 2 = csc x. Figure 4.55 shows a parabola intersecting one of the infinite number of U-shaped curves that make up the graph of the cosecant function. In fact, the parabola intersects all of those curves that lie above the x-axis, since the parabola must spread out to cover the entire domain of y = x 2, which is all real numbers! The cosecant curves do not keep spreading out, as they are hemmed in by asymptotes. That means that the U-shaped curves in the cosecant function are not parabolas. SOLUTION There is no algebraic attack that looks hopeful, so we solve this equa- tion graphically. The intersection point of the graphs of y = x 2 and y = csc x that has the smallest positive x-coordinate is shown in Figure 4.55. We use the grapher to determine that x L 1.068. [–6.5, 6.5] by [–3, 3] FIGURE 4.55 A graphical solution of a trigonometric equation. (Example 3) Now try Exercise 39. To close this section, we summarize the properties of the six basic trigonometric functions in tabular form. The “n” that appears in several places should be understood as taking on all possible integer values: 0, ⫾ 1, ⫾2, ⫾3, Á . Summary: Basic Trigonometric Functions Function Period Domain sin x cos x tan x cot x sec x csc x 2p 2p p p 2p 2p All reals All reals x Z p/2 + np x Z np x Z p/2 + np x Z np Range 3 - 1, 14 3 - 1, 14 All reals All reals 1 - q , -1] ´ 31, q 2 1- q , -1] ´ 31, q 2 Asymptotes None None x = p/2 + np x = np x = p/2 + np x = np Zeros np p/2 + np np p/2 + np None None Even/Odd Odd Even Odd Odd Even Odd 6965_CH04_pp319-402.qxd 1/14/10 1:50 PM Page 365 SECTION 4.5 QUICK REVIEW 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant 365 (For help, go to Sections 1.2, 2.6, and 4.3.) Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–4, state the period of the function. 1. y = cos 2x 2. y = sin 3x 1 1 3. y = sin x 4. y = cos x 3 2 In Exercises 5–8, find the zeros and vertical asymptotes of the function. x + 5 x - 3 5. y = 6. y = x + 4 x - 1 7. y = x + 1 1x - 221x + 22 8. y = x + 2 x1x - 32 In Exercises 9 and 10, tell whether the function is odd, even, or neither. 1 9. y = x 2 + 4 10. y = x SECTION 4.5 EXERCISES In Exercises 1–4, identify the graph of each function. Use your understanding of transformations, not your graphing calculator. 3. Graphs of csc x and 3 csc 2x are shown. y 1. Graphs of one period of csc x and 2 csc x are shown. 10 8 6 4 2 y 10 8 6 4 2 y1 –π y2 –π y2 π y1 π x x –10 4. Graphs of cot x and cot 1x - 0.52 + 3 are shown. y 2. Graphs of two periods of 0.5 tan x and 5 tan x are shown. 10 8 6 4 y 10 8 6 4 2 –π y1 –π π y2 x y2 π –4 –6 –8 –10 y1 –8 –10 x In Exercises 5–12, describe the graph of the function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function. 5. y = tan 2x 6. y = - cot 3x 7. y = sec 3x 8. y = csc 2x 9. y = 2 cot 2x 11. y = csc 1x/22 10. y = 3 tan 1x/22 12. y = 3 sec 4x 6965_CH04_pp319-402.qxd 366 1/14/10 1:50 PM Page 366 CHAPTER 4 Trigonometric Functions In Exercises 13–16, match the trigonometric function with its graph. Then give the Xmin and Xmax values for the viewing window in which the graph is shown. Use your understanding of transformations, not your graphing calculator. 39. csc x = 2, 0 … x … 2p 40. tan x = 0.3, 0 … x … 2p 41. Writing to Learn The figure shows a unit circle and an angle t whose terminal side is in Quadrant III. y x2 + y2 = 1 P1(–a, –b) [?, ?] by [–10, 10] (b) [?, ?] by [–10, 10] (a) t t– π x P2(a, b) [?, ?] by [–10, 10] (d) [?, ?] by [–10, 10] (c) 13. y = - 2 tan x 14. y = cot x 15. y = sec 2x 16. y = - csc x In Exercises 17–20, analyze each function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 17. ƒ1x2 = cot x 18. ƒ1x2 = sec x 20. ƒ1x2 = tan 1x/22 19. ƒ1x2 = csc x In Exercises 21–28, describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. 21. y = 3 tan x 22. y = - tan x 23. y = 3 csc x 24. y = 2 tan x 1 25. y = - 3 cot x 2 p 27. y = - tan x + 2 2 26. y = - 2 sec (a) If the coordinates of point P2 are 1a, b2, explain why the coordinates of point P1 on the circle and the terminal side of angle t - p are 1- a, - b2. b (b) Explain why tan t = . a (c) Find tan 1t - p2, and show that tan t = tan1t - p2. (d) Explain why the period of the tangent function is p. (e) Explain why the period of the cotangent function is p. 42. Writing to Learn Explain why it is correct to say y = tan x is the slope of the terminal side of angle x in standard position. P is on the unit circle. y 1 x 2 P(cos x, sin x) x 28. y = 2 tan px - 2 x x In Exercises 29–34, solve for x in the given interval. You should be able to find these numbers without a calculator, using reference triangles in the proper quadrants. 29. sec x = 2, 0 … x … p/2 30. csc x = 2, p/2 … x … p 31. cot x = - 13, p/2 … x … p 32. sec x = - 12, p … x … 3p/2 43. Periodic Functions Let ƒ be a periodic function with period p. That is, p is the smallest positive number such that 33. csc x = 1, 2p … x … 5p/2 ƒ1x + p2 = ƒ1x2 34. cot x = 1, - p … x … - p/2 In Exercises 35–40, use a calculator to solve for x in the given interval. p 0 … x … 35. tan x = 1.3, 2 p 0 … x … 36. sec x = 2.4, 2 3p … x … 2p 37. cot x = - 0.6, 2 3p 38. csc x = - 1.5, p … x … 2 for any value of x in the domain of ƒ. Show that the reciprocal 1/ƒ is periodic with period p. 44. Identities Use the unit circle to give a convincing argument for the identity. (a) sin 1t + p2 = - sin t (b) cos 1t + p2 = - cos t (c) Use (a) and (b) to show that tan 1t + p2 = tan t. Explain why this is not enough to conclude that the period of tangent is p. 6965_CH04_pp319-402.qxd 1/14/10 1:50 PM Page 367 SECTION 4.5 45. Lighthouse Coverage The Bolivar Lighthouse is located on a small island 350 ft from the shore of the mainland as shown in the figure. (a) Express the distance d as a function of the angle x. (b) If x is 1.55 rad, what is d ? Graphs of Tangent, Cotangent, Secant, and Cosecant 54. Multiple Choice sects the graph of y = (A) x. (B) x 2. The graph of y = sec x never inter(C) csc x. (D) cos x. (E) sin x. 55. Multiple Choice tion y = k csc x? If k Z 0, what is the range of the func- (A) 3-k, k4 (B) 1- k, k2 (C) 1- q , -k2 ´ 1k, q 2 350 ft 367 (D) 1- q , -k4 ´ 3k, q 2 (E) 1- q , - 1/k4 ´ 31/k, q 2 x d 56. Multiple Choice The graph of y = csc x has the same set of asymptotes as the graph of y = 46. Hot-Air Balloon A hot-air balloon over Albuquerque, New Mexico, is being blown due east from point P and traveling at a constant height of 800 ft. The angle y is formed by the ground and the line of vision from P to the balloon. This angle changes as the balloon travels. (a) Express the horizontal distance x as a function of the angle y. (b) When the angle is p/20 rad, what is its horizontal distance from P? (c) An angle of p/20 rad is equivalent to how many degrees? Wind blowing due east (A) sin x. (B) tan x. (D) sec x. (E) csc 2x. Explorations In Exercises 57 and 58, graph both ƒ and g in the 3-p, p4 by 3- 10, 104 viewing window. Estimate values in the interval 3-p, p4 for which ƒ 7 g. 57. ƒ1x2 = 5 sin x and g1x2 = cot x 58. ƒ1x2 = - tan x and g1x2 = csc x 59. Writing to Learn Graph the function ƒ1x2 = - cot x on the interval 1-p, p2. Explain why it is correct to say that ƒ is increasing on the interval 10, p2, but it is not correct to say that ƒ is increasing on the interval 1 -p, p2. 60. Writing to Learn Graph functions ƒ1x2 = - sec x and g1x2 = 800 ft (C) cot x. 1 x - 1p/22 simultaneously in the viewing window 30, p4 by 3- 10, 104. Discuss whether you think functions ƒ and g are equivalent. 61. Write csc x as a horizontal translation of sec x. P y 62. Write cot x as the reflection about the x-axis of a horizontal translation of tan x. x In Exercises 47–50, find approximate solutions for the equation in the interval - p 6 x 6 p. 47. tan x = csc x 48. sec x = cot x 49. sec x = 5 cos x 50. 4 cos x = tan x Standardized Test Questions Extending the Ideas 63. Group Activity Television Coverage A television camera is on a platform 30 m from the point on High Street where the Worthington Memorial Day Parade will pass. Express the distance d from the camera to a particular parade float as a function of the angle x, and graph the function over the interval -p/2 6 x 6 p/2. High Street 51. True or False The function ƒ1x2 = tan x is increasing on the interval 1- q , q 2. Justify your answer. 52. True or False If x = a is an asymptote of the secant function, then cot a = 0. Justify your answer. You should answer these questions without using a calculator. 53. Multiple Choice The graph of y = cot x can be obtained by a horizontal shift of the graph of y = (A) - tan x. (B) - cot x. (C) sec x. (D) tan x. (E) csc x. Float d Camera x 30 m 6965_CH04_pp319-402.qxd 368 1/14/10 1:50 PM Page 368 CHAPTER 4 Trigonometric Functions 64. What’s in a Name? The word sine comes from the Latin word sinus, which means “bay” or “cove.” It entered the language through a mistake (variously attributed to Gerardo of Cremona or Robert of Chester) in translating the Arabic word “jiba” (chord) as if it were “jaib” (bay). This was due to the fact that the Arabs abbreviated their technical terms, much as we do today. Imagine someone unfamiliar with the technical term “cosecant” trying to reconstruct the English word that is abbreviated by “csc.” It might well enter their language as their word for “cascade.” The names for the other trigonometric functions can all be explained. (a) Cosine means “sine of the complement.” Explain why this is a logical name for cosine. (b) In the figure below, BC is perpendicular to OC, which is a radius of the unit circle. By a familiar geometry theorem, BC is tangent to the circle. OB is part of a secant that intersects the unit circle at A. It lies along the terminal side of an angle of t radians in standard position. Write the coordinates of A as functions of t. where h is the height of the liquid in the tube, r (rho) is the density of the liquid, g = 9.8 m/sec2 is the acceleration due to gravity, r is the radius of the tube, and f (phi) is the angle of contact between the tube and the liquid’s surface. Whole blood has a surface tension of 0.058 N/m (newton per meter) and a density of 1050 kg/m3. Suppose that blood rises to a height of 1.5 m in a capillary blood vessel of radius 4.7 * 10 - 6 m. What is the contact angle between the capillary vessel and the blood surface? 31N = 11kg # m2/sec24 66. Advanced Curve Fitting A researcher has reason to believe that the data in the table below can best be described by an algebraic model involving the secant function: y = a sec1bx2 Unfortunately, her calculator will do only sine regression. She realizes that the following two facts will help her: 1 1 1 = = cos 1bx2 y a a sec 1bx2 and cos 1bx2 = sin abx + (c) Use similar triangles to find length BC as a trig function of t. (d) Use similar triangles to find length OB as a trig function of t. (e) Use the results from parts (a), (c), and (d) to explain where the names “tangent, cotangent, secant,” and “cosecant” came from. y B D C x 65. Capillary Action A film of liquid in a thin (capillary) tube has surface tension g (gamma) given by g = 1 hrgr sec f, 2 1 1 p = sin abx + b. y a 2 (b) Store the x-values in the table in L1 in your calculator and the y-values in L2. Store the reciprocals of the y-values in L3. Then do a sine regression for L3 11/y2 as a function of L1 1x2. Write the regression equation. (d) Write the secant model: y = a sec 1bx2 Does the curve fit the 1L1, L22 scatter plot? 1 t (a) Use these two facts to show that (c) Use the regression equation in (b) to determine the values of a and b. A O p b 2 x 1 2 3 4 y 5.0703 5.2912 5.6975 6.3622 x 5 6 7 8 y 7.4359 9.2541 12.716 21.255