4.7 Inverse Trigonometric. Functions

Transcription

1 78 CHAPTER 4 Trigonometric Functions What ou ll learn about Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric Functions Applications of Inverse Trigonometric Functions... and wh Inverse trig functions can be used to solve trigonometric equations. 4.7 Inverse Trigonometric Functions Inverse Sine Function You learned in Section.4 that each function has an inverse relation, and that this inverse relation is a function onl if the original function is one-to-one. The si basic trigonometric functions, being periodic, fail the horizontal line test for one-to-oneness rather spectacularl. However, ou also learned in Section.4 that some functions are important enough that we want to stud their inverse behavior despite the fact that the are not one-to-one. We do this b restricting the domain of the original function to an interval on which it is one-to-one, then finding the inverse of the restricted function. (We did this when defining the square root function, which is the inverse of the function = restricted to a nonnegative domain.) If ou restrict the domain of = sin to the interval -p/, p/4, as shown in Figure 4.69a, the restricted function is one-to-one. The inverse sine function = sin - is the inverse of this restricted portion of the sine function (Figure 4.69b). [, ] b [.,.] (a) [.5,.5] b [.7,.7] (b) FIGURE 4.69 The (a) restriction of = sin is one-to-one and (b) has an inverse, = sin -. π B the usual inverse relationship, the statements = sin - and = sin are equivalent for -values in the restricted domain -p/, p/4 and -values in -, 4. This means that sin - can be thought of as the angle between -p/ and p/ whose sine is. Since angles and directed arcs on the unit circle have the same measure, the angle sin - is also called the arcsine of. π Inverse Sine Function (Arcsine Function) The unique angle in the interval -p/, p/4 such that sin = is the inverse sine (or arcsine) of, denoted sin - or arcsin. The domain of = sin - is -, 4 and the range is -p/, p/4. FIGURE 4.70 The values of = sin - will alwas be found on the right-hand side of the unit circle, between -p/ and p/. It helps to think of the range of = sin - as being along the right-hand side of the unit circle, which is traced out as angles range from -p/ to p/ (Figure 4.70).

2 SECTION 4.7 Inverse Trigonometric Functions 79 EXAMPLE Evaluating sin - without a Calculator Find the eact value of each epression without a calculator. (a) (b) (c) sin - a p sin- a sin- a - b b b (d) (e) sin - asin a 5p sin- asin a p 9 bb 6 bb sin - / = p/6. (Eam- sin - - / = -p/. FIGURE 4.7 ple a) FIGURE 4.7 (Eample b) SOLUTION (a) Find the point on the right half of the unit circle whose -coordinate is / and draw a reference triangle (Figure 4.7). We recognize this as one of our special ratios, and the angle in the interval -p/, p/4 whose sine is / is p/6. Therefore sin - a b = p 6. (b) Find the point on the right half of the unit circle whose -coordinate is - / and draw a reference triangle (Figure 4.7). We recognize this as one of our special ratios, and the angle in the interval -p/, p/] whose sine is - / is -p/. Therefore sin - a - b = - p. (c) sin - p/ does not eist, because the domain of sin - is -, 4 and p/ 7. (d) Draw an angle of p/9 in standard position and mark its -coordinate on the -ais (Figure 4.7). The angle in the interval -p/, p/4 whose sine is this number is p/9. Therefore sin - asin a p 9 bb = p 9. (e) Draw an angle of 5p/6 in standard position (notice that this angle is not in the interval -p/, p/4) and mark its -coordinate on the -ais. (See Figure 4.74 on the net page.) The angle in the interval -p/, p/4 whose sine is this number is p - 5p/6 = p/6. Therefore sin - asin a 5p 6 bb = p 6. Now tr Eercise. sin π 9 FIGURE 4.7 sin - sin p/9 = p/9. (Eample d) EXAMPLE Evaluating sin - with a Calculator Use a calculator in radian mode to evaluate these inverse sine values: (a) sin (b) sin - sin.49p SOLUTION (a) (b) sin = Á L sin - sin.49p = Á L -.59 Although this is a calculator answer, we can use it to get an eact answer if we are alert enough to epect a multiple of p. Divide the answer b p: Ans/p = Therefore, we conclude that sin - sin.49p = -0.49p. You should also tr to work Eample b without a calculator. It is possible! Now tr Eercise 9.

3 80 CHAPTER 4 Trigonometric Functions 5π sin 6 Inverse Cosine and Tangent Functions If ou restrict the domain of = cos to the interval 0, p4, as shown in Figure 4.75a, the restricted function is one-to-one. The inverse cosine function = cos - is the inverse of this restricted portion of the cosine function (Figure 4.75b). FIGURE 4.74 sin - sin 5p/6 = p/6. (Eample e) What about the Inverse Composition Rule? Does Eample e violate the Inverse Composition Rule of Section.4? That rule guarantees that ƒ - ƒ = for ever in the domain of ƒ. Keep in mind, however, that the domain of f might need to be restricted in order for ƒ - to eist. That is certainl the case with the sine function. So Eample e does not violate the Inverse Composition Rule, because that rule does not appl at = 5p/6. It lies outside the (restricted) domain of sine. [, 4] b [.4,.4] (a) [, ] b [,.5] (b) FIGURE 4.75 The (a) restriction of = cos is one-to-one and (b) has an inverse, = cos -. B the usual inverse relationship, the statements = cos - and = cos are equivalent for -values in the restricted domain 0, p4 and -values in -, 4. This means that cos - can be thought of as the angle between 0 and p whose cosine is. The angle cos - is also the arccosine of. Inverse Cosine Function (Arccosine Function) The unique angle in the interval 0, p4 such that cos = is the inverse cosine (or arccosine) of, denoted cos or arccos. The domain of = cos - is -, 4 and the range is 0, p4. FIGURE 4.76 The values of = cos - will alwas be found on the top half of the unit circle, between 0 and p. 0 It helps to think of the range of = cos - as being along the top half of the unit circle, which is traced out as angles range from 0 to p (Figure 4.76). If ou restrict the domain of = tan to the interval -p/, p/, as shown in Figure 4.77a, the restricted function is one-to-one. The inverse tangent function = tan - is the inverse of this restricted portion of the tangent function (Figure 4.77b). [, ] b [, ] (a) [4, 4] b [.8,.8] (b) FIGURE 4.77 The (a) restriction of = tan is one-to-one and (b) has an inverse, = tan -.

4 SECTION 4.7 Inverse Trigonometric Functions 8 π B the usual inverse relationship, the statements = tan - and = tan are equivalent for -values in the restricted domain -p/, p/ and -values in - q, q. This means that tan - can be thought of as the angle between -p/ and p/ whose tangent is. The angle tan - is also the arctangent of. π FIGURE 4.78 The values of = tan - will alwas be found on the right-hand side of the unit circle, between (but not including) -p/ and p/. Inverse Tangent Function (Arctangent Function) The unique angle in the interval -p/, p/ such that tan = is the inverse tangent (or arctangent) of, denoted tan or arctan. The domain of = tan - is - q, q and the range is -p/, p/. It helps to think of the range of = tan - as being along the right-hand side of the unit circle (minus the top and bottom points), which is traced out as angles range from -p/ to p/ (noninclusive) (Figure 4.78). FIGURE 4.79 cos - - / = p/4. (Eample a) EXAMPLE Evaluating Inverse Trig Functions without a Calculator Find the eact value of the epression without a calculator. (a) cos - a - b (b) tan - (c) cos - cos -. SOLUTION (a) Find the point on the top half of the unit circle whose -coordinate is - / and draw a reference triangle (Figure 4.79). We recognize this as one of our special ratios, and the angle in the interval 0, p4 whose cosine is - / is p/4. Therefore cos - a - b = p 4. (b) Find the point on the right side of the unit circle whose -coordinate is times its -coordinate and draw a reference triangle (Figure 4.80). We recognize this as one of our special ratios, and the angle in the interval -p/, p/ whose tangent is is p/. Therefore tan - = p. (c) Draw an angle of -. in standard position notice that this angle is not in the interval 0, p4 and mark its -coordinate on the -ais (Figure 4.8). The angle in the interval 0, p4 whose cosine is this number is.. Therefore cos - cos -. =.. Now tr Eercises 5 and 7.

5 8 CHAPTER 4 Trigonometric Functions cos (.) FIGURE 4.80 (Eample b) tan - = p/. FIGURE 4.8 cos - cos -. =.. (Eample c) EXAMPLE 4 Describing End Behavior Describe the end behavior of the function = tan -. SOLUTION We can get this information most easil b considering the graph of = tan -, remembering how it relates to the restricted graph of = tan. (See Figure 4.8.) [, ] b [, ] (a) [4, 4] b [.8,.8] (b) FIGURE 4.8 The graphs of (a) = tan (restricted) and (b) = tan -. The vertical asmptotes of = tan are reflected to become the horizontal asmptotes of = tan -. (Eample 4) When we reflect the graph of = tan about the line = to get the graph of = tan -, the vertical asmptotes = p/ become horizontal asmptotes = p/. We can state the end behavior accordingl: p lim : -q tan- = - and lim : +q tan- = p. Now tr Eercise. What about Arccot, Arcsec, and Arccsc? Because we alread have inverse functions for their reciprocals, we do not reall need inverse functions for cot, sec, and csc for computational purposes. Moreover, the decision of how to choose the range of arcsec and arccsc is not as straightforward as with the other functions. See Eercises 6, 7, and 7. Composing Trigonometric and Inverse Trigonometric Functions We have alread seen the need for caution when appling the Inverse Composition Rule to the trigonometric functions and their inverses (Eamples e and c). The following equations are alwas true whenever the are defined: sin sin - = cos (cos - = tan tan - =

6 SECTION 4.7 Inverse Trigonometric Functions 8 On the other hand, the following equations are true onl for -values in the restricted domains of sin, cos, and tan: sin - sin = cos - cos = tan - tan = An even more interesting phenomenon occurs when we compose inverse trigonometric functions of one kind with trigonometric functions of another kind, as in sin tan -. Surprisingl, these trigonometric compositions reduce to algebraic functions that involve no trigonometr at all! This curious situation has profound implications in calculus, where it is sometimes useful to decompose nontrigonometric functions into trigonometric components that seem to come out of nowhere. Tr Eploration. EXPLORATION Finding Inverse Trig Functions of Trig Functions In the right triangle shown to the right, the angle u is measured in radians.. Find tan u.. Find tan -.. Find the hpotenuse of the triangle as a function of. 4. Find sin tan - as a ratio involving no trig functions. 5. Find sec tan - as a ratio involving no trig functions. 6. If 6 0, then tan - is a negative angle in the fourth quadrant (Figure 4.8). Verif that our answers to parts (4) and (5) are still valid in this case. (, ) FIGURE 4.8 If 6 0, then u = tan - is an angle in the fourth quadrant. (Eploration ) FIGURE 4.84 A triangle in which u = sin -. (Eample 5) EXAMPLE 5 Composing Trig Functions with Arcsine Compose each of the si basic trig functions with sin - and reduce the composite function to an algebraic epression involving no trig functions. SOLUTION This time we begin with the triangle shown in Figure 4.84, in which u = sin -. (This triangle could appear in the fourth quadrant if were negative, but the trig ratios would be the same.) The remaining side of the triangle (which is cos u) can be found b the Pthagorean Theorem. If we denote the unknown side b s, we have s + = s = - s = - Note the ambiguous sign, which requires a further look. Since sin - is alwas in Quadrant I or IV, the horizontal side of the triangle can onl be positive. Therefore, we can actuall write s unambiguousl as -, giving us the triangle in Figure FIGURE 4.85 If u = sin -, then cos u = -. Note that cos u will be positive because sin - can onl be in Quadrant I or IV. (Eample 5) (continued)

7 84 CHAPTER 4 Trigonometric Functions We can now read all the required ratios straight from the triangle: sin sin - = cos sin - = - tan sin - = - csc sin - = sec sin - = - cot sin - = - Now tr Eercise FIGURE 4.86 The diagram for the stadium screen. (Eample 6) Angle (degrees) Repla Screen Viewing Angle (Dodger Stadium) Distance (feet) FIGURE 4.87 Viewing angle u as a function of distance from the wall. (Eample 6) Applications of Inverse Trigonometric Functions When an application involves an angle as a dependent variable, as in u = ƒ, then to solve for, it is natural to use an inverse trigonometric function and find = ƒ - u. EXAMPLE 6 Calculating a Viewing Angle The bottom of a 0-foot repla screen at Dodger Stadium is 45 feet above the plaing field. As ou move awa from the wall, the angle formed b the screen at our ee changes. There is a distance from the wall at which the angle is the greatest. What is that distance? SOLUTION Model The angle subtended b the screen is represented in Figure 4.86 b u, and u = u. Since, it follows that u = tan - - u tan u = 65/ 65/. Similarl, u = tan - 45/. Thus, u = tan tan- 45. Solve Graphicall Figure 4.87 shows a graph of u that reflects degree mode. The question about distance for maimum viewing angle can be answered b finding the -coordinate of the maimum point of this graph. Using grapher methods we see that this maimum occurs when L 54 feet. Therefore the maimum angle subtended b the repla screen occurs about 54 feet from the wall. Now tr Eercise 55. QUICK REVIEW 4.7 (For help, go to Section 4..) Eercise numbers with a gra background indicate problems that the authors have designed to be solved without a calculator. In Eercises 4, state the sign (positive or negative) of the sine, cosine, and tangent in the quadrant.. Quadrant I. Quadrant II. Quadrant III 4. Quadrant IV In Eercises 50, find the eact value. 5. sin p/6 6. tan p/4 7. cos p/ 8. sin p/ 9. sin -p/6 0. cos -p/

8 SECTION 4.7 Inverse Trigonometric Functions 85 SECTION 4.7 EXERCISES In Eercises, find the eact value... sin - sin- a a - b b. tan cos - 5. cos - a 6. tan - b 7. tan cos - a - b 9. sin - a - b 0. tan - -. cos - 0. sin - In Eercises 6, use a calculator to find the approimate value. Epress our answer in degrees.. sin arcsin tan cos In Eercises 70, use a calculator to find the approimate value. Epress our result in radians. 7. tan tan sin cos In Eercises and, describe the end behavior of the function.. = tan -. = tan - In Eercises, find the eact value without a calculator.. cos sin - / 4. sin tan - 5. sin - cos p/4 6. cos - cos 7p/4 7. cos sin - / 8. sin tan arcsin cos p/ 0. arccos tan p/4. cos tan -. tan - cos p In Eercises 6, analze each function for domain, range, continuit, increasing or decreasing behavior, smmetr, boundedness, etrema, asmptotes, and end behavior.. ƒ = sin - 4. ƒ = cos - 5. ƒ = tan - 6. ƒ = cot - (See graph in Eercise 67.) In Eercises 740, use transformations to describe how the graph of the function is related to a basic inverse trigonometric graph. State the domain and range. 7. ƒ = sin - 8. g = cos - 9. h = 5 tan - / 40. g = arccos / In Eercises 446, find the solution to the equation without a calculator. 4. sin sin - = 4. cos - cos = 4. sin = 44. tan = cos cos - = / 46. sin - sin = p/0 In Eercises 475, find an algebraic epression equivalent to the given epression. (Hint: Form a right triangle as done in Eample 5.) 47. sin tan cos tan tan arcsin 50. cot arccos 5. cos arctan 5. sin arccos 5. Group Activit Viewing Angle You are standing in an art museum viewing a picture. The bottom of the picture is ft above our ee level, and the picture is ft tall. Angle u is formed b the lines of vision to the bottom and to the top of the picture. Picture (a) Show that u = tan - a 4. b - tan- a b (b) Graph u in the 0, 54 b 0, 554 viewing window using degree mode. Use our grapher to show that the maimum value of u occurs approimatel 5. ft from the picture. (c) How far (to the nearest foot) are ou standing from the wall if u = 5? 54. Group Activit Analsis of a Lighthouse A rotating beacon L stands m across the harbor from the nearest point P along a straight shoreline. As the light rotates, it forms an angle u as shown in the figure, and illuminates a point Q on the same shoreline as P. Q P (a) Show that u = tan - a b. (b) Graph u in the viewing window -0, 04 b -90, 904 using degree mode. What do negative values of represent in the problem? What does a positive angle represent? A negative angle? (c) Find u when = 5. L

9 86 CHAPTER 4 Trigonometric Functions 55. Rising Hot-Air Balloon The hot-air balloon festival held each ear in Phoeni, Arizona, is a popular event for photographers. Jo Silver, an award-winning photographer at the event, watches a balloon rising from ground level from a point 500 ft awa on level ground. 500 ft (a) Write u as a function of the height s of the balloon. (b) Is the change in u greater as s changes from 0 ft to 0 ft, or as s changes from 00 ft to 0 ft? Eplain. (c) Writing to Learn In the graph of this relationship shown here, do ou think that the -ais represents the height s and the -ais angle u, or does the -ais represent angle u and the -ais height s? Eplain. [0, 500] b [5, 80] 56. Find the domain and range of each of the following functions. (a) ƒ = sin sin - (b) g = sin - + cos - (c) h = sin - sin (d) k = sin cos - (e) q = cos - sin Standardized Test Questions 57. True or False sin sin - = for all real numbers. Justif our answer. 58. True or False The graph of = arctan has two horizontal asmptotes. Justif our answer. You should answer these questions without using a calculator. 59. Multiple Choice cos - a - b = 7p p p (A) - (B) - (C) (D) p (E) 5p Multiple Choice sin - sin p = (A) -p (B) -p (C) 0 (D) p (E) p s 6. Multiple Choice sec tan - = (A) (B) csc (C) + (D) - (E) sin cos 6. Multiple Choice The range of the function ƒ = arcsin is (A) - q, q. (B) -,. (C) -, 4. (D) 0, p4. (E) -p/, p/4. Eplorations 6. Writing to Learn Using the format demonstrated in this section for the inverse sine, cosine, and tangent functions, give a careful definition of the inverse cotangent function. Hint: The range of = cot - is 0, p Writing to Learn Use an appropriatel labeled triangle to eplain wh sin - + cos - = p/. For what values of is the left-hand side of this equation defined? 65. Graph each of the following functions and interpret the graph to find the domain, range, and period of each function. Which of the three functions has points of discontinuit? Are the discontinuities removable or nonremovable? (a) = sin - sin (b) = cos - cos (c) = tan - tan Etending the Ideas 66. Practicing for Calculus Epress each of the following functions as an algebraic epression involving no trig functions. (a) cos sin - (b) sec tan - (c) sin cos - (d) -csc cot - (e) tan sec Arccotangent on the Calculator Most graphing calculators do not have a button for the inverse cotangent. The graph is shown below. Find an epression that ou can put into our calculator to produce a graph of = cot -. [, ] b [, 4]

10 SECTION 4.7 Inverse Trigonometric Functions Advanced Decomposition Decompose each of the following algebraic functions b writing it as a trig function of an arctrig function. (a) - (b) (c) Use elementar transformations and the arctangent function to construct a function with domain all real numbers that has horizontal asmptotes at = 4 and = Avoiding Ambiguities When choosing the right triangle in Eample 5, we used a hpotenuse of. It is sometimes necessar to use a variable quantit for the hpotenuse, in which case it is a good idea to use rather than, just in case is negative. (All of our definitions of the trig functions have involved triangles in which the hpotenuse is assumed to be positive.) (a) If we use the triangle below to represent u = sin - /, eplain wh side s must be positive regardless of the sign of. (b) Use the triangle in part (a) to find tan sin - /. (c) Using an appropriate triangle, find sin cos - /. (a) The graph on the left has one horizontal asmptote. What is it? (b) The graph on the right has two horizontal asmptotes. What are the? (c) Which of these graphs is also the graph of = cos - /? (d) Which of these graphs is increasing on both connected intervals? 7. Defining Arccosecant The range of the cosecant function is - q, -4, q, which must become the domain of the arccosecant function. The graph of = arccsc must therefore be the union of two unbroken curves. Two possible graphs with the correct domain are shown below. 4 (, π/) (, π/) (, π/) 4 (, π/) s 7. Defining Arcsecant The range of the secant function is - q, -4, q, which must become the domain of the arcsecant function. The graph of = arcsec must therefore be the union of two unbroken curves. Two possible graphs with the correct domain are shown below. (a) The graph on the left has one horizontal asmptote. What is it? (b) The graph on the right has two horizontal asmptotes. What are the? (c) Which of these graphs is also the graph of = sin - /? (d) Which of these graphs is decreasing on both connected intervals? (, π) (, π) (, 0) (, 0)