Inverse Trigonometric Functions 343
Total Page：16
File Type：pdf, Size：1020Kb
333202_0407.qxd 12/7/05 11:10 AM Page 343 Section 4.7 Inverse Trigonometric Functions 343 4.7 Inverse Trigonometric Functions What you should learn •Evaluate and graph the Inverse Sine Function inverse sine function. Recall from Section 1.9 that, for a function to have an inverse function, it must •Evaluate and graph the be one-to-one—that is, it must pass the Horizontal Line Test. From Figure 4.71, other inverse trigonometric you can see that y ϭ sin x does not pass the test because different values of x functions. yield the same y-value. •Evaluate and graph the compositions of y trigonometric functions. y = sin x 1 Why you should learn it x −π π You can use inverse trigonometric −1 functions to model and solve real-life problems. For instance, sin x has an inverse function in Exercise 92 on page 351, an on this interval. inverse trigonometric function can be used to model the angle FIGURE 4.71 of elevation from a television camera to a space shuttle launch. However, if you restrict the domain to the interval Ϫ͞2 ≤ x ≤ ͞2 (corresponding to the black portion of the graph in Figure 4.71), the following properties hold. 1. On the interval ͓Ϫ͞2, ͞2͔, the function y ϭ sin x is increasing. 2. On the interval ͓Ϫ͞2, ͞2͔, y ϭ sin x takes on its full range of values, Ϫ1 ≤ sin x ≤ 1. 3. On the interval ͓Ϫ͞2, ͞2͔, y ϭ sin x is one-to-one. So, on the restricted domain Ϫ͞2 ≤ x ≤ ͞2,y ϭ sin x has a unique inverse function called the inverse sine function. It is denoted by y ϭ arcsin x or y ϭ sinϪ1 x. The notation sinϪ1 x is consistent with the inverse function notation f Ϫ1͑x͒. The arcsin x notation (read as “the arcsine of x”) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin x means the angle (or arc) whose sine is x. Both notations, arcsin x and sinϪ1 x, are commonly used in mathematics, so remember that sinϪ1 x denotes the inverse NASA sine function rather than 1͞sin x. The values of arcsin x lie in the interval Ϫ͞2 ≤ arcsin x ≤ ͞2. The graph of y ϭ arcsin x is shown in Example 2. Definition of Inverse Sine Function The inverse sine function is defined by When evaluating the inverse sine y ϭ arcsin x if and only if sin y ϭ x function, it helps to remember where Ϫ1 ≤ x ≤ 1 and Ϫ͞2 ≤ y ≤ ͞2. The domain of y ϭ arcsin x is the phrase “the arcsine of x is ͓Ϫ1, 1͔, and the range is ͓Ϫ͞2, ͞2͔. the angle (or number) whose sine is x.” 333202_0407.qxd 12/7/05 11:10 AM Page 344 344 Chapter 4 Trigonometry Example 1 Evaluating the Inverse Sine Function If possible, find the exact value. As with the trigonometric functions, much of the work 1 Ί3 a.arcsinϪ b.sinϪ1 c. sinϪ1 2 with the inverse trigonometric 2 2 functions can be done by exact calculations rather than by cal- Solution culator approximations. Exact 1 a. Because sinϪ ϭϪ for Ϫ ≤ y ≤ , it follows that calculations help to increase 6 2 2 2 your understanding of the inverse functions by relating 1 arcsinϪ ϭϪ . Angle whose sine is Ϫ1 them to the right triangle 2 6 2 definitions of the trigonometric Ί3 functions. b. Because sin ϭ for Ϫ ≤ y ≤ , it follows that 3 2 2 2 Ί Ϫ 3 sin 1 ϭ . Angle whose sine is Ί3͞2 2 3 c. It is not possible to evaluate y ϭ sinϪ1 x when x ϭ 2 because there is no angle whose sine is 2. Remember that the domain of the inverse sine function is ͓Ϫ1, 1͔. Now try Exercise 1. You may wish to illustrate the reflections Example 2 Graphing the Arcsine Function of y ϭ sin x and y ϭ arcsin x about the ϭ line y x. Consider using a graphing Sketch a graph of utility to do this. y ϭ arcsin x. Solution By definition, the equations y ϭ arcsin x and sin y ϭ x are equivalent for y Ϫ͞2 ≤ y ≤ ͞2. So, their graphs are the same. From the interval ͓Ϫ͞2, ͞2͔, you can assign values to y in the second equation to make a table π 1, of values. Then plot the points and draw a smooth curve through the points. π ( 2 ) 2 2 π Ϫ Ϫ Ϫ , y 0 ()2 4 2 4 6 6 4 2 π 1, Ί2 1 1 Ί2 (0, 0) ()2 6 x ϭ sin y Ϫ1 Ϫ Ϫ 01 x 2 2 2 2 π 1 − 1, − ()2 6 y = arcsin x The resulting graph for y ϭ arcsin x is shown in Figure 4.72. Note that it is the 2 π − , − reflection (in the line y ϭ x) of the black portion of the graph in Figure 4.71. Be π ( ) − 2 4 sure you see that Figure 4.72 shows the entire graph of the inverse sine function. π 2 −1, − ϭ ͓Ϫ ͔ ( 2 ) Remember that the domain of y arcsin x is the closed interval 1, 1 and the range is the closed interval ͓Ϫ͞2, ͞2͔. FIGURE 4.72 Now try Exercise 17. 333202_0407.qxd 12/7/05 11:10 AM Page 345 Section 4.7 Inverse Trigonometric Functions 345 Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval 0 ≤ x ≤ , as shown in Figure 4.73. y y = cos x x −π π π 2π −1 2 cos x has an inverse function on this interval. FIGURE 4.73 Consequently, on this interval the cosine function has an inverse function—the inverse cosine function—denoted by y ϭ arccos x or y ϭ cosϪ1 x. Similarly, you can define an inverse tangent function by restricting the domain of y ϭ tan x to the interval ͑Ϫ͞2, ͞2͒. The following list summarizes the definitions of the three most common inverse trigonometric functions. The remaining three are defined in Exercises 101–103. Definitions of the Inverse Trigonometric Functions You may need to point out to your Function Domain Range students that the range for each of these functions is different. Students y ϭ arcsin x if and only if sin y ϭ x Ϫ1 ≤ x ≤ 1 Ϫ ≤ y ≤ should know these ranges well to 2 2 ensure that their answers are within the correct range. Referencing the y ϭ arccos x if and only if cos y ϭ x Ϫ1 ≤ x ≤ 1 0 ≤ y ≤ graphs of the inverse trigonometric functions may also be helpful. y ϭ arctan x if and only if tan y ϭ x Ϫϱ < x < ϱ Ϫ < y < 2 2 The graphs of these three inverse trigonometric functions are shown in Figure 4.74. y y y π 2 π π 2 y = arcsin x yx= arctan yx= arccos x π x −1 1 2 −2 −1 12 π π x − − − 2 2 1 1 DOMAIN: ͓Ϫ1, 1͔ DOMAIN: ͓Ϫ1, 1͔ DOMAIN: ͑Ϫϱ, ϱ͒ ͓Ϫ ͔ ͓ ͔ ͑Ϫ ͒ RANGE: 2, 2 RANGE: 0, RANGE: 2, 2 FIGURE 4.74 333202_0407.qxd 12/8/05 8:25 AM Page 346 346 Chapter 4 Trigonometry Example 3 Evaluating Inverse Trigonometric Functions Find the exact value. Ί2 a.arccos b. cosϪ1͑Ϫ1͒ 2 c.arctan 0 d. tanϪ1͑Ϫ1͒ Solution a. Because cos͑͞4͒ ϭ Ί2͞2, and ͞4 lies in ͓0, ͔, it follows that Ί2 arccos ϭ . Angle whose cosine is Ί2͞2 2 4 b. Because cos ϭϪ1, and lies in ͓0, ͔, it follows that Ϫ cos 1͑Ϫ1͒ ϭ . Angle whose cosine is Ϫ1 c. Because tan 0 ϭ 0, and 0 lies in ͑Ϫ͞2, ͞2͒, it follows that arctan 0 ϭ 0. Angle whose tangent is 0 d. Because tan͑Ϫ͞4͒ ϭϪ1, and Ϫ͞4 lies in ͑Ϫ͞2, ͞2͒, it follows that Ϫ tan 1͑Ϫ1͒ ϭϪ . Angle whose tangent is Ϫ1 4 Now try Exercise 11. Example 4 Calculators and Inverse Trigonometric Functions Use a calculator to approximate the value (if possible). a. arctan͑Ϫ8.45͒ b. sinϪ1 0.2447 c. arccos 2 Solution Function Mode Calculator Keystrokes a. arctan͑Ϫ8.45͒ RadianTANϪ1 ͧ ͧ؊ͨ 8.45 ͒ ENTER ͑Ϫ ͒ Ϸ Ϫ It is important to remember that From the display, it follows that arctan 8.45 1.453001. the domain of the inverse sine b. sinϪ1 0.2447 RadianSINϪ1 ͧ 0.2447 ͒ ENTER function and the inverse cosine Ϫ From the display, it follows that sin 1 0.2447 Ϸ 0.2472103. function is ͓Ϫ1, 1͔, as indicated in Example 4(c). c. arccos 2 RadianCOSϪ1 ͧ 2 ͒ ENTER In real number mode, the calculator should display an error message because the domain of the inverse cosine function is ͓Ϫ1, 1͔. Now try Exercise 25. In Example 4, if you had set the calculator to degree mode, the displays would have been in degrees rather than radians. This convention is peculiar to calculators. By definition, the values of inverse trigonometric functions are always in radians. 333202_0407.qxd 12/7/05 11:10 AM Page 347 Section 4.7 Inverse Trigonometric Functions 347 Compositions of Functions Recall from Section 1.9 that for all x in the domains of f and f Ϫ1, inverse func- tions have the properties f ͑ f Ϫ1͑x͒͒ ϭ x and f Ϫ1͑ f ͑x͒͒ ϭ x.