4.7 INVERSE TRIGONOMETRIC FUNCTIONS

Transcription

1 Section 4.7 Inverse Trigonometric Functions INVERSE TRIGONOMETRIC FUNCTIONS NASA What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse trigonometric functions. Evaluate and graph the compositions of trigonometric functions. Wh ou should learn it You can use inverse trigonometric functions to model and solve real-life problems. For instance, in Eercise 06 on page 49, an inverse trigonometric function can be used to model the angle of elevation from a television camera to a space shuttle launch. When evaluating the inverse sine function, it helps to remember the phrase the arcsine of is the angle (or number) whose sine is. Inverse Sine Function Recall from Section.9 that, for a function to have an inverse function, it must be one-to-one that is, it must pass the Horizontal Line Test. From Figure 4.7, ou can see that sin does not pass the test because different values of ield the same -value. FIGURE 4.7 However, if ou restrict the domain to the interval (corresponding to the black portion of the graph in Figure 4.7), the following properties hold.. On the interval,, the function sin is increasing.. On the interval,, sin takes on its full range of values, sin.. On the interval,, sin is one-to-one. So, on the restricted domain, sin has a unique inverse function called the inverse sine function. It is denoted b arcsin or sin has an inverse function on this interval. sin. = sin The notation sin is consistent with the inverse function notation f. The arcsin notation (read as the arcsine of ) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin means the angle (or arc) whose sine is. Both notations, arcsin and sin, are commonl used in mathematics, so remember that sin denotes the inverse sine function rather than sin. The values of arcsin lie in the interval arcsin. The graph of arcsin is shown in Eample. Definition of Inverse Sine Function The inverse sine function is defined b arcsin if and onl if where and. The domain of arcsin is,, and the range is,. sin

2 4 Chapter 4 Trigonometr Eample Evaluating the Inverse Sine Function As with the trigonometric functions, much of the work with the inverse trigonometric functions can be done b eact calculations rather than b calculator approimations. Eact calculations help to increase our understanding of the inverse functions b relating them to the right triangle definitions of the trigonometric functions. If possible, find the eact value. a. b. sin c. sin arcsin a. Because sin for it follows that 6, arcsin Angle whose sine is b. Because sin for it follows that, sin. 6. Angle whose sine is c. It is not possible to evaluate sin when because there is no angle whose sine is. Remember that the domain of the inverse sine function is,. Now tr Eercise 5. Eample Graphing the Arcsine Function,, 6, FIGURE (0, 0) (, ), 6 = arcsin (, 4 ) Sketch a graph of arcsin. B definition, the equations arcsin and sin are equivalent for. So, their graphs are the same. From the interval,, ou can assign values to in the second equation to make a table of values. Then plot the points and draw a smooth curve through the points. The resulting graph for arcsin is shown in Figure 4.7. Note that it is the reflection (in the line ) of the black portion of the graph in Figure 4.7. Be sure ou see that Figure 4.7 shows the entire graph of the inverse sine function. Remember that the domain of arcsin is the closed interval, and the range is the closed interval,. Now tr Eercise. 4 sin

3 Section 4.7 Inverse Trigonometric Functions 4 Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval 0, as shown in Figure 4.7. = cos FIGURE 4.7 Consequentl, on this interval the cosine function has an inverse function the inverse cosine function denoted b arccos or cos has an inverse function on this interval. cos. Similarl, ou can define an inverse tangent function b restricting the domain of tan to the interval,. The following list summarizes the definitions of the three most common inverse trigonometric functions. The remaining three are defined in Eercises 5 7. Definitions of the Inverse Trigonometric Functions Function Domain Range arcsin if and onl if sin arccos if and onl if cos 0 arctan if and onl if tan < < < < The graphs of these three inverse trigonometric functions are shown in Figure = arcsin = arccos = arctan DOMAIN:, RANGE:, FIGURE 4.74 DOMAIN:, RANGE: 0, DOMAIN:, RANGE:,

4 44 Chapter 4 Trigonometr Eample Evaluating Inverse Trigonometric Functions Find the eact value. a. arccos b. c. arctan 0 d. cos tan a. Because cos4, and 4 lies in 0,, it follows that arccos Angle whose cosine is b. Because cos, and lies in 0,, it follows that cos. Angle whose cosine is c. Because tan 0 0, and 0 lies in,, it follows that arctan Angle whose tangent is 0 d. Because tan4, and 4 lies in,, it follows that tan 4. Now tr Eercise 5. Angle whose tangent is Eample 4 Calculators and Inverse Trigonometric Functions Use a calculator to approimate the value (if possible). a. arctan8.45 b. sin c. arccos WARNING / CAUTION Remember that the domain of the inverse sine function and the inverse cosine function is,, as indicated in Eample 4(c). Function Mode Calculator Kestrokes a. arctan8.45 Radian 8.45 From the displa, it follows that arctan c. arccos Radian COS TAN ENTER b. sin Radian SIN ENTER From the displa, it follows that sin ENTER In real number mode, the calculator should displa an error message because the domain of the inverse cosine function is,. Now tr Eercise 9. In Eample 4, if ou had set the calculator to degree mode, the displas would have been in degrees rather than radians. This convention is peculiar to calculators. B definition, the values of inverse trigonometric functions are alwas in radians.

5 Section 4.7 Inverse Trigonometric Functions 45 You can review the composition of functions in Section.8. Compositions of Functions Recall from Section.9 that for all in the domains of f and f, inverse functions have the properties f f and f f. Inverse Properties of Trigonometric Functions If and, then sinarcsin and arcsinsin. If and 0, then cosarccos and arccoscos. If is a real number and < <, then tanarctan and arctantan. Keep in mind that these inverse properties do not appl for arbitrar values of and. For instance, arcsin sin In other words, the propert arcsinsin arcsin is not valid for values of outside the interval,.. Eample 5 Using Inverse Properties If possible, find the eact value. a. tanarctan5 b. arcsin sin 5 c. coscos a. Because 5 lies in the domain of the arctan function, the inverse propert applies, and ou have tanarctan5 5. b. In this case, 5 does not lie within the range of the arcsine function,. However, 5 is coterminal with 5 which does lie in the range of the arcsine function, and ou have arcsin sin 5 arcsin sin. c. The epression coscos is not defined because cos is not defined. Remember that the domain of the inverse cosine function is,. Now tr Eercise 49.

6 46 Chapter 4 Trigonometr Eample 6 shows how to use right triangles to find eact values of compositions of inverse functions. Then, Eample 7 shows how to use right triangles to convert a trigonometric epression into an algebraic epression. This conversion technique is used frequentl in calculus. = 5 u = arccos Angle whose cosine is FIGURE () = 4 u = arcsin 5 5 Angle whose sine is 5 FIGURE 4.76 ( ( Eample 6 Find the eact value. a. b. tan arccos Evaluating Compositions of Functions a. If ou let u arccos then cos u,. Because cos u is positive, u is a first-quadrant angle. You can sketch and label angle u as shown in Figure Consequentl, tan arccos b. If ou let u arcsin 5, then sin u 5. Because sin u is negative, u is a fourthquadrant angle. You can sketch and label angle u as shown in Figure Consequentl, cos arcsin 5 cos arcsin 5 opp 5 tan u adj. Now tr Eercise 57. cos u adj hp 4 5. Eample 7 Some Problems from Calculus Write each of the following as an algebraic epression in. a. sinarccos, 0 b. cotarccos, 0 < u = arccos Angle whose cosine is FIGURE 4.77 () If ou let u arccos, then cos u, where. Because ou can sketch a right triangle with acute angle u, as shown in Figure From this triangle, ou can easil convert each epression to algebraic form. a. b. cos u adj hp sinarccos sin u opp hp 9, cotarccos cot u adj opp 9, Now tr Eercise < In Eample 7, similar arguments can be made for -values ling in the interval, 0.