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Chapter 4 Trigonometric Functions © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved 1 SECTION 4.6 Inverse Trigonometric Functions OBJECTIVES 1 2 3 4 5 Graph and apply the inverse sine function. Graph and apply the inverse cosine function. Graph and apply the inverse tangent function. Evaluate inverse trigonometric functions using a calculator. Find exact values of composite functions involving the inverse trigonometric functions. © 2010 Pearson Education, Inc. All rights reserved 2 INVERSE SINE FUNCTION If we restrict the domain of y = sin x to the interval , , 2 2 then it is a oneto-one function and its inverse is also a function. © 2010 Pearson Education, Inc. All rights reserved 3 INVERSE SINE FUNCTION The inverse function for y = sin x, x 2 2 is called the inverse sine, or arcsine, The graph is function. obtained by reflecting the graph of y = sin x, for x 2 2 in the line y = x. © 2010 Pearson Education, Inc. All rights reserved 4 INVERSE SINE FUNCTION y = sin–1 x means sin y = x, where –1 ≤ x ≤ 1 and 2 y 2 . Read y = sin–1 x as “y equals inverse sine at x.” The domain of y = sin–1 x is [–1, 1]. The range of y = sin–1 x is , . 2 2 © 2010 Pearson Education, Inc. All rights reserved 5 EXAMPLE 1 Finding the Exact Value for y = sin–1 x Find the exact values of y. 1 3 1 1 1 b. y sin c. y sin 3 a. y sin 2 2 Solution 3 3 a. y sin means sin y , y . 2 2 2 2 3 Since sin and is in the interval 3 2 3 3 1 we have y sin . , , 2 2 2 3 1 © 2010 Pearson Education, Inc. All rights reserved 6 EXAMPLE 1 Finding the Exact Value for y = sin–1 x Solution continued 1 1 b. y sin means sin y , y . 2 2 2 2 1 Since sin and is in the interval 6 2 6 1 1 we have y sin . , , 2 2 2 6 1 c. Since 3 is not in the domain of the inverse sine function, which is [–1, 1], sin–1 3 does not exist. © 2010 Pearson Education, Inc. All rights reserved 7 INVERSE COSINE FUNCTION If we restrict the domain of y = cos x to the interval [0, π], then it is a oneto-one function and its inverse is also a function. © 2010 Pearson Education, Inc. All rights reserved 8 INVERSE COSINE FUNCTION The inverse function for y = cos x, 0 x is called the inverse cosine, or arccosine, function. The graph is obtained by reflecting the graph of y = cos x, with 0 x in the line y = x. © 2010 Pearson Education, Inc. All rights reserved 9 INVERSE COSINE FUNCTION y = cos–1 x means cos y = x, where –1 ≤ x ≤ 1 and 0 y . Read y = cos–1 x as “y equals inverse cosine at x.” The domain of y = cos–1 x is [–1, 1]. The range of y = cos–1 x is 0, . © 2010 Pearson Education, Inc. All rights reserved 10 EXAMPLE 2 Finding the Exact Value for cos–1 x Find the exact values of y. 1 2 1 1 b. y cos a. y cos 2 2 Solution 2 2 a. y cos means cos y , 0 y . 2 2 2 Since cos and 0 , 4 2 4 2 1 we have y cos . 2 4 1 © 2010 Pearson Education, Inc. All rights reserved 11 EXAMPLE 2 Finding the Exact Value for cos–1 x Solution continued 1 1 b. y cos means cos y , 0 y . 2 2 1 2 1 2 Since cos and 0 , 3 2 3 1 2 we have y cos . 2 3 1 © 2010 Pearson Education, Inc. All rights reserved 12 INVERSE TANGENT FUNCTION If we restrict the domain of y = tan x to the interval , then , 2 2 it is a one-toone function and its inverse is also a function. © 2010 Pearson Education, Inc. All rights reserved 13 INVERSE TANGENT FUNCTION The inverse function for y = tan x, x 2 2 is called the inverse tangent, or arctangent, function. The graph is obtained by reflecting the graph of y = tan x, with x 2 2 line y = x. © 2010 Pearson Education, Inc. All rights reserved , in the 14 INVERSE TANGENT FUNCTION y = tan–1 x means tan y = x, where –∞ ≤ x ≤ ∞ and y . 2 2 Read y = tan–1 x as “y equals inverse tangent at x.” The domain of y = tan–1 x is [–∞, ∞]. The range of y = tan–1 x is , . 2 2 © 2010 Pearson Education, Inc. All rights reserved 15 EXAMPLE 3 Finding the Exact Value for tan–1 x Find the exact values of y. 1 a. y tan 0 b. y tan 1 3 Solution a. y tan 1 0 Since tan 0 0 and 2 0 2 , 1 we have y tan 0 0. © 2010 Pearson Education, Inc. All rights reserved 16 EXAMPLE 3 Finding the Exact Value for tan–1 x Solution continued b. y tan 1 3 Since tan 3 and , 3 2 3 2 we have y tan 1 3 3 . © 2010 Pearson Education, Inc. All rights reserved 17 INVERSE COTANGENT FUNCTION y = cot–1 x means cot y = x, where –∞ ≤ x ≤ ∞ and 0 y . INVERSE COSECANT FUNCTION y = csc–1 x means csc y = x, where |x| ≥ 1 and 2 y 2 , y 0. INVERSE SECANT FUNCTION y = sec–1 x means sec y = x, where |x| ≥ 1 and 0 y , y . 2 © 2010 Pearson Education, Inc. All rights reserved 18 EXAMPLE 4 Finding the Exact Value for csc–1 x Find the exact for y = csc−1 2. Solution Since and we have y = csc−1 2 = © 2010 Pearson Education, Inc. All rights reserved 19 USING A CALCULATOR WITH INVERSE TRIGONOMETRIC FUNCTIONS To find csc–1 1 x find sin . x To find sec–1 1 x find cos . x 1 1 1 . To find x start by finding tan x If x ≥ 0, this is the correct value. If x < 0, add π to get the correct value. cot–1 © 2010 Pearson Education, Inc. All rights reserved 1 20 EXAMPLE 5 Using a Calculator to Find the Values of Inverse Functions Use a calculator to find the value of y in radians rounded to four decimal places. a. y sin 1 0.75 b. y cot 1 2.8 c. y = cot−1 (−2.3) Solution Set the calculator to Radian mode. a. y sin 1 0.75 0.8481 1 1 1 b. y cot 2.8 tan 0.3430 2.8 c. y = cot−1 (−2.3) = π + tan−1 ≈ 2.7315 © 2010 Pearson Education, Inc. All rights reserved 21 EXAMPLE 6 Using a Calculator to Find the Values of Inverse Functions Use a calculator to find the value of y in degrees rounded to four decimal places. a. y tan 1 0.99 b. y sec 1 25 c. y = cot−1 (−1.3) Solution Set the calculator to Degree mode. 1 a. y sin 0.99 44.7121 1 1 1 b. y sec 25 cos 87.7076 25 c. y = cot−1 (−1.3) = 180º + tan−1 ≈ 142.4314º © 2010 Pearson Education, Inc. All rights reserved 22 COMPOSITION OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS © 2010 Pearson Education, Inc. All rights reserved 23 EXAMPLE 7 Finding the Exact Value of sin−1 (sin x) and cos−1 (cos x) Find the exact value of Solution a. Because we have © 2010 Pearson Education, Inc. All rights reserved 24 EXAMPLE 7 Finding the Exact Value of sin−1 (sin x) and cos−1 (cos x) Solution continued is not in the interval [0, π], but b. cos . So, © 2010 Pearson Education, Inc. All rights reserved 25 EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Find the exact value of 1 1 1 2 b. sin cos a. cos tan 4 3 Solution a. Let represent the radian measure of the 2 angle in , , with tan . 2 2 3 Since tan is positive, must be positive, 1 2 tan and 0 . 3 2 © 2010 Pearson Education, Inc. All rights reserved 26 EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Solution continued 2 tan 3 1 y 2 tan . x 3 So x = 3 and y = 2. r x 2 y2 32 2 2 9 4 13 3 3 3 13 1 2 cos tan cos 3 r 13 13 © 2010 Pearson Education, Inc. All rights reserved 27 EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Find the exact value of 1 1 b. sin cos 4 Solution b. Let represent the radian measure of the 1 angle in 0, , with cos . 4 Since cos is negative, we have 1 1 cos and . 4 2 © 2010 Pearson Education, Inc. All rights reserved 28 EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Solution continued x 1 1 1 cos cos r 4 4 So x = –1 and r = 4. 2 2 2 r x y 4 1 y 2 2 2 15 y 2 y 15 y 15 1 1 sin cos sin 4 r 4 © 2010 Pearson Education, Inc. All rights reserved 29 EXAMPLE 9 Finding the Rotation Angle for a Security Camera A security camera is to be installed 20 feet away from the center of a jewelry counter. The counter is 30 feet long. What angle, to the nearest degree, should the camera rotate through so that it scans the entire counter? © 2010 Pearson Education, Inc. All rights reserved 30 EXAMPLE 9 Finding the Rotation Angle for a Security Camera Solution The counter center , the camera , and a counter end form a right triangle. The angle at vertex A is where θ is the angle through which the camera rotates. © 2010 Pearson Education, Inc. All rights reserved 31 EXAMPLE 9 Finding the Rotation Angle for a Security Camera Solution continued Set the camera to 74º rotate through to scan the entire counter. © 2010 Pearson Education, Inc. All rights reserved 32