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6.4 Inverse Trigonometric Functions and Right Triangles Copyright © Cengage Learning. All rights reserved. Objectives ► The Inverse Sine, Inverse Cosine, and Inverse Tangent Functions ► Solving for Angles in Right Triangles ► Evaluating Expressions Involving Inverse Trigonometric Functions 2 Solving for Angles in Right Triangles 3 Example 3 – Finding an Angle in a Right Triangle Find the angle in the triangle shown in Figure 2. Figure 2 Solution: Since is the angle opposite the side of length 10 and the hypotenuse has length 50, we have sin sin Now we can use sin–1 to find : = sin–1 11.5 Definition of sin–1 Calculator (in degree mode) 4 Evaluating Expressions Involving Inverse Trigonometric Functions 5 Example 7 – Composing Trigonometric Functions and Their Inverses Find cos(sin–1 ). Solution 1: Let = sin–1 . Then is the number in the interval [– /2, /2] whose sine is . Let's interpret as an angle and draw a right triangle with as one of its acute angles, with opposite side 3 and hypotenuse 5 (see Figure 6). cos = Figure 6 6 Example 7 – Solution 1 cont’d The remaining leg of the triangle is found by the Pythagorean Theorem to be 4. From the figure we get cos(sin–1 ) = cos = So cos(sin–1 ) = . 7 Example 7 – Solution 2 cont’d It's easy to find sin(sin–1 ). In fact, by the cancellation properties of inverse functions, this value is exactly . To find cos(sin–1 ), we first write the cosine function in terms of the sine function. Let u = sin–1 . Since – /2 u /2, cos u is positive, and we can write the following: cos u = Cos2 u + sin2 u = 1 u= Property of inverse functions: 8 Example 7 – Solution 2 cont’d Calculate So cos(sin–1 ) = . 9