Copyright © 2017, 2013, 2009 Pearson Education, Inc. 6 Inverse Circular Functions and Trigonometric Equations Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1

Equations Involving Inverse Trigonometric Functions 6.4 Equations Involving Inverse Trigonometric Functions Solution for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

Solve y = 3 cos 2x for x, where x is restricted to the interval Example 1 SOLVING AN EQUATION FOR A SPECIFIED VARIABLE Solve y = 3 cos 2x for x, where x is restricted to the interval Divide by 3. Definition of arccosine Multiply by Because y = 3 cos 2x for x has period π, the restriction ensures that this function is one-to-one and has a one-to-one relationship.

Example 2 CHECK Solution set: {1} SOLVING AN EQUATION INVOLVING AN INVERSE TRIGONOMETRIC FUNCTION Divide by 2. Definition of arcsine CHECK Solution set: {1}

Example 3 and for u in quadrant I, SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS and for u in quadrant I, Substitute. Definition of arccosine

Example 3 SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS (continued) Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side.

Isolate one inverse function on one side of the equation: Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY Isolate one inverse function on one side of the equation: Definition of arcsine The arccosine function yields angles in quadrants I and II, so, by definition,

From equation (1) and by the definition of the arcsine function, Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.) Sine sum identity From equation (1) and by the definition of the arcsine function, and u lies in quadrant I.

From the triangle, we have Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.) From the triangle, we have Substitute. Multiply by 2. Subtract x. Square each side. Distribute, then add 3x2.

Now check the solution in the original equation. Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.) Divide by 4. Take the square root of each side. Choose the positive root because x > 0. Now check the solution in the original equation.