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Download Equations Involving Inverse Trigonometric Functions
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Equations Involving Inverse Trigonometric Functions In this section, we use the inverse relationships listed below to solve equations: For if and only if For and if and only if For and if and only if and ( EXAMPLE 1: Solve the equation ) for x in the interval [ ( Solution: First, multiply both sides by 4: Since , ] ) , and we can use the inverse cosine relationship: ( ) . ( Finally, divide by 3: ( EXAMPLE 2: Solve the equation ) ) for x in the interval Solution: First, subtract 2 from both sides and multiply by −1: ( ) ( ) ( ) ( ) , so we can use the inverse tangent relationship: ( ( Finally, subtract π from both sides: EXAMPLE 3: Solve the equation Solution: First, divide both sides by 3: ( ) ( for x. ) ( Now use the inverse sine relationship: Finally, multiply by 2 and add 1: ) √ √ ) √ ) EXAMPLE 4: Find the exact solutions: Solution: Add to both sides: ( Using the inverse cosine relationship: Let . Then ) with , and . √ By the Pythagorean identity relating sine and cosine, √ Using the addition identity for sine, we can rewrite the equation as ( ) which implies that √ Thus, the equation becomes ( Squaring both sides: √ √ √ √ √ ) Since squaring both sides of the equation may introduce extraneous solutions, we check both solutions: First, check Next, check so . ( ) ( You can verify this using a graphing calculator: ) is a solution. ( ) so is not a solution