5.3 Solving Trigonometric Equations

Simple trigonometric equation Solve Solutions: General Solutions:

Techniques to Solve Trigonometric Equations: Combining like terms Factoring Extracting square roots Factoring quadratic types equations Rewriting with single variable Converting to quadratic type Functions of multiple angles Using inverse functions

Combining Like terms Example 1:

Extracting Square Roots

Factoring Solve Can we divide both sides by cot x ? No Solutions Why cant we divide by cotx? No Solutions General Solution

Trigonometric Equations of Quadratic type Quadratic in sin x: Quadratic in csc x: To solve quadratic equations, factor or use Quadratic Formula

Factoring Quadratic Equation Set each factor equal to zero

Changing to a Single Trigonometric Function Solve Use Pythagorean Identity Factor

Changing to a Single Trigonometric Function (continued) Set each factor equal to zero Solve for x General Solution

Converting to Quadratic Type Square both sides Produces extraneous solutions Verify possible solutions Solve Square both sides Pythagorean Identity

Converting to Quadratic Type (continued) Combine like terms Factor Set factors equal to zero and solve for x

Verifying Solutions: cos x+1 = sin x Check for Substitute π/2 for x Solution checks Solution does not check

Verifying Solutions: cos x+1 = sin x (continued) Check for Of the three possible solutions x=3π/2 is extraneous In the interval [0,2 π], there are two solutions: x = π/2 and x = π Solution checks

The end