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Name __________________________________ Date ___________________ LESSON 5.4 STUDY GUIDE GOAL: Use trigonometric sum and difference formulas. EXAMPLE 1: Evaluate a trigonometric expression Find the exact value of cos 75 Solution cos 75 (45 30) cos 45 cos 30 sin 45 sin 30 2 3 2 1 2 2 2 2 6 2 4 Substitute 45° + 30° for 75°. Sum formula for cosine Evaluate. Simplify. Exercises for Example 1: Using the Sum and Difference Formulas Find the exact value of the expression. (answer in radical form not decimal form) 1. cos 15 2. tan 15 3. sin 75 4. cos 105 5. sin 12 6. cos 5 12 7. tan 13 12 8. 7 12 sin EXAMPLE 2: Find sin(a b) given that sin a 4 π 12 3π with 0 < a < and cos b With < b < 5 2 2 13 Solution Using a Pythagorean identity and quadrant signs gives cos a 3 and sin b 5 . 5 13 sin (a b) sin a cos b cos a sin b 4 12 3 5 5 14 5 13 33 65 Difference formula for sine Substitute. Simplify. Exercises for Example 2 1 Evaluate the expression given that cos a with < a < and sin b 2 3 2 3 π with 0 < b < . 2 9. sin(a b) 10. cos(a b) 11. tan(a b) 12. sin(a b) EXAMPLE 3: Simplify an expression Simplify the expression sin x π. 2 Solution sin x π sin x cos cos x sin 2 2 2 (sin x)(0) (cos x)(l) cos x Sum formula for sine Evaluate. Simplify. Exercises for Example 3 Simplify the expression. 13. cos x π 2 14. tan(x 2) EXAMPLE 4: Solve a trigonometric equation Solve: sin (x ) sin (x ) 1 for 0 x < 2. Solution sin(x ) sin(x ) 1 Write equation. sin x cos cos x sin sin x cos cos x sin 1 Use formulas. (sin x) (1) (cos x) (0) (sin x) (1) (cos x) (0) 1 Evaluate. 2 sin x 1 Simplify. 5π 1 π In the interval 0 x < 2, the solutions of x sin1 are and . 6 6 2 Exercise for Example 4 16. Solve sin x π sin x π = 1 for 0 x <2. 4 4