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EXAMPLE 1 Evaluate a trigonometric expression Find the exact value of (a) sin 15° and (b) tan 7π . 12 Substitute 60° – 45° a. sin 15° = sin (60° – 45°) for 15°. = sin 60° cos 45° – cos 60° sin 45° Difference formula for sine = 3 ( 2 )– 1 ( 2 ) 2 2 2 2 = 6– 2 4 Evaluate. Simplify. EXAMPLE 1 Evaluate a trigonometric expression b. tan 7π = tan ( π + π ) 12 3 4 π π tan 3 + tan 4 = π π 1 – tan 3 tan 4 Substitute π + π For 7π . 3 12 4 Sum formula for tangent. 3 +1 =1– 3 1 Evaluate. = –2 – 3 Simplify. EXAMPLE 2 Use a difference formula 4 3π Find cos (a – b) given that cos a = – 5 with π < a < 2 and sin b = 5 with 0 < b < π . 13 2 SOLUTION Using a Pythagorean identity and quadrant signs gives sin a = – 3 and cos b = 12 . 13 5 cos (a – b) = cos a cos b + sin a sin b Difference formula for cosine 3 5 4 12 Substitute. = – ( ) + (– )( ) 5 13 5 13 63 Simplify. = – 65 GUIDED PRACTICE for Examples 1 and 2 Find the exact value of the expression. 1. sin 105° 3. ANSWER 6 + 4 2. 2+ 3 4. ANSWER 6 – 4 ANSWER 2 cos 75° 2 tan 5π 12 cos π 12 ANSWER 2 + 4 6 GUIDED PRACTICE 5. for Examples 1 and 2 8 π Find sin (a – b) given that sin a = 17 with 0 < a < 2 3π and cos b = – 24 with π < b < . 2 25 ANSWER 87 – 425 EXAMPLE 3 Simplify an expression Simplify the expression cos (x + π). cos (x + π) = cos x cos π – sin x sin π Sum formula for cosine = (cos x)(–1) – (sin x)(0) Evaluate. = – cos x Simplify. EXAMPLE 4 Solve a trigonometric equation Solve sin ( x + π ) + sin ( x – π ) = 1 for 0 ≤ x < 2π. 3 3 Write equation. sin ( x + π ) + sin ( x – π ) = 1 3 3 Use formulas. sin x cos π + cos x sin π + sin x cos π – cos x sin π = 1 3 3 3 3 1 3 1 3 sin x + cos x + sin x – 2 2 2 2 cos x = 1 Evaluate. sin x = 1 Simplify. ANSWER In the interval 0 ≤ x <2π, the only solution is x = π . 2 EXAMPLE 5 Solve a multi-step problem Daylight Hours The number h of hours of daylight for Dallas, Texas, and Anchorage, Alaska, can be approximated by the equations below, where t is the time in days and t = 0 represents January 1. On which days of the year will the two cities have the same amount of daylight? Dallas: h1 = 2 sin ( π t – 1.35) + 12.1 182 Anchorage: h2 = –6cos ( π t ) + 12.1 182 EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Solve the equation h1 = h2 for t. 2 sin ( π t – 1.35) + 12.1 = – 6 cos ( π t ) + 12.1 182 182 sin ( π t – 1.35) = – 3 cos ( π t ) 182 182 sin ( π t ) cos 1.35 – cos ( π t ) sin 1.35 = – 3 cos ( π t ) 182 182 182 sin ( π t ) (0.219) – cos ( π t ) (0.976) = – 3 cos ( π t ) 182 182 182 0.219 sin ( π t ) = – 2.024 cos ( π t ) 182 182 EXAMPLE 5 Solve a multi-step problem tan ( π t ) = – 9.242 182 πt –1 (– 9.242) + nπ = tan 182 πt – 1.463 + nπ 182 t STEP 2 – 84.76 + 182n Find the days within one year (365 days) for which Dallas and Anchorage will have the same amount of daylight. t – 84.76 + 182(1) 97, or on April 8 t – 84.76 + 182(2) 279, or on October 7 GUIDED PRACTICE for Examples 3, 4, and 5 Simplify the expression. 6. sin (x + 2π) ANSWER sin x 7. cos (x – 2π) ANSWER cos x 8. tan (x – π) ANSWER tan x GUIDED PRACTICE for Examples 3, 4, and 5 9. Solve 6 cos ( π t ) + 5 = – 24 sin ( π t + 22) + 5 for 0 ≤ t < 2π. 75 75 ANSWER about 5.65