Simplify an expression 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.

Solve a trigonometric equation EXAMPLE 4 Solve a trigonometric equation Solve sin ( x + ) + sin ( x – ) = 1 for 0 ≤ x < 2π. π 3 Write equation. sin ( x + ) + sin ( x – ) π 3 = 1 Use formulas. sin x cos + cos x sin + sin x cos – cos x sin π 3 = 1 sin x + cos x + sin x – cos x 1 2 3 = 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 Anchorage: h2 π t 182 = 2 sin ( – 1.35) + 12.1 π t 182 = –6cos ( ) + 12.1

EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Solve the equation h1 = h2 for t. 2 sin ( – 1.35) + 12.1 π t 182 π t 182 = – 6 cos ( ) + 12.1 sin ( – 1.35) π t 182 π t 182 = – 3 cos ( ) sin ( ) cos 1.35 – cos ( ) sin 1.35 π t 182 π t 182 = – 3 cos ( ) sin ( ) (0.219) – cos ( ) (0.976) π t 182 π t 182 = – 3 cos ( ) 0.219 sin ( ) π t 182 π t 182 = – 2.024 cos ( )

EXAMPLE 5 Solve a multi-step problem π t 182 tan ( ) = – 9.242 π t 182 = tan –1 (– 9.242) + nπ π t 182 – 1.463 + nπ t – 84.76 + 182n STEP 2 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) 8. tan (x – π) tan x ANSWER sin x ANSWER 7. cos (x – 2π) cos x ANSWER

GUIDED PRACTICE for Examples 3, 4, and 5 π t 75 π t 75 9. Solve 6 cos ( ) + 5 = – 24 sin ( + 22) + 5 for 0 ≤ t < 2π. about 5.65 ANSWER