# EXAMPLE 3 Simplify an expression Simplify the expression cos (x + π). Sum formula for cosine cos (x + π) = cos x cos π – sin x sin π Evaluate. = (cos x)(–1)

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EXAMPLE 3 Simplify an expression Simplify the expression cos (x + π). Sum formula for cosine cos (x + π) = cos x cos π – sin x sin π Evaluate. = (cos x)(–1) – (sin x)(0) Simplify. = – cos x

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

EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Solve the equation h 1 = h 2 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 π t 182 = – 3 cos ( ) sin ( ) (0.219) – cos ( ) (0.976) π t 182 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) sin x ANSWER Sum formula for sine sin (x + 2π) = sin x cos 2π + cos x sin 2π Evaluate. = (sin x)(1) – (cos x)(0) Simplify. = sin x

GUIDED PRACTICE for Examples 3, 4, and 5 Simplify the expression. 7. cos (x – 2π) cos x ANSWER Difference formula for cosine cos (x – 2π) = cos x cos 2π + sin x sin 2π Evaluate. = (cos x)(1) + (sin x)(0) Simplify. = cos x

GUIDED PRACTICE for Examples 3, 4, and 5 Simplify the expression. 8. tan (x – π) tan x ANSWER Difference formula for tangent Evaluate. Simplify. = tan x tan (x – π) = tan x – tan π 1 + tan x tan π tan x – 0 1 + tan x ( 0 )

GUIDED PRACTICE for Examples 3, 4, and 5 9. Solve 6 cos ( ) + 5 = – 24 sin ( + 22) + 5 for 0 ≤ t < 2π. π t 75 π t 75 π t 75 π t 75 6 cos ( ) + 5 = – 24 sin ( + 22) + 5 π t 75 π t 75 6 cos ( ) = – 24 sin ( + 22) π t 75 π t 75 6 cos ( ) = – 24 (sin ( ) cos (22) + cos( )sin(22)) π t 75 π t 75 π t 75 6 cos ( ) + 24(cos( )(sin 22))= – 24 (sin ( ) cos (22)) π t 75 π t 75 π t 75 6 cos ( ) – 0.212(cos( ))= 23.99 (sin ( )) π t 75 π t 75 π t 75 5.788(cos( ))= 23.99 (sin ( ))

GUIDED PRACTICE for Examples 3, 4, and 5 about 5.65 ANSWER = π t 75 cos( ) sin ( ) π t 75 5.79 23.99 = 5.79 23.99 π t 75 tan ( ) 0.2413 = tan -1 ( ) π t 75 π t 75 0.2367 = 17.76 = πt 5.65 ≈ t

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