1. 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.

2. 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

3. 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)
π t
182
= –6cos ( )

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

5. EXAMPLE 5
Solve a multi-step problem
π t
182
tan ( )
= – 9.242
π t
182
= tan –1 (– 9.242) + nπ
π t
182
– nπ t
– n
STEP 2
Find the days within one year (365 days) for which Dallas and Anchorage will have the same amount of daylight. t
– (1)
97, or on April 8 t
– (2)
279, or on October 7

6. 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

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