1. Verify a trigonometric identity
EXAMPLE 5
Verify a trigonometric identity
Verify the identity cos 3x = 4 cos3 x – 3 cos x.
cos 3x
= cos (2x + x)
Rewrite cos 3x as cos (2x + x).
= cos 2x cos x – sin 2x sin x
Use a sum formula.
= (2 cos2 x – 1) cos x – (2 sin x cos x) sin x
Use double-angle formulas.
= 2 cos3 x – cos x – 2 sin2 x cos x
Multiply.
= 2 cos3 x – cos x – 2(1 – cos2 x) cos x
Use a Pythagorean identity.
= 2 cos3 x – cos x – 2 cos x + 2 cos3 x
Distributive property
= 4 cos3 x – 3 cos x
Combine like terms.

2. Solve a trigonometric equation
EXAMPLE 6
Solve a trigonometric equation
Solve sin 2x + 2 cos x = 0 for 0 ≤ x <2π.
SOLUTION
sin 2x + 2 cos x
= 0
Write original equation.
2 sin x cos x + 2 cos x
= 0
Use a double-angle formula.
2 cos x (sin x + 1)
= 0
Factor.
Set each factor equal to 0 and solve for x.
2 cos x
= 0
sin x + 1
= 0
cos x
= 0
sin x
= – 1 π 2 3π
= , 3π 2 = x x

3. EXAMPLE 6
Solve a trigonometric equation
CHECK
Graph the function y = sin 2x + 2 cos x on a graphing calculator. Then use the zero feature to find the x-values on the interval 0 ≤ x <2π for which y = 0. The two x-values are: x π 2 = x
and 3π 2 =
1.57
4.71

4. Find a general solution
EXAMPLE 7
Find a general solution x 2
Find the general solution of 2 sin = 1.
2 sin x 2
= 1
Write original equation.
2 sin x 2 1 =
Divide each side by 2. x 2
= nπ or π 6 5π
+ 2nπ x 2
General solution for x
= nπ or π 3 5π
+ 4nπ
General solution for x

5. GUIDED PRACTICE
for Examples 5, 6, and 7
Verify the identity.
sin 3x = 3 sin x – 4 sin3 x
SOLUTION
Sin 3x = sin (2x + x)
= sin 2x cos x + cos 2x sin x
= 2sin x cos x cos x + (1 _ 2 sin2 x) sin x
= 2 sin x cos2 x + sin x _ 2 sin3 x
= 2 sin x(1 _ sin2 x) + sin x _ 2 sin3 x
= 2 sin x _ 2 sin3 x + sin x _ 2sin3 x
= 3 sin x _ 4 sin3 x

6. GUIDED PRACTICE
for Examples 5, 6, and 7
Verify the identity.
cos 10x = 2 cos2 5x
SOLUTION
1 + cos 10x = cos2(5x) – 1
= 2 cos2 5x

7. GUIDED PRACTICE
for Examples 5, 6, and 7
Solve the equation.
tan 2x + tan x = 0 for 0 ≤ x <2π.
2 tan x
+ tan x = 0
1 – tan2 x
2 tan x + tan x – tan3 x = 0(1 – tan2 x)
tan x (3 – tan2 x) = 0
tan x = 0 OR (3 – tan2 x) = 0
tan x = 0 OR tan x = 3
ANSWER
0, , , , , π 3 2π 4π 5π

8. GUIDED PRACTICE
for Examples 5, 6, and 7
Solve the equation. x 2
cos = 0 x 2
2 cos = 0 =
1 + cos x 2 =
1 + cos x 2 – 1 =
1 + cos x 2 1 4
cos x – 1 2 =
ANSWER
+ 4n π or n π 4π 3 8π