EXAMPLE 5 Verify a trigonometric identity

Verify the identity cos 3x = 4 cos3 x – 3 cos x.

Rewrite cos 3x as cos (2x + x).

cos 3x = cos (2x + x)

Use a sum formula.= cos 2x cos x – sin 2x sin x

Use double-angle formulas.

= (2 cos2 x – 1) cos x – (2 sin x cos x) sin x

Multiply.= 2 cos3 x – cos x – 2 sin2 x cos x

Use a Pythagorean identity.

= 2 cos3 x – cos x – 2(1 – cos2 x) cos x

Distributive property= 2 cos3 x – cos x – 2 cos x + 2 cos3 x

Combine like terms.

= 4 cos3 x – 3 cos x

EXAMPLE 6 Solve a trigonometric equation

Solve sin 2x + 2 cos x = 0 for 0 ≤ x <2π.

SOLUTION

Write original equation.sin 2x + 2 cos x = 0

Use a double-angle formula.2 sin x cos x + 2 cos x = 0

Factor.2 cos x (sin x + 1) = 0

Set each factor equal to 0 and solve for x.

2 cos x = 0

cos x = 0

x π2

3π2= ,

sin x + 1 = 0

sin x = –1

x3π2=

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 π2 = 1.57 xand 3π

2 = 4.71

EXAMPLE 7 Find a general solution

x2

Find the general solution of 2 sin = 1.

Write original equation.2 sin x2 = 1

Divide each side by 2.2 sin x2

12=

x2

General solution forx2 = + 2nπ orπ

6

5π6 + 2nπ

General solution for xx = + 4nπ orπ3

5π3 + 4nπ

GUIDED PRACTICE for Examples 5, 6, and 7

Verify the identity.11. sin 3x = 3 sin x – 4 sin3 x

SOLUTION

Sin 3x = sin (2x + x)

= sin 2x cos x + cos 2x sin x

= 2sin xcos xcos x + (1 – 2 sin2 x) sin x

= 2 sin xcos2 x + sin x – 2 sin3 x

= 2 sin x(1 – sin2 x) + sin x – 2sin3 x

= 2 sin x – sin3 x + sin x – 2sin3 x

= 3 sin x – 4 sin3 x

GUIDED PRACTICE for Examples 5, 6, and 7

Verify the identity.12. 1 + cos 10x = 2 cos2 5x

SOLUTION

1 + cos 10x = 1 + 2 cos2(5x) – 1 = 2 cos2 5x

GUIDED PRACTICE for Examples 5, 6, and 7

Solve the equation.13. tan 2x + tan x = 0 for 0 ≤ x <2π.

ANSWER

0, , , , , π3

2π3 π 4π

35π3

x2

14. 2 cos + 1 = 0

ANSWER

+ 4n π or + 4n π4π3

8π3