EXAMPLE 5 Verify a trigonometric identity Verify the

EXAMPLE 6 Solve a trigonometric equation Solve sin 2 x + 2 cos x

EXAMPLE 6 Solve a trigonometric equation CHECK Graph the function y = sin 2

EXAMPLE 7 Find a general solution Find the general solution of 2 sin x

GUIDED PRACTICE for Examples 5, 6, and 7 Verify the identity. 11. sin 3

GUIDED PRACTICE for Examples 5, 6, and 7 Verify the identity. 12. 1 +

GUIDED PRACTICE for Examples 5, 6, and 7 Solve the equation. 13. tan 2

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

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

EXAMPLE 6 Solve a trigonometric equation CHECK Graph the function y = sin 2 x + 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 3π 1. 57 and x = 2 4. 71

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

GUIDED PRACTICE for Examples 5, 6, and 7 Verify the identity. 11. sin 3 x = 3 sin x – 4 sin 3 x SOLUTION Sin 3 x = sin (2 x + x) = sin 2 x cos x + cos 2 x sin x = 2 sin xcos x + (1 – 2 sin 2 x) sin x = 2 sin xcos 2 x + sin x – 2 sin 3 x = 2 sin x(1 – sin 2 x) + sin x – 2 sin 3 x = 2 sin x – sin 3 x + sin x – 2 sin 3 x = 3 sin x – 4 sin 3 x

GUIDED PRACTICE for Examples 5, 6, and 7 Verify the identity. 12. 1 + cos 10 x = 2 cos 2 5 x SOLUTION 1 + cos 10 x = 1 + 2 cos 2(5 x) – 1 = 2 cos 2 5 x

GUIDED PRACTICE for Examples 5, 6, and 7 Solve the equation. 13. tan 2 x + tan x = 0 for 0 ≤ x <2π. ANSWER 0, 4π 5π π 2π , , π, , 3 3 14. 2 cos x + 1 = 0 2 ANSWER 4π 8π + 4 n π or + 4 n π 3 3