Verify a trigonometric identity

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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.

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

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

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 = + 2nπ or π 6 5π + 2nπ x 2 General solution for x = + 4nπ or π 3 5π + 4nπ General solution for x

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, , , , , π 3 2π 4π 5π x 2 14. 2 cos + 1 = 0 ANSWER + 4n π or + 4n π 4π 3 8π