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EXAMPLE 5

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 ). Rewrite cos 3 x as cos (2 x + x ). = cos 2 x cos x – sin 2 x sin x. Use a sum formula.

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EXAMPLE 5

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  1. EXAMPLE 5 Verify a trigonometric identity Verify the identity cos 3x =4 cos3 x – 3 cos x. cos 3x = cos (2x + x) Rewrite cos 3xas cos (2x + x). = cos 2xcos x – sin 2xsinx Use a sum formula. = (2 cos2x – 1) cosx –(2 sin x cosx) sinx Use double-angle formulas. = 2 cos3x – cosx – 2 sin2xcosx Multiply. = 2 cos3x – cos x – 2(1– cos2 x) cosx Use a Pythagorean identity. = 2 cos3x – cosx – 2 cosx + 2 cos3x Distributive property Combine like terms. = 4 cos3x – 3 cosx

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

  3. Graph the function y = sin 2x + 2 cosx 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: 3π 2 π 1.57 and x x = = 4.71 2 EXAMPLE 6 Solve a trigonometric equation CHECK

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

  5. for Examples 5, 6, and 7 GUIDED PRACTICE Verify the identity. 11. sin 3x = 3 sinx – 4 sin3x SOLUTION Sin 3x = sin (2x + x) = sin 2x cos x + cos 2x sin x = 2sin xcos xcos x + (1 –2 sin2x) sin x = 2 sin xcos2x + sin x –2 sin3x = 2 sin x(1 – sin2x) + sin x – 2sin3x = 2 sin x – sin3x + sin x – 2sin3x = 3 sin x – 4 sin3x

  6. for Examples 5, 6, and 7 GUIDED PRACTICE Verify the identity. 12. 1 + cos 10x = 2 cos2 5x SOLUTION 1 + cos 10x = 1 + 2 cos2(5x) – 1 = 2 cos2 5x

  7. ANSWER 4π 4π 5π 8π 2π 3 3 3 3 3 0, , , , , π π x 3 2 14. 2 cos+ 1 = 0 ANSWER + 4n π or + 4n π for Examples 5, 6, and 7 GUIDED PRACTICE Solve the equation. 13. tan 2x + tan x =0for0 ≤ x <2π.

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