What is ${cos}^{5} θ - 6 {sin}^{3} θ$ $cos^5theta-6sin^3theta$ in terms of non-exponential trigonometric functions?

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What is ${cos}^{5} θ - 6 {sin}^{3} θ$ $cos^5theta-6sin^3theta$ in terms of non-exponential trigonometric functions?

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Answer 1 · 2015-12-18T14:39:31

Use these trig identities to transform ${cos}^{5} x$ $cos^5 x$ and ${sin}^{3} x$ $sin^3 x$
$cos 2 x = 2 {cos}^{2} x - 1 = 1 - 2 {sin}^{2} x$ $cos 2x = 2cos^2 x - 1 = 1 - 2sin^2 x$
${sin}^{2} x + {cos}^{2} x = 1$ $sin^2 x + cos ^2 x = 1$

(1) --> ${cos}^{5} x = cos x ({cos}^{2} x) ({cos}^{2} x) =$ $cos^5 x = cos x(cos^2 x)(cos^2 x) =$
$= cos x (1 - sin x) (1 + sin x) (\frac{1 + cos 2 x}{2})$ $= cos x(1 - sin x)(1 + sin x)((1 + cos 2x)/2)$
(2) --> $6 {sin}^{3} x = 6 sin x ({sin}^{2} x) = 6 sin x (1 - cos x) (1 + cos x) .$ $6sin^3 x = 6sin x(sin^2 x) = 6sin x(1 - cos x)(1 + cos x).$
f(x) = (1) + (2)
$f (x) = (\frac{1}{2}) cos x (1 - sin x) (1 + sin x) (1 + cos 2 x) - 6 sin x (1 - cos x) (1 + cos x)$ $f(x) = (1/2)cos x(1 - sin x)(1 + sin x)(1 + cos 2x) - 6sin x(1 - cos x)(1 + cos x)$

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What is ${cos}^{5} θ - 6 {sin}^{3} θ$ $cos^5theta-6sin^3theta$ in terms of non-exponential trigonometric functions?

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What is cos5θ−6sin3θcos^5theta-6sin^3theta in terms of non-exponential trigonometric functions?

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What is ${cos}^{5} θ - 6 {sin}^{3} θ$ $cos^5theta-6sin^3theta$ in terms of non-exponential trigonometric functions?