What is $cos^4theta+sin^3theta$ in terms of non-exponential trigonometric functions?

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Answer 1 · 2016-07-08T08:26:30

$=1/8(3+4cos2theta+2cos4 theta+6sintheta-2sin3theta)$

We will use the identities

$color(red)(>2cos^2theta=(1+cos2theta))$

$color(blue)(>4sin^3theta=3sintheta-sin3theta)$

The given expression

$=cos^4theta+sin^3theta$

$=1/4(4cos^4theta+4sin^3theta)$

$=1/4((2cos^2theta)^2+3sintheta-sin3theta)$

$=1/4((1+cos2theta)^2+3sintheta-sin3theta)$

$=1/4(1+2cos2theta+cos^2 2theta+3sintheta-sin3theta)$

$=1/4(1+2cos2theta+1/2(1+cos4 theta)+3sintheta-sin3theta)$

$=1/4(3/2+2cos2theta+cos4 theta+3sintheta-sin3theta)$

$=1/8(3+4cos2theta+2cos4 theta+6sintheta-2sin3theta)$

What is cos^4theta+sin^3thetacos^4theta+sin^3theta in terms of non-exponential trigonometric functions?