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

1 Answer
Jul 8, 2016

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

Explanation:

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)