# Trigonometric constants

1) Express (cosx)^n in terms of sums of coskx

2) Express (sinx)^n in terms of sums of coskx or sinkx

3) Express cosnx in terms of sums of (cosx)^k

4) Express sinnx in terms of sums of (sinx)^k

5) Derive exact trigonometric identities for sin(k*pi) for k being:

1/12, 1/10,1/8,1/6,1/5,1/4,3/10,1/3,3/8,2/5,5/12

https://brainmass.com/math/trigonometry/powers-sine-cosine-exact-trigonometric-constants-82284

#### Solution Preview

Please see attached

Powers of Sine and Cosine

1) It can be useful to express in terms of sums of :

First for even powers:

Conclude

Or

There is a corollary from this, integrating the left hand side is a lot harder than integrating the left hand side:

Conclude

2) Now for odd powers:

Conclude

Or

There is a corollary from this, integrating the left hand side is a lot harder than integrating the left hand side:

Conclude ...

#### Solution Summary

This provides several examples of working with exact trigonometric constants. The expert derives the exact trigonometric identities for the functions.