Question

What is `cos^3theta-cos^2theta+costheta` in terms of non-exponential trigonometric functions?

Answer 1

`cos^3 theta - cos^2 theta + cos theta = frac{cos 3theta}{4} - frac{cos 2theta}{2} + frac{7 cos theta}{4} - 1/2`

Explanation:

Do you know the compound angle identities for `cos theta`?

`cos 2theta -= 2cos^2 theta -1`

`cos 3theta -= 4cos^3 theta - 3cos theta`

Change the subject of the formula.

`cos^2 theta -= frac{cos 2theta + 1}{2}`

`cos^3 theta -= frac{cos 3theta + 3cos theta}{4}`

So substitute them into your expression.

`cos^3 theta - cos^2 theta + cos theta`

`= frac{cos 3theta + 3cos theta}{4} - frac{cos 2theta + 1}{2} + cos theta`

`= frac{cos 3theta}{4} - frac{cos 2theta}{2} + frac{7 cos theta}{4} - 1/2`

Answer 2

`f(x) = cos x((1 + cos 2x)/2 - cos x + 1)`

Explanation:

Put cos x in common factor -->
`f(x) = cos x(cos^2 x - cos x + 1)`.
Replace `cos^2 x` by `((1 + cos 2x)/2)`
`f(x) = cos x[(1 - cos 2x)/2 - cos x + 1] =`