Question

How do you rewrite `cos 3theta` in terms of only `costheta` and `sintheta`?

Answer 1

Since `cos3theta` has an odd coefficient, the double angle formula doesn't really work at first; fortunately, it derives from the additive angle formulas, which we can use:

`\mathbf(sin(upmv) = sinucosv pm cosusinv)`
`\mathbf(cos(upmv) = cosucosv ∓ sinusinv)`

Thus:

`cos3theta`

`= cos(theta+2theta)`

`= costhetacolor(red)(cos2theta) - sinthetacolor(red)(sin2theta)`

Next, we still have `cos2theta` and `sin2theta` present.

So, we have to rewrite `cos2theta` and `sin2theta` as follows, using the "double angle" formula (which is really the additive angle formula for `color(green)(u = v)`):

`color(green)(cos(theta+theta)) = costhetacostheta - sinthetasintheta`

`= color(green)(cos^2theta - sin^2theta)`

`color(green)(sin(theta+theta)) = sinthetacostheta + costhetasintheta`

`= color(green)(2sinthetacostheta)`

Thus, we end up with:

`color(blue)(cos3theta)`

`= costheta(color(green)(cos^2theta - sin^2theta)) - sintheta(color(green)(2sinthetacostheta))`

`= cos^3theta - sin^2thetacostheta - 2sin^2thetacostheta`

`= color(blue)(cos^3theta - 3sin^2thetacostheta)`