# Express $\sin4\theta$ in terms of powers of $\sin\theta$ and $\cos\theta$

As far as I know $\sin4\theta$ = $4\sin\theta \cos\theta$, but I don't know if that's correct or what to do from there?

• Try $\theta= \pi/4$ or $45^\circ$ to check your answer – Henry Oct 15 '14 at 11:09

$$\sin 4\theta = \sin (2\times 2\theta) = 2 \sin 2\theta \cos 2\theta=2(2\sin\theta \cos \theta)(\cos^2\theta- \sin^2 \theta) \\= 4 \sin \theta \cos \theta (\cos ^2 \theta - \sin ^2 \theta)$$
$\sin 2\theta=2\sin \theta \cos \theta.$
$\cos 2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$