Evaluate $\frac{d}{d x}\left(\sin x-\frac{4}{3} \sin ^{3} x\right)$ when $x=15$.
Solution: Of course this problem can be done by brute force, differentiating and then using the half angle formula to find $\sin$ and $\cos$ of 15 , but there is a quicker way. $e^{i x}=\cos x+i \sin x$, so $\sin (3 x)$ is the imaginary part of $(\cos x+i \sin x)^{3}$, which is $3 \cos ^{2} x \sin x-\sin ^{3} x=3 \sin x-4 \sin ^{3} x$, so the expression we are differentiating is just $\frac{1}{3} \sin (3 x)$. Hence the derivative is $\cos (3 x)$, and $\cos 45=\frac{\sqrt{2}}{2}$.