Harmonic form: R sin(x + α)

The idea

$a\sin x + b\cos x$ can be written as $R\sin(x + \alpha)$ with $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \frac{b}{a}$. Its maximum is $R$ and its minimum $-R$, which makes equations and extremes quick.

1. Expand $R\sin(x + \alpha)$ as $R\sin x\cos\alpha + R\cos x\sin\alpha$ and match coefficients.
2. $R = \sqrt{a^2 + b^2}$; $\tan\alpha = \frac{R\sin\alpha}{R\cos\alpha}$.
3. Maximum $R$ when the sine is $1$; minimum $-R$ when it is $-1$.

Watch out Check the signs of $R\cos\alpha$ and $R\sin\alpha$ to put $\alpha$ in the right quadrant.