What is the maximum value of $f(x) = 3\sin x + 4\cos x$?

Explanation

1. 01

   $a\sin x + b\cos x$ can be written $\sqrt{a^{2} + b^{2}}\,\sin(x + \alpha)$, so it runs from $-\sqrt{a^{2} + b^{2}}$ to $\sqrt{a^{2} + b^{2}}$.

2. 02

   $\sqrt{9 + 16} = 5$.