Mathematics

The key insight is that $a\sin x + b\cos x$ traces out a sinusoidal wave as $x$ varies, and you can rewrite it as a single sine (or cosine) function with a shifted phase. The maximum value of any sinusoid $R\sin(x + \phi)$ is simply its amplitude $R$. So find the amplitude by combining the coefficients $a$ and $b$ using the Pythagorean theorem—the maximum occurs when the two components align perfectly.

The key insight is that $a\sin x + b\cos x$ traces out a sinusoidal wave as $x$ varies, and you can rewrite it as a single sine (or cosine) function with a shifted phase. The maximum value of any sinusoid $R\sin(x + \phi)$ is simply its amplitude $R$. So find the amplitude by combining the coefficients $a$ and $b$ using the Pythagorean theorem—the maximum occurs when the two components align perfectly.