Bound for $\left|\sin(x) +\cos(x)\right|$ I'm taking a numerical analysis class and i'm needing to bound $\left|\sin(x) + \cos(x)\right|$ quite often. So far i've been putting that this is always $\leq |1 + 1| = 2$. Is this the minimal bound? I sort of get a feeling that it might be $\sqrt{2}$. If anyone knows and could give an explanation that would be great.

$$\cos x+\sin x=\sqrt2(\cos x\cos\frac\pi4+\sin x\sin\frac\pi4)=\sqrt2\cos(x-\frac\pi4)$$
so $|\cos x+\sin x|\le\sqrt2$.
More generally, the maximum value of $|a\cos x+b\sin x|$ is $\sqrt{a^2+b^2}$.
