Combining sine and cosine

Consider the graph of $y = \sin x + \cos x$. Combining the two terms gives a single sine, $\sin x + \cos x = \sqrt{2}\,\sin\left(x + \dfrac{\pi}{4}\right)$.

This has amplitude $\sqrt{2}$ and is $y = \sin x$ shifted left by $\dfrac{\pi}{4}$. The value stays between $-\sqrt{2}$ and $\sqrt{2}$, reaching its maximum $\sqrt{2}$ when $x + \dfrac{\pi}{4} = \dfrac{\pi}{2}$, that is $x = \dfrac{\pi}{4}$. The upper and lower lines $y = \pm\sqrt{2}$ show the amplitude, and the large dot marks the maximum point $\left(\dfrac{\pi}{4}, \sqrt{2}\right)$.