Find the period of f(x) = |sin x| + |cos x|.

Step 1. A quarter turn. Turning through $\dfrac{\pi}{2}$ sends the point $(a, b)$ to $(-b, a)$. So $\cos\left(x + \dfrac{\pi}{2}\right) = -\sin x$ and $\sin\left(x + \dfrac{\pi}{2}\right) = \cos x$.

Step 2. $\dfrac{\pi}{2}$ is a period. Taking sizes, $f\left(x + \dfrac{\pi}{2}\right) = |\cos x| + |\sin x| = f(x)$.

Step 3. Nothing smaller. $f(x)^2 = 1 + 2|\sin x||\cos x|$, so $f(x) = 1$ only where $\sin x = 0$ or $\cos x = 0$, at multiples of $\dfrac{\pi}{2}$. A period $T$ would need $f(T) = f(0) = 1$, so $T$ is a multiple of $\dfrac{\pi}{2}$.