Find the angle of elevation of the Sun when the length of the shadow of a vertical pole is $\sqrt{3}$ times its height.

Let the height of the vertical pole be $h$ and the length of its shadow be $x$.
Given that:

$$x=\sqrt{3}h$$

Let $\theta$ be the angle of elevation of the Sun.

$$\tan \theta =\frac{\text{Height}}{\text{Shadow}}=\frac{h}{x}$$

$$\tan \theta =\frac{h}{\sqrt{3}h}=\frac{1}{\sqrt{3}}$$

Since $\tan 30^{\circ }=\frac{1}{\sqrt{3}}$, we have:

$$\theta =30^{\circ }$$