🗼 Height & Distance — Basics

Height and base are the two legs of the triangle, joined by the tangent of the elevation angle:

$\tan\theta=\dfrac{\text{height}}{\text{base}}$.

Learn the three standard values cold: $\tan30^\circ=\dfrac{1}{\sqrt3}$, $\tan45^\circ=1$, $\tan60^\circ=\sqrt3$.

• At $45^\circ$ the legs are equal — height : base $=1:1$. The triangle is a perfect right isosceles, so height equals distance.

• At $30^\circ$ and $60^\circ$ the legs are in the ratio $1:\sqrt3$. At $60^\circ$ the height is the long side ($\text{height}=\sqrt3\times\text{base}$); at $30^\circ$ the base is the long side.

Use sin or cos only when the slant line itself (a ladder, a kite string) is involved: $\sin\theta=\dfrac{\text{height}}{\text{slant}}$ and $\cos\theta=\dfrac{\text{base}}{\text{slant}}$.

• Broken tree: when a tree snaps and the top tips over to touch the ground, the bent-over piece is the slanted hypotenuse and the standing stump is the height. So total height $=\text{stump}+\text{broken part}$ — never report just one piece.