Right-Angled Trigonometry, Elevation, Depression, and Bearings

Surveyors, sailors, and pilots have used triangles to fix positions for centuries. In IB AI, right-angled trigonometry is the workhorse for navigation and surveying problems, and the language of bearings and angles of elevation and depression connects the math to real measurement.

The core ratios

In a right triangle, with respect to an angle $\theta$, we use SOH-CAH-TOA: $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}.$$ To find a missing side, choose the ratio that involves the side you want and a side you know. To find a missing angle, use the inverse functions $\arcsin$, $\arccos$, $\arctan$ on your GDC.

Angles of elevation and depression

The angle of elevat