Finding an Angle

SOH-CAH-TOA also runs backwards. If you know two sides of a right-angled triangle, their ratio fixes the angle \theta — you just have to undo the sine, cosine or tangent.

On a calculator the undo buttons are sin⁻¹, cos⁻¹ and tan⁻¹ (the inverse trig functions). For instance, if \sin\theta = \dfrac{\text{opp}}{\text{hyp}}, then

\theta = \sin^{-1}\!\left(\frac{\text{opp}}{\text{hyp}}\right).

The recipe is always the same: form the ratio of the two sides you know, then press the matching inverse button.

To find an unknown angle \theta from two known sides:

form the ratio of the two sides you know;
pick the matching inverse: opposite & hypotenuse → \sin^{-1} (SOH), adjacent & hypotenuse → \cos^{-1} (CAH), opposite & adjacent → \tan^{-1} (TOA);
apply it to the ratio to get \theta.

Worked example

We know the opposite (3) and the adjacent (4), so the matching ratio is the tangent. Step through the figure to undo it with \tan^{-1}.