Inverse Trig: Finding Missing Angles

The three-step recipe

Step 1: label the two known sides relative to the angle you want — opposite, adjacent, or hypotenuse. Step 2: pick the ratio that uses those two sides (SOH-CAH-TOA) and write the equation, for example $\tan x^\circ = \dfrac{6}{8}$. Step 3: apply the matching inverse function to both sides: $x = \tan^{-1}\left(\dfrac{6}{8}\right) \approx 36.9^\circ$.

The inverse function must match the ratio you built. If you set up a cosine ratio, you need $\cos^{-1}$ — pressing $\sin^{-1}$ on a cosine ratio gives the other acute angle of the triangle, which is the most common wrong answer on these problems.

Building the ratio correctly

Everything is labeled from the angle you are solving for. The side across from it is opposite; the leg touching it is adjacent; the side across from the right angle is the hypotenuse no matter what.

Then match the pair of known sides: opposite and hypotenuse call for $\sin^{-1}$, adjacent and hypotenuse call for $\cos^{-1}$, opposite and adjacent call for $\tan^{-1}$. Keep the order inside the fraction right — the ratio is always written with the same side on top as in the original definition, so $\tan x^\circ$ is opposite over adjacent, never adjacent over opposite. The figure below labels each side by its role relative to angle $\theta$.

Checking your answer

The two acute angles of a right triangle add to $90^\circ$, so if your angle is across from the shorter leg it should come out less than $45^\circ$, and if it is across from the longer leg it should be more than $45^\circ$. A ten-second size check like this catches nearly every mixed-up ratio.

You can also verify directly: take the sine, cosine, or tangent of your answer and confirm it reproduces the ratio you started with.

Worked examples

Try one yourself

Common questions