In Solving Right Triangles you knew an angle and found sides. Now go the other way: from two sides, find the angle. The ratio is easy to compute; the inverse trig functions turn a ratio back into an angle.
Why it works
For acute angles, each ratio goes with exactly one angle: as $A$ grows from $0^\circ$ to $90^\circ$, $\sin A$ grows from 0 to 1 and never repeats a value. So "the acute angle whose sine is $0.6$" names a single angle. We write it $\sin^{-1}(0.6)$, read "inverse sine of 0.6." Likewise $\cos^{-1}$ and $\tan^{-1}$.
Key idea
If $\sin A = r$, then $A = \sin^{-1}(r)$; the same pattern holds for $\cos^{-1}$ and $\tan^{-1}$. Choose the ratio that uses the two sides you know.
Worked example: Two legs
A right triangle has legs $5$ and $8$. Find the angle opposite the leg of length $5$, to the nearest degree.
- Opposite $5$, adjacent $8$: $\tan A = \frac{5}{8} = 0.625$.
- $A = \tan^{-1}(0.625) \approx 32.0^\circ$.
Answer: $A \approx 32^\circ$
Worked example: A leg and the hypotenuse
In a right triangle, the leg opposite angle $A$ is $7$ and the hypotenuse is $12$. Find $m\angle A$ to the nearest degree.
- $\sin A = \frac{7}{12} \approx 0.5833$.
- $A = \sin^{-1}(0.5833) \approx 35.7^\circ \approx 36^\circ$.
Answer: $36^\circ$
Worked example: Both acute angles
A ramp rises $1$ m over a horizontal run of $12$ m. Find the angle the ramp makes with the ground and the angle at the top.
- $\tan A = \frac{1}{12}$, so $A = \tan^{-1}\left(\frac{1}{12}\right) \approx 4.8^\circ$.
- The other acute angle is $90 - 4.8 = 85.2^\circ$.
Answer: About $4.8^\circ$ and $85.2^\circ$
Check an inverse trig answer by going forward: if $A = \tan^{-1}(0.625) \approx 32.0^\circ$, then $\tan 32.0^\circ \approx 0.625$. Also check that the angle is sensible. The longer leg is opposite the larger acute angle, so if the opposite side is shorter than the adjacent side, the angle must be less than $45^\circ$.
Common mistake
Reading $\sin^{-1}$ as $\frac{1}{\sin}$. Here the $-1$ means "inverse function," like undoing, not "reciprocal." $\sin^{-1}(0.5) = 30^\circ$, while $\frac{1}{\sin 0.5^\circ}$ is about $115$. On a calculator, use the $\sin^{-1}$ key (often SHIFT then SIN), in degree mode.