Finding an Angle

A skate ramp rises 3 feet over a run of 4 feet. How steep is it? You could measure the angle with a protractor… if you could somehow get one inside the wood. But you don't need to. The two side lengths already fix the angle — and a calculator can hand it to you in one keypress.

This is SOH-CAH-TOA run backwards. Going forwards, an angle gives you a ratio of sides. Going backwards, a ratio of sides gives you the angle. The buttons that reverse the trip are the inverse trig functions — sin⁻¹, cos⁻¹ and tan⁻¹ — and each one answers a single question: "what angle has this ratio?"

For instance, if \sin\theta = \dfrac{\text{opp}}{\text{hyp}}, then

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

The recipe never changes: form the ratio of the two sides you know, then press the matching inverse button. For the ramp, opposite 3 over adjacent 4 gives \tan\theta = 0.75, so \theta = \tan^{-1}(0.75) \approx 36.9^\circ — a good, honest hill.

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 1 — the tangent (opposite & adjacent)

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}.

Notice the shape of the working: ratio first, inverse second. \tan\theta = \tfrac34 = 0.75, then \theta = \tan^{-1}(0.75) \approx 36.9^\circ. On the calculator that is 0.75, then the shift (or 2nd) key, then tan.

Worked example 2 — the sine (opposite & hypotenuse)

A 5\,\text{m} ladder leans against a wall. Its foot is such that the top reaches 4\,\text{m} up the wall. What angle does the ladder make with the ground?

The 4\,\text{m} height is opposite the angle, and the 5\,\text{m} ladder is the hypotenuse. Opposite over hypotenuse is the sine, so:

\sin\theta = \frac{4}{5} = 0.8 \quad\Longrightarrow\quad \theta = \sin^{-1}(0.8) \approx 53.1^\circ.

Better than a guess — and safe: firefighters aim for a ladder angle of about 75^\circ, so this one is a touch flat.

Worked example 3 — the slope of a hill

A road sign warns of a "1 in 5" gradient: the road climbs 1\,\text{m} vertically for every 5\,\text{m} travelled along the ground. What angle is that, really?

Rise is opposite, run is adjacent — that pairing is the tangent again:

\tan\theta = \frac{1}{5} = 0.2 \quad\Longrightarrow\quad \theta = \tan^{-1}(0.2) \approx 11.3^\circ.

Just over 11^\circ — gentler than it sounds, but on a wet day in a heavy truck, quite enough. The very same sum tells a surveyor the pitch of a roof or a pilot the angle of descent onto a runway from the height lost and the distance flown.

Use the inverse, not the plain function. To go from a ratio back to an angle you must press \sin^{-1} / \cos^{-1} / \tan^{-1} — the shift-sin button. Pressing plain \sin does the opposite job (angle → ratio) and gives nonsense. This is the single most common slip.
\sin^{-1} is NOT \tfrac{1}{\sin}. The little -1 here means "the inverse function", not a reciprocal power. It's a genuinely confusing notation: \sin^{-1}(0.5) = 30^\circ, whereas \tfrac{1}{\sin 0.5} is something else entirely.
Keep the calculator in DEGREES. If your answer comes out like 0.6435 instead of 36.9^\circ, the calculator is in radians — flip the mode to DEG and try again.

Whenever a machine needs to answer "what angle points there?", it is quietly running an inverse tangent. A phone's tilt sensor measures how gravity splits between two directions and takes \tan^{-1} of the ratio to know the screen's angle. A game's aiming system turns the gap between you and a target (\Delta y over \Delta x) into an aim angle the same way. A robot arm figures out how far to rotate each joint from the position it wants to reach.

Programmers use a beefed-up version called \operatorname{atan2}(y, x) — the same idea as \tan^{-1}(y/x), but clever enough to give the correct answer in all four quadrants (and never divide by zero when x = 0). It's one of the most-called functions in graphics, navigation and robotics. A surveyor finding the angle of a slope and a pilot finding the angle of descent are doing, by hand, exactly what your phone does sixty times a second.