Right-angled trigonometry (SOHCAHTOA)

Labelling opposite, adjacent and hypotenuse from the angle you are using, choosing the right ratio, finding sides and angles, the exact values that must be known by heart, and angles of elevation and depression.

Geometry & measures · Pythagoras & right-angled trigonometry

Right-angled trigonometry (SOHCAHTOA)

Why it works

All triangles with the same angles are similar, so in a right-angled triangle the ratio of any two sides depends only on the angle — not on how big the triangle is. Those fixed ratios are named:

\sin\theta = \frac{\text{opp}}{\text{hyp}}, \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}

remembered as SOH CAH TOA.

Label from the angle you are using, every time. The hypotenuse never moves (opposite the right angle), but "opposite" and "adjacent" swap over depending on which angle you're working from. So: mark the angle, label the side facing it opp, the remaining short side adj, and only then choose the ratio containing the two sides you care about.

Finding a side vs finding an angle.

Side: substitute and rearrange, e.g. $\sin 35° = \frac{x}{12}$ gives

x = 12\sin 35°

. If the unknown is on the bottom (

\cos 40° = \frac{9}{x}

), swap it with the trig value:

x = \frac{9}{\cos 40°}

Angle: use the inverse functions $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$

— e.g.

\tan\theta = \frac{5}{7}

gives

\theta = \tan^{-1}\left(\frac{5}{7}\right) = 35.5°

(1 dp).

Your calculator must be in degrees. A "wrong by a lot" answer is nearly always radians mode. And don't round part-way: keep the full display value in the calculator until the final answer.

Exact values you are expected to know (they appear on the non-calculator paper):

$\theta$	$0°$	$30°$	$45°$	$60°$	$90°$
$\sin\theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt2}{2}$	$\frac{\sqrt3}{2}$	$1$
$\cos\theta$	$1$	$\frac{\sqrt3}{2}$	$\frac{\sqrt2}{2}$	$\frac{1}{2}$	$0$
$\tan\theta$	$0$	$\frac{\sqrt3}{3}$	$1$	$\sqrt3$	—

(

\tan 90°

is undefined — the adjacent side has shrunk to nothing.)

Elevation and depression are measured from the horizontal. The angle of elevation looks up from horizontal; the angle of depression looks down from horizontal. They are equal to each other for the same line of sight (alternate angles), which is what lets you drop the angle of depression into the triangle at the bottom. Watch out for the extra step in "eye height" questions: the trig gives the height above eye level, so add the observer's height at the end.