Right-Triangle Trigonometry — AP® Precalculus Notes

Unit 3 builds sine and cosine as functions of any angle. They start life as something smaller: ratios of the sides of a right triangle. This page is the algebra you need for that first step.

Naming the sides

In a right triangle the side across from the right angle is the hypotenuse; it is always the longest side. The other two sides, the legs, get their names from the acute angle you are working with, called $\theta$ here:

the opposite side is the leg across from $\theta$ ;
the adjacent side is the leg that touches $\theta$ (the hypotenuse touches it too, but it already has a name).

A right triangle with the right angle at the lower right corner and the angle theta at the lower left corner. The longest side, across from the right angle, is the hypotenuse. The vertical side across from theta is the side opposite theta. The horizontal side that touches theta and is not the hypotenuse is the side adjacent to theta. To the right: sine of theta is opposite over hypotenuse, cosine of theta is adjacent over hypotenuse, tangent of theta is opposite over adjacent (SOH CAH TOA). — **Figure 1.** Opposite, adjacent and hypotenuse, seen from the angle $\theta$ . LevlPrep original diagram.

Move to the other acute angle and the two legs swap names. Always ask "opposite to which angle?"

Three ratios that depend only on the angle

Any two right triangles with the same acute angle $\theta$ are similar: one is a scaled copy of the other. Scaling multiplies every side by the same number, so the ratio of any two sides does not change. Each ratio is therefore a property of the angle alone, and each has a name. These are the trigonometric ratios:

Ratio	Definition	Memory aid
$\sin\theta$	$\frac{\text{opposite}}{\text{hypotenuse}}$	SOH
$\cos\theta$	$\frac{\text{adjacent}}{\text{hypotenuse}}$	CAH
$\tan\theta$	$\frac{\text{opposite}}{\text{adjacent}}$	TOA

Because the hypotenuse is the longest side, $\sin\theta$ and $\cos\theta$ are between $0$ and $1$ for an acute angle. $\tan\theta$ can be any positive number. When the adjacent side lies flat, $\tan\theta$ is the rise over the run of the hypotenuse: its slope.

Worked example 1. A right triangle has legs $5$ and $12$ . The angle $A$ is across from the side $5$ . Find $\sin A$ , $\cos A$ and $\tan A$ .

First the hypotenuse: $\sqrt{5^2 + 12^2} = \sqrt{169} = 13$ .

From $A$ : opposite $= 5$ , adjacent $= 12$ , hypotenuse $= 13$ . So $\sin A = \frac{5}{13}$ , $\cos A = \frac{12}{13}$ , $\tan A = \frac{5}{12}$ .

Finding a missing side

Know one angle and one side? Choose the ratio that uses the side you know and the side you want, write an equation, and solve it.

Worked example 2. A ramp's sloped surface is 20 feet long and makes a $35^\circ$ angle with the ground. Use $\sin 35^\circ \approx 0.5736$ to find how high the ramp rises.

The rise is opposite the angle and the 20 feet is the hypotenuse, so use sine: $\sin 35^\circ = \frac{h}{20}$ .

Multiply both sides by 20: $h$ $= 20 \sin 35^\circ$ $\approx 20(0.5736)$ $= 11.472$ feet.

If the unknown side is on the bottom of the ratio, you divide instead: with $\cos\theta = \frac{6}{x}$ , multiply by $x$ to get $x\cos\theta = 6$ , then $x = \frac{6}{\cos\theta}$ .

The same idea gives formulas. If a hypotenuse has length $r$ , then the opposite side is $r\sin\theta$ and the adjacent side is $r\cos\theta$ . These two expressions come back in the next topics, where they become the coordinates of a point on a circle.