Right-Triangle Trigonometry

SOH-CAH-TOA: find sides and angles with sine, cosine, and tangent.

The idea

Right-triangle trigonometry relates an acute angle of a right triangle to ratios of its sides. One acute angle $\theta$ determines the triangle's shape completely — the right angle and the angle sum fix the third angle — so any two right triangles sharing $\theta$ are scaled copies of each other. Scaling changes every length but no ratio of two lengths, so a ratio of sides depends on $\theta$ alone.

Name the sides relative to $\theta$: the hypotenuse is the side across from the right angle, the leg touching $\theta$ is the adjacent leg, and the leg across from $\theta$ is the opposite leg. The three ratios, remembered as SOH-CAH-TOA, are $\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}.$

Each ratio links the angle to two particular sides, so an angle and one side determine another side. The inverse trigonometric functions $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$ run the other way: a ratio is a bare number, not an angle, and the inverse returns the angle that produced it.

Ways to work on it

• Walkthrough. Name the sides, read the ratios, and recover an angle with inverse trig.
• Practice. Read a sine, cosine, or tangent ratio off a right triangle.
• Hardest. Solve a right triangle: find a missing angle and a missing side.

Not sure where to start? Take the ten-question placement test.