20.2 Trigonometric ratios (SOHCAHTOA) — Learn

Prerequisites: right-angled triangles, fractions, and a calculator in degree mode.

Objectives: select and use sine, cosine, or tangent to find a missing side or angle.

For an acute angle $\theta$, label the sides from that angle. The opposite side is across from it, the adjacent side touches it, and the hypotenuse is opposite the right angle. The ratios are $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

SOHCAHTOA is a memory aid: sine uses opposite and hypotenuse, cosine uses adjacent and hypotenuse, and tangent uses opposite and adjacent.

Example 1. Find the opposite side when $\theta=30^\circ$ and the hypotenuse is $10$ cm.

Example 2. Find the angle when the opposite side is $7$ cm and the adjacent side is $9$ cm.

The inverse tangent key returns an angle from a ratio. Use degree mode.

Watch out. The side labels depend on the chosen angle. Mark opposite and adjacent only after locating $\theta$, then check that your ratio uses the known sides.

Progress check

For the garden diagonal from 20.1, identify a possible angle and the sides you would need to know before using a trigonometric ratio.

You should now be able to

• label the sides from a chosen angle;
• choose a ratio to find a missing side or use an inverse ratio to find an angle;
• use degree mode and check that the answer fits the triangle.