SOH CAH TOA: Sine, Cosine and Tangent in Right-Angled Triangles

Why side ratios depend only on the angle, how to label opposite, adjacent and hypotenuse, how to choose the right ratio, and how to find a missing side or a missing angle.

3 min read Supports 3 lessons

Take two right-angled triangles with the same $30°$ angle, one tiny and one huge. They are the same shape, one an enlargement of the other, so the [[ratios|ratios]] of their sides are identical. In every right-angled triangle with a $30°$ angle, the side opposite it is exactly half the longest side.

Key idea

In a right-angled triangle, the ratios of the sides depend only on the angle, not on the size of the triangle. Those ratios are called sine, cosine and tangent.

Labelling the sides

Label the sides relative to the angle you care about, $\theta$ :

Hypotenuse: the longest side, opposite the right angle.

Opposite: the side across from $\theta$ .

Adjacent: the side next to $\theta$ that is not the hypotenuse.

Tip

Opposite and adjacent swap if you switch to the other acute angle. The hypotenuse never changes.

SOH CAH TOA

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

Finding a missing side

Decide which side you know and which you want, pick the ratio that uses those two, and solve.

$\text{A 5 m ramp makes a } 20^\circ \text{ angle with the ground.}$
How high does it rise?

1/4

Watch out

If the unknown is on the bottom, as in $\cos 40° = \frac{8}{x}$ , the answer is $x = \frac{8}{\cos 40°}$ . Multiply by $x$ first, then divide.

Finding a missing angle

If you know two sides and want the angle, use the inverse functions $\sin^{-1}$ , $\cos^{-1}$ and $\tan^{-1}$ . They take a ratio and return the angle.

Worked example

A ladder $6$ m long reaches $5$ m up a wall. What angle does it make with the ground?

1
Opposite $5$ , hypotenuse $6$ : sine.
$\sin\theta = \frac{5}{6}$

Watch out

Make sure your calculator is in degree mode when angles are in degrees.

Exact values

Two special triangles give exact values worth knowing. Half an equilateral triangle gives $\sin 30° = \frac12$ and $\cos 30° = \frac{\sqrt3}{2}$ . A right-angled isosceles triangle gives $\sin 45° = \cos 45° = \frac{\sqrt2}{2}$ and $\tan 45° = 1$ .

Key idea

Right-angled triangles only allow angles below $90°$ . The unit circle extends sine and cosine to every angle, which is what turns them into the wave-shaped functions used throughout science.

Common questions

What does SOH CAH TOA stand for?: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
How do I know whether to use sin, cos or tan?: Label the sides relative to the angle, note which side you know and which you want, and choose the ratio that contains exactly those two sides.

Practise soh cah toa

The SOH CAH TOA lesson has worked examples and unlimited practice that adapts to you.

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Common questions

What does SOH CAH TOA stand for?

Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

How do I know whether to use sin, cos or tan?

Label the sides relative to the angle, note which side you know and which you want, and choose the ratio that contains exactly those two sides.