SOH CAH TOA — Trigonometry Made Simple

Learn SOHCAHTOA step by step — how to find missing sides and angles in right-angled triangles using sin, cos and tan. With 6 worked examples.

Trigonometry sounds complicated, but at GCSE level it's really just about right-angled triangles. You learn three ratios and a memory trick: SOH CAH TOA.

Labelling the Sides

Every right-angled triangle has three sides relative to a chosen angle (not the right angle):

Hypotenuse (H) — the longest side, always opposite the right angle
Opposite (O) — the side directly opposite your chosen angle
Adjacent (A) — the side next to your chosen angle (that isn't the hypotenuse)

SOH CAH TOA

Sin = O ÷ H | Cos = A ÷ H | Tan = O ÷ A

Memory trick: "Some Old Horses Can Always Hear Their Owners Approach".

Finding a Missing Side

Worked Example 1 — Find the opposite side

Angle = 35°, Hypotenuse = 10 cm. Find the Opposite.

We have the angle, H, and want O → use SOH
Sin 35° = O ÷ 10
O = 10 × Sin 35° = 10 × 0.574 = 5.74 cm

Worked Example 2 — Find the hypotenuse

Angle = 50°, Adjacent = 8 cm. Find the Hypotenuse.

We have A and want H → use CAH
Cos 50° = 8 ÷ H
H = 8 ÷ Cos 50° = 8 ÷ 0.643 = 12.44 cm

Worked Example 3 — Find the adjacent side

Angle = 40°, Opposite = 6 cm. Find the Adjacent.

We have O and want A → use TOA
Tan 40° = 6 ÷ A
A = 6 ÷ Tan 40° = 6 ÷ 0.839 = 7.15 cm

Finding a Missing Angle

Use the inverse function on your calculator: sin⁻¹, cos⁻¹, or tan⁻¹.

Worked Example 4 — Find the angle

Opposite = 5 cm, Hypotenuse = 13 cm. Find the angle.

We have O and H → use SOH
Sin θ = 5 ÷ 13 = 0.385
θ = sin⁻¹(0.385) = 22.6°

Worked Example 5

Adjacent = 7, Opposite = 4. Find the angle.

O and A → use TOA
Tan θ = 4 ÷ 7 = 0.571
θ = tan⁻¹(0.571) = 29.7°

Worked Example 6

Adjacent = 9, Hypotenuse = 12. Find the angle.

A and H → use CAH
Cos θ = 9 ÷ 12 = 0.75
θ = cos⁻¹(0.75) = 41.4°

How to Choose Which Ratio

Ask yourself: which two sides am I dealing with?

You have / want	Use
O and H	SOH (Sin)
A and H	CAH (Cos)
O and A	TOA (Tan)

Try It Yourself

Angle = 60°, Hypotenuse = 14 cm. Find the Opposite. (Answer: 12.12 cm)
Opposite = 8 cm, Adjacent = 6 cm. Find the angle. (Answer: 53.1°)
Angle = 28°, Opposite = 5 cm. Find the Hypotenuse. (Answer: 10.65 cm)