Trigonometry (SOHCAHTOA)

GCSE Maths · Geometry

Trigonometry

Trigonometry links the angles and sides of a right-angled triangle using three ratios: sine (sin), cosine (cos) and tangent (tan). It lets you find a missing side or angle when Pythagoras alone can't.

Labelling the sides

Relative to the angle you're working with (θ):

Hypotenuse (H) — the longest side, opposite the right angle.
Opposite (O) — the side opposite the angle θ.
Adjacent (A) — the side next to the angle θ (between θ and the right angle).

SOHCAHTOA

This memory aid gives the three ratios:

SOH → sin θ = Opposite ÷ Hypotenuse
CAH → cos θ = Adjacent ÷ Hypotenuse
TOA → tan θ = Opposite ÷ Adjacent

Finding a missing side

1. Label the sides O, A, H relative to the given angle.

2. Choose the ratio that uses the two sides involved (the one you know and the one you want).

3. Substitute and rearrange.

Example: angle 30°, hypotenuse 10, find the opposite side.

Uses O and H → sin. sin 30 = O ÷ 10 → O = 10 × sin 30 = 10 × 0.5 = 5.

Finding a missing angle

Use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) on your calculator.

Example: opposite = 4, adjacent = 3, find the angle.

Uses O and A → tan. tan θ = 4 ÷ 3 → θ = tan⁻¹(4 ÷ 3) = 53.1°.

Exact trig values (learn these)

θ	sin	cos	tan
0°	0	1	0
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3
90°	1	0	—

Worked example

A right-angled triangle has an angle of 40° and an adjacent side of 8 cm. Find the hypotenuse.

Uses A and H → cos. cos 40 = 8 ÷ H → H = 8 ÷ cos 40 = 8 ÷ 0.766 ≈ 10.4 cm. ✓

Common mistakes

Labelling opposite/adjacent wrongly — they depend on which angle you're using.
Forgetting to use the inverse function when finding an angle.
Having the calculator in the wrong mode — make sure it's in degrees.

Exam tips

Write SOHCAHTOA and label O, A, H every time.
Rearrange carefully: to find the side on the bottom of the ratio, you divide.
Learn the exact values table — non-calculator papers require them.

Key facts to remember

SOH CAH TOA: sin = O/H, cos = A/H, tan = O/A (right-angled triangles).
Label O, A, H relative to the angle; pick the ratio with the two relevant sides.
Use sin⁻¹/cos⁻¹/tan⁻¹ to find angles; know the exact values for 0, 30, 45, 60, 90°.