GCSE Trigonometry: SOHCAHTOA Explained with Examples

What Is Trigonometry?

Trigonometry is the study of relationships between angles and sides in triangles. In GCSE Maths, you'll focus on right-angled triangles, using the three trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios let you find missing sides or angles when you know at least one angle and one side.

• Only applies directly to right-angled triangles at GCSE Foundation
• Higher tier extends to the sine rule and cosine rule for any triangle
• The three ratios are sin, cos, and tan

SOHCAHTOA — The Key Memory Aid

SOHCAHTOA is the mnemonic that defines the three trig ratios in terms of the sides of a right-angled triangle. Label your sides first: Hypotenuse (H) is always opposite the right angle; Opposite (O) is the side opposite the angle you're working with; Adjacent (A) is the side next to that angle (not the hypotenuse).

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

Finding a Missing Side

To find a missing side, identify which ratio connects the angle you know, the side you know, and the side you want. Rearrange to solve for the unknown side.

Finding a Missing Angle

To find a missing angle, use the inverse trig function (sin⁻¹, cos⁻¹, or tan⁻¹) on your calculator. This is sometimes written as arcsin, arccos, or arctan.

Exact Trigonometric Values

For Higher tier, you must know the exact values for sin, cos, and tan of 0°, 30°, 45°, 60°, and 90° without a calculator. These come up in non-calculator papers.

• sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = cos 45° = 1/√2, tan 45° = 1
• sin 60° = √3/2, cos 60° = ½, tan 60° = √3
• sin 90° = 1, cos 90° = 0, tan 90° = undefined