Trigonometry · step 1 of 13

Right-triangle ratios: sin, cos & tan

In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent — the mnemonic SOH CAH TOA. "Opposite" and "adjacent" are relative to the angle you are working with, so the same side changes role when you switch angles, and that is where most errors originate.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Label relative to the angle in useThe hypotenuse is fixed, but opposite and adjacent swap when you move to the other acute angle.
Choose the ratio containing your two known quantitiesOne unknown and one known side, or two known sides for an angle.
Solve for a side or use an inverse for an angleMultiply or divide for a side; use arcsin, arccos or arctan for an angle.
Check plausibilityThe hypotenuse is the longest side, and both acute angles lie strictly between 0° and 90°.

Worked example

A right triangle has hypotenuse 10 and one acute angle 30°. Find the side opposite that angle and the third side.

Opposite: sin 30° = opp/10, so opp = 10 sin 30°.
sin 30° = 1/2 exactly, so opp = 5.
Adjacent: cos 30° = adj/10, and cos 30° = √3/2.
So adj = 10 × √3/2 = 5√3 ≈ 8.66.

Answer. Opposite 5, adjacent 5√3 ≈ 8.66 — the standard 30–60–90 triangle scaled by 5.

Where marks get dropped

These are the specific errors that cost credit on right-triangle ratios: sin, cos & tan questions — QED's rubric penalises each of them separately.

Using the wrong side as "adjacent". It is the non-hypotenuse side touching your angle, and it swaps when you change angle.
Leaving the calculator in radians when the question is in degrees. This produces plausible-looking but wrong numbers.
Applying SOH CAH TOA to a non-right triangle. Those need the sine or cosine rule instead.

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Right-triangle ratios: sin, cos & tan — frequently asked questions

How do I know which ratio to use?

List what you know and what you want. If the hypotenuse is involved use sine or cosine; if only the two shorter sides, use tangent.

What are the exact values worth memorising?

sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, and the cosines in reverse order. tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Why is the hypotenuse always longest?

Because it is opposite the largest angle, the right angle. Consequently sine and cosine of an acute angle are always between 0 and 1.

The rest of Trigonometry

Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.

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