Trigonometry in Right-Angled Triangles (SOH CAH TOA) — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the three trigonometric ratios and how to use them in right-angled triangles. By the end of this guide you should be able to label the sides of a triangle relative to a given angle, choose the correct ratio, and use it to find either a missing side or a missing angle.

You should also be able to work with the exact values of sin, cos and tan at the standard angles, apply the ratios to problems involving angles of elevation and depression, and know when to reach for trigonometry rather than Pythagoras' theorem.

The organising idea is that the labelling comes first, and it decides everything else. The names "opposite" and "adjacent" are not fixed properties of a triangle — they depend entirely on which angle you are working from. Once the three sides are labelled relative to that angle, the ratio you need is whichever one contains the two sides the question mentions, and the arithmetic follows automatically. Students who go wrong in this topic almost always went wrong at the labelling stage, before any calculation began.

Key terms and definitions

Hypotenuse — the side opposite the right angle. This one never changes, whichever angle you work from.

Opposite — the side facing the angle you are using.

Adjacent — the side next to that angle which is not the hypotenuse.

θ (theta) — the usual symbol for the angle being used.

SOH CAH TOA — the memory aid holding the three ratios: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

Inverse function — sin⁻¹, cos⁻¹ and tan⁻¹, used to get from a ratio back to the angle that produced it.

Angle of elevation — the angle measured upwards from the horizontal. Angle of depression is measured downwards from the horizontal.

Core concepts

Labelling relative to the angle

The hypotenuse is fixed: it is always opposite the right angle. The other two names move.

Stand at the angle you have been given. The side directly across the triangle from it is the opposite. The remaining side, the one you are standing next to, is the adjacent.

Work from the other acute angle instead and those two names swap over. This is why a question that changes which angle it asks about changes which ratio you need, even though the triangle has not moved.

Choosing the ratio

Write down which two sides the question involves — the one you know and the one you want — and pick the ratio containing exactly those two.

Opposite and hypotenuse → sin. Adjacent and hypotenuse → cos. Opposite and adjacent → tan.

Notice that the third side never appears. If a question gives you the opposite and asks for the adjacent, the hypotenuse is irrelevant, and trying to find it first is wasted work.

Finding a side

Substitute into the ratio and then solve the resulting equation, exactly as you would any other equation.

If the unknown is on the top of the fraction, multiply. For an angle of 30° and a hypotenuse of 10, the opposite is found from sin 30° = opposite/10, so opposite = 10 × sin 30° = 5.

If the unknown is on the bottom, divide. For an angle of 30° and an opposite of 5, the hypotenuse is found from sin 30° = 5/hypotenuse, so hypotenuse = 5 ÷ sin 30° = 10.

That is the second decision in the topic, and it is made by looking at where the unknown sits, not by memorising two separate formulas.

Finding an angle

When both sides are known, form the ratio as a number and then apply the inverse function.

If the opposite is 6 and the adjacent is 8, then tan θ = 6/8 = 0.75, so θ = tan⁻¹(0.75) = 36.9° to one decimal place.

The inverse functions are usually the SHIFT or 2nd function of the sin, cos and tan keys. Make sure the calculator is in degrees mode: an answer in radians looks plausible but is wrong, and it is worth checking that sin 30° gives exactly 0.5 before starting.

Exact values worth knowing

Some angles have exact values that the specification expects you to recall without a calculator.

sin 30° = 1/2 and cos 60° = 1/2. cos 30° = √3/2 and sin 60° = √3/2. sin 45° = cos 45° = 1/√2. tan 45° = 1, tan 30° = 1/√3 and tan 60° = √3. Also sin 0° = 0, cos 0° = 1 and tan 0° = 0.

The value tan 45° = 1 is worth understanding rather than memorising: at 45° the triangle is isosceles, so the opposite and adjacent are equal and their ratio must be 1.

Elevation and depression

An angle of elevation is measured upwards from the horizontal, as when looking up at the top of a building. An angle of depression is measured downwards from the horizontal, as when looking down from a cliff to a boat.

Both are measured from the horizontal, never from the vertical — that is the usual error. In a question about a person looking up at a tower, the horizontal ground is the adjacent side and the height of the tower is the opposite.

Trigonometry or Pythagoras?

The two tools do different jobs, and the choice takes a moment to make.

If the question involves an angle, whether given or wanted, use trigonometry. If it involves three sides and no angle at all, use Pythagoras' theorem.

A question giving two sides and asking for the third has no angle in it, so Pythagoras is faster and exact.

Worked examples

Example 1: Finding a side with the unknown on top

A right-angled triangle has a hypotenuse of 12 cm and an angle of 40°. Find the length of the side opposite that angle, to one decimal place.

The two sides involved are the opposite and the hypotenuse, so the ratio is sin.

sin 40° = opposite/12, so opposite = 12 × sin 40°.

That gives 12 × 0.6428 = 7.7 cm to one decimal place.

Sense check: the opposite must be shorter than the hypotenuse, and 7.7 is less than 12. ✓

Example 2: Finding a side with the unknown on the bottom

The side opposite an angle of 25° measures 7 cm. Find the hypotenuse, to one decimal place.

Again the opposite and hypotenuse are involved, so use sin.

sin 25° = 7/hypotenuse. The unknown is underneath, so divide: hypotenuse = 7 ÷ sin 25°.

That gives 7 ÷ 0.4226 = 16.6 cm to one decimal place.

Sense check: the hypotenuse is the longest side, and 16.6 is greater than 7. ✓ An answer of 3 cm would have signalled that the calculation was done the wrong way round.

Example 3: Finding an angle

A ladder of length 6 m rests with its foot 2 m from a wall. Find the angle it makes with the ground, to one decimal place.

The ladder is the hypotenuse and the 2 m along the ground is adjacent to the angle we want, so the ratio is cos.

cos θ = 2/6 = 0.3333.

Apply the inverse: θ = cos⁻¹(0.3333) = 70.5° to one decimal place.

Sense check: a steep ladder should give a large angle, and 70.5° is consistent with a base much shorter than the ladder.

Common mistakes and how to avoid them

Labelling before choosing the angle. Opposite and adjacent depend on which angle you are using. Fix the angle first, then label.

Choosing the ratio by habit. Identify the two sides the question actually involves and pick the ratio containing both.

Multiplying when you should divide. Look at where the unknown sits in the fraction: on top, multiply; underneath, divide.

Forgetting the inverse function when finding an angle. tan θ = 0.75 is not the answer; θ = tan⁻¹(0.75) is.

Leaving the calculator in radians. Check that sin 30° returns 0.5 before you begin.

Measuring elevation or depression from the vertical. Both angles are measured from the horizontal.

Rounding partway through. Keep full accuracy in the calculator and round only the final answer.

Exam technique for "Trigonometry in Right-Angled Triangles"

Label all three sides relative to the given angle before writing any equation. It takes seconds and prevents the error that accounts for most lost marks here.

Write the chosen ratio with the numbers substituted — "sin 40° = x/12" — as its own line. That substitution is where the method mark sits, and it is awarded even if the calculator work afterwards is wrong.

Say which ratio you are using. Naming sin, cos or tan makes the method visible.

Sense-check every side you find: the hypotenuse is the longest side, and any other side must be shorter than it.

Watch the required accuracy, and give angles in degrees with the degree symbol.

When a question gives three sides and no angle, use Pythagoras' theorem instead — it is quicker and gives an exact answer.

Quick revision summary

Label first. The hypotenuse is opposite the right angle and never moves. The opposite faces the angle you are using and the adjacent sits beside it, so both swap if you change angles.

SOH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Choose the ratio containing the two sides the question mentions.

To find a side: if the unknown is on top, multiply; if it is underneath, divide. So opposite = hyp × sin θ, but hyp = opposite ÷ sin θ.

To find an angle: form the ratio as a decimal, then apply sin⁻¹, cos⁻¹ or tan⁻¹.

Know the exact values: sin 30° = cos 60° = 1/2, cos 30° = sin 60° = √3/2, sin 45° = cos 45° = 1/√2, and tan 45° = 1.

Angles of elevation and depression are both measured from the horizontal.

Use trigonometry when an angle is involved; use Pythagoras' theorem when the question has three sides and no angle.