Trigonometry at GCSE: SOH CAH TOA explained

Trigonometry is one of those GCSE Maths topics that sounds intimidating but actually follows a very predictable pattern. Once you learn the three ratios and practise the method a few times, you can answer almost any right-angled triangle question the exam throws at you.

This guide covers everything you need: The three trig ratios (SOH CAH TOA), how to pick the right one, worked examples for finding missing sides and angles, the exact trig values you need for higher tier, and a checklist you can use every time you see a trig question.

SOH CAH TOA is a mnemonic that helps you remember the three trigonometric ratios. Each group of three letters tells you the ratio between two sides of a right-angled triangle.

SOH means sin θ = opposite / hypotenuse. CAH means cos θ = adjacent / hypotenuse. TOA means tan θ = opposite / adjacent.

The angle θ (theta) is the angle you are working with – not the right angle. The hypotenuse is always the longest side, sitting opposite the right angle. The opposite side is the one directly across from the angle θ. The adjacent side is the one next to the angle θ that is not the hypotenuse.

Getting these labels right is the single most important step. If you label the sides correctly, the rest is just substitution and rearranging.

Every trig question gives you one side and one angle (or two sides), and asks you to find something missing. The trick is to look at which two sides are involved – the one you know and the one you want – and then pick the ratio that connects them.

If you have the opposite and the hypotenuse, use sin (SOH). If you have the adjacent and the hypotenuse, use cos (CAH). If you have the opposite and the adjacent, use tan (TOA).

A quick way to decide: Label all three sides, then cross out the side you neither know nor need. The two sides left tell you which ratio to use. For example, if you know the adjacent and want the hypotenuse, cross out the opposite – you are left with adjacent and hypotenuse, so use cos.

Suppose you have a right-angled triangle where the angle is 35° and the hypotenuse is 12 cm. You want to find the length of the opposite side.

Step 1: Label the sides. The hypotenuse is 12 cm. The side opposite the 35° angle is the one you want. The adjacent side is not involved.

Step 2: Choose the ratio. You have the hypotenuse and want the opposite, so use SOH – sin θ = opposite / hypotenuse.

Step 3: Substitute. sin 35° = x / 12.

Step 4: Rearrange. Multiply both sides by 12 to get x = 12 × sin 35°.

Step 5: Calculate. Using your calculator, sin 35° = 0.5736 (to 4 decimal places), so x = 12 × 0.5736 = 6.88 cm (to 2 decimal places).

That is the entire method. Every missing-side question follows the same five steps: label, choose, substitute, rearrange, calculate.

Now suppose you have a right-angled triangle where the opposite side is 7 cm and the adjacent side is 10 cm. You want to find the angle θ.

Step 1: Label the sides. You have the opposite (7 cm) and the adjacent (10 cm). The hypotenuse is not involved.

Step 2: Choose the ratio. Opposite and adjacent means tan – TOA – so tan θ = opposite / adjacent.

Step 3: Substitute. tan θ = 7 / 10 = 0.7.

Step 4: Use the inverse function. To find the angle itself, you need inverse tan (written tan⁻¹ on your calculator). So θ = tan⁻¹(0.7).

Step 5: Calculate. θ = tan⁻¹(0.7) = 34.99°, which rounds to 35.0° (to 1 decimal place).

The key difference from finding a side is step 4. Whenever you are finding an angle, you use the inverse trig function – sin⁻¹, cos⁻¹, or tan⁻¹ – to go from a ratio back to an angle. On most calculators, you access these by pressing shift or 2nd function before the sin, cos, or tan button.

The inverse trig functions reverse what sin, cos, and tan do. If sin takes an angle and gives you a ratio, then sin⁻¹ takes a ratio and gives you an angle.

sin⁻¹ is also called arcsin. cos⁻¹ is also called arccos. tan⁻¹ is also called arctan. You might see either name in textbooks, but your calculator will just show sin⁻¹, cos⁻¹, and tan⁻¹.

You only need inverse trig when the question asks you to find a missing angle. If you are finding a missing side, you will not need them – you will rearrange the formula algebraically instead.

At higher tier, you are expected to know the exact values of sin, cos, and tan for three specific angles: 30°, 45°, and 60°. These come up in non-calculator questions and in proofs, so you cannot rely on your calculator for them.

The values come from two special triangles. The 45° values come from an isosceles right-angled triangle with two sides of length 1 and a hypotenuse of √2. The 30° and 60° values come from an equilateral triangle with side length 2, cut in half to create a right-angled triangle with sides 1, √3, and 2.

You do not need to derive them in the exam, but understanding where they come from makes them much easier to remember. If you forget a value, you can sketch the triangle and read it off.

A useful pattern to spot: The sin values for 0°, 30°, 45°, 60°, and 90° go 0, 1/2, √2/2, √3/2, 1. The cos values are the same sequence in reverse. If you remember the sin column, you automatically know the cos column.

For tan, notice that tan 45° = 1 (because opposite and adjacent are the same length in a 45° right-angled triangle), tan 30° is less than 1, and tan 60° is greater than 1. Tan 90° is undefined because you would be dividing by zero.

The most common error is mislabelling the sides. Students often confuse the adjacent and opposite sides, especially when the triangle is rotated or drawn in an unusual orientation. Always identify the right angle first, then label the hypotenuse (opposite the right angle), and finally work out which of the remaining sides is opposite and which is adjacent relative to the angle you are using.

Another frequent mistake is forgetting to use the inverse function when finding an angle. If you get an answer like 0.7 when you expected an angle in degrees, you probably forgot to press sin⁻¹, cos⁻¹, or tan⁻¹.

Rounding too early also costs marks. Keep at least four decimal places during your working and only round your final answer to the degree of accuracy the question asks for. If the question says "give your answer to 1 decimal place", only round at the very end.

Trigonometry at GCSE is really about one skill applied in different contexts: Labelling the sides, picking the right ratio, and doing some straightforward algebra. The questions change, but the method does not.

At foundation tier, you need to find missing sides and missing angles using SOH CAH TOA. At higher tier, you also need to know the exact values for 30°, 45°, and 60° and be comfortable using them without a calculator. Some higher-tier questions combine trigonometry with Pythagoras' theorem, surds, or algebraic proof – but the trig part still follows the same steps.

The best way to get confident is to practise. Work through a few questions slowly with the checklist above, and you will quickly find that the method becomes second nature.