GCSE Trigonometry: SOHCAHTOA Explained with Examples

What Is Trigonometry and Where It Appears on the Paper

Trigonometry is the study of the relationship between the angles and the side lengths of a triangle. At GCSE you work almost entirely with right-angled triangles, using three ratios: sine (sin), cosine (cos) and tangent (tan). Given one angle and one side, these ratios let you calculate any other side. Given two sides, they let you calculate any angle.

Every board tests it, on both tiers. On Foundation papers a SOHCAHTOA question is typically worth 2 to 4 marks and usually asks for a single missing side or angle. On Higher papers trigonometry gets combined with Pythagoras, bearings, area and 3D shapes, and the exact-value questions appear on the non-calculator paper. It sits in the same strand as Pythagoras' theorem, and the two are revised together on our Pythagoras and trigonometry topic page.

The good news is that every SOHCAHTOA question, however it is dressed up, comes down to the same three steps: label the sides, choose the ratio, solve the equation. This guide takes each step in turn.

Step One: Label the Sides Correctly

Most trigonometry mistakes happen before any calculation starts, at the labelling stage. The three sides of a right-angled triangle are named relative to the angle you are working with, so the same physical side can be "opposite" in one question and "adjacent" in another.

The hypotenuse (H) is the easiest: it is always the longest side, and it is always opposite the right angle. It never changes, whichever angle you use. The opposite (O) is the side directly across from the angle you are using, the one the angle "points at". The adjacent (A) is the side next to your angle that is not the hypotenuse.

Get into the habit of writing H, O and A on the diagram before doing anything else. Examiners cannot give you method marks for a correct ratio applied to the wrong sides, so ten seconds of labelling protects the whole question.

Step Two: Choose the Ratio with SOHCAHTOA

SOHCAHTOA is the mnemonic that stores all three ratios: SOH means $\sin\theta = \frac{O}{H}$, CAH means $\cos\theta = \frac{A}{H}$, and TOA means $\tan\theta = \frac{O}{A}$.

To choose the right one, look at which two sides the question involves: the side you know and the side you want (or, for an angle question, the two sides you know). Tick them off against the letters. Know the hypotenuse and want the opposite? That is O and H, so sin. Know the opposite and the adjacent? That is O and A, so tan. Exactly one ratio fits any pair of sides, so this step is mechanical once the labelling is right.

Many students like the formula-triangle version: write the three letters of the chosen ratio in a triangle with the top letter over the bottom two, cover the quantity you want, and what remains tells you whether to multiply or divide. Use whichever version you can do quickly under pressure; they are the same maths.

Finding a Missing Side (Unknown on Top)

When the side you want sits on top of the ratio (the O in sin or tan, or the A in cos), the rearrangement is a single multiplication. This is the friendlier of the two side-finding cases: write the ratio, substitute what you know, multiply.

Always write the ratio equation down before reaching for the calculator. On a 3-mark question, boards typically award a method mark for the correct ratio and substitution, so a slip on the calculator still leaves you with most of the marks if the working is on the page.

Finding a Missing Side (Unknown on the Bottom)

When the unknown side is on the bottom of the ratio, you must divide instead of multiply, and this is where a large share of exam errors happen. If $\cos 40^\circ = \frac{12}{h}$, then $h = \frac{12}{\cos 40^\circ}$, not $12 \times \cos 40^\circ$.

A quick sanity check catches the error every time: $\cos 40^\circ \approx 0.766$, which is less than 1, so dividing by it must make the answer bigger than 12. If you are finding a hypotenuse, your answer must be the longest side; if it comes out shorter than a given side, you multiplied when you should have divided.

Finding a Missing Angle with Inverse Functions

To find an angle from two known sides, you need the inverse trigonometric functions: $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ (also written arcsin, arccos, arctan). These undo the ratio: if $\tan\theta = 0.75$, then $\theta = \tan^{-1}(0.75)$. On most calculators they sit above the sin, cos and tan keys, accessed with SHIFT or 2nd.

Two practical warnings. First, check your calculator is in degrees mode (look for a small D on screen). In radians mode $\tan^{-1}(0.75)$ gives 0.6435, which is not a GCSE answer, and the working looks identical, so the error is easy to miss. Second, keep the fraction exact inside the inverse function rather than rounding early: $\tan^{-1}(6 \div 8)$ is safer than rounding 0.75 first (harmless here, but with a value like $5 \div 7$ early rounding shifts the final angle).

Exact Values for 30°, 45° and 60°

Both tiers now require certain trig values without a calculator: sin, cos and tan of $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$ (tan $90^\circ$ is undefined). These turn up on the non-calculator paper, usually as a short question in the form "write down the exact value" or embedded in a triangle problem.

If memorising a table feels brittle, learn the two triangles they come from instead. Cutting an equilateral triangle of side 2 in half gives a right-angled triangle with sides 1, $\sqrt{3}$ and 2, which produces all the $30^\circ$ and $60^\circ$ values. A right-angled isosceles triangle with legs of 1 has hypotenuse $\sqrt{2}$, which produces the $45^\circ$ values. Sketching one of these on the exam takes twenty seconds and is more reliable than a half-remembered list.

For example: a right-angled triangle has an angle of $60^\circ$ and the adjacent side is 5 cm. The opposite side is $5 \times \tan 60^\circ = 5\sqrt{3}$ cm, and on a non-calculator paper you leave the answer exactly in that surd form.

When SOHCAHTOA Fails: Triangles Without a Right Angle

SOHCAHTOA only works when the triangle has a right angle, because the three ratios are defined using the hypotenuse, and only right-angled triangles have one. If a question gives you a triangle with angles like $50^\circ$, $60^\circ$ and $70^\circ$ and no right angle marked, SOHCAHTOA is the wrong tool, and forcing it will produce a confident-looking wrong answer.

Higher tier has the tools for this situation: the sine rule ($\frac{a}{\sin A} = \frac{b}{\sin B}$) links each side to the angle opposite it, and the cosine rule ($a^2 = b^2 + c^2 - 2bc\cos A$) handles the cases the sine rule cannot, such as two sides with the angle between them. There is also the area formula $\frac{1}{2}ab\sin C$. These are separate techniques with their own practice questions, but the habit that matters here is the check itself: before writing SOH, CAH or TOA, confirm the right angle is actually marked on the diagram.

On Foundation you will not be asked about non-right-angled triangles in this way, so if you are on Foundation and a trig question appears, it is safe to assume SOHCAHTOA (or Pythagoras) is the intended method.

Exam Technique and How to Practise

Trigonometry rewards routine. The questions vary their context (ladders, ramps, bearings, shadows) but not their structure, so the fastest way to improve is to drill the three-step routine until choosing the ratio is automatic. Pair it with Pythagoras practice, because papers love combining them in one diagram: use Pythagoras when you know two sides and want the third with no angle involved, and trigonometry the moment an angle other than the right angle enters the question. Our guide to Pythagoras' theorem covers that side of the strand.

On marks: show the ratio, the substitution and the rearrangement as separate lines. Round only at the final step, and give at least 3 significant figures unless the question specifies. If the question says "exact value", a rounded decimal scores nothing; leave the surd in.

Exam Ladder generates trigonometry questions at every difficulty with fully worked solutions, so you can practise the routine on fresh numbers each time, and mock exams are graded on real exam-board grade boundaries so you can see what your trigonometry marks are worth. Check how long you have left with the exam calendar, and start with a 7-day free trial.