The Sine and Cosine Rule: When Right-Angle Trig Isn't Enough

Quick answer: Use the sine rule when you have a matching pair — a side and the angle directly opposite it. Use the cosine rule when you have either all three sides (to find an angle) or two sides and the angle between them (to find the third side). SOHCAHTOA only works on right-angled triangles, so the moment a triangle has no right angle, one of these two rules is your tool.

When I first hit non-right-angled triangles in IGCSE, my instinct was to force SOHCAHTOA onto everything, get an impossible answer, and blame my calculator. The real problem was simpler: SOH CAH TOA needs a right angle. No right angle, no SOHCAHTOA — that is literally why the sine and cosine rules exist.

Here is the part that made me relax about them: on Cambridge IGCSE and the current GCSE papers (AQA, Edexcel and OCR), all three formulae are printed for you on the formula sheet. The exam is not testing whether you memorised them. It is testing whether you can choose the right one and substitute into it correctly. Get the choice right and these become some of the most dependable marks on the whole Higher paper.

First: label the triangle properly

Both rules depend on one convention, and skipping it is where most wrong answers begin. Side a is opposite angle A, side b is opposite angle B, side c is opposite angle C. Lowercase letters for sides, capitals for angles, and every side sits directly across the triangle from its matching angle — never next to it. Ten seconds spent labelling first saves you from the classic disaster of pairing a side with the angle it touches instead of the angle it faces.

The sine rule: when you have a matching pair

Reach for the sine rule when you have (or want) a matching pair — a side and the angle opposite it — plus one more known value.

• To find a side: a / sin A = b / sin B = c / sin C
• To find an angle: sin A / a = sin B / b = sin C / c

They are the same equation flipped upside down. Put what you are looking for on top: sides on top for a side, sines on top for an angle. You only ever use two of the three fractions at once — the pair you have and the pair you want.

The cosine rule: when there is no matching pair

If you do not have a matching pair, the sine rule cannot even start, and you switch to the cosine rule. Two situations trigger it:

• Two sides and the included angle (the angle between them), and you want the third side: a^2 = b^2 + c^2 - 2bc cos A
• All three sides, and you want an angle: cos A = (b^2 + c^2 - a^2) / (2bc)

The word that matters is included. For the "find a side" version, the known angle must sit between the two known sides. If it does not, you are back to the sine rule.

The whole decision in three questions

1. Is it right-angled? Use SOHCAHTOA (or Pythagoras for a missing side).
2. Not right-angled — do you have a matching pair (a side and its opposite angle)? Use the sine rule.
3. No matching pair (three sides, or two sides and the angle between them)? Use the cosine rule.

Run those three questions in order and you will never pick the wrong tool. If choosing is the bit that trips you up, throw a triangle at the Math solver and watch which rule it picks and why.

Don't forget the area rule

There is a third formula that lives with these two, and it is free marks when it turns up: the area of any triangle is (1/2) ab sin C — two sides and the included angle between them. Same "included angle" idea as the cosine rule. If a question hands you two sides and the angle wedged between them and asks for area, this is the one.

Worked example: cosine rule, then sine rule

Triangle ABC. Angle A = 62°, side b = 7 cm and side c = 9 cm. Find side a, then find angle B.

Step 1 — choose the rule. I have two sides (b and c) and the angle between them (A). That is two sides and the included angle, so it is the cosine rule.

Step 2 — substitute into a^2 = b^2 + c^2 - 2bc cos A:

a^2 = 7^2 + 9^2 - 2 × 7 × 9 × cos 62°.

Step 3 — work it out (calculator in degree mode):

a^2 = 49 + 81 - 126 × cos 62° = 130 - 126 × 0.4695 = 130 - 59.15 = 70.85.

Step 4 — square root: a = 8.42 cm (3 s.f.). Notice I kept the full value in the calculator and only rounded at the very end.

Step 5 — now find angle B. I have a matching pair (side a with angle A), so I switch to the sine rule with sines on top:

sin B / b = sin A / a, so sin B = 7 × sin 62° / 8.417 = 6.181 / 8.417 = 0.7343.

Step 6 — B = arcsin(0.7343) = 47.2° (3 s.f.). Because side b is shorter than side a, angle B must be acute, so there is no ambiguity here. If I wanted angle C, I would just do 180° - 62° - 47.2° = 70.8°.

The ambiguous case: the trap that costs the most

Here is the one that catches people out. When you use the sine rule to find an angle, your calculator only ever gives you the acute answer — but sometimes an obtuse angle fits the triangle just as well. This happens when you are given two sides and an angle that is not between them (the SSA case).

Mini-example: angle A = 30°, side a = 5, side b = 8. Find angle B.

sin B = b sin A / a = 8 × sin 30° / 5 = 8 × 0.5 / 5 = 0.8.

Your calculator says B = 53.1°. But B = 180° - 53.1° = 126.9° also has a sine of 0.8. Check both: 30 + 53.1 = 83.1° and 30 + 126.9 = 156.9°, both under 180°, so both make a real triangle. There are genuinely two answers.

The rule: after any sine-rule angle, ask whether the obtuse partner (180° minus your answer) also fits. If the angles still add to less than 180°, both are valid and a full-mark answer gives both. If the obtuse version would push the total over 180°, discard it. The cosine rule never has this problem — its cos value comes out negative for an obtuse angle, so it hands you the correct angle directly.

Common mistakes that quietly lose marks

• Calculator in radian mode. If your answers look wild, check the little R or D at the top of the screen. It must say D for degrees.
• Rounding too early. Keep full accuracy through the working and round only the final answer, usually to 3 significant figures. Rounding mid-way drifts the answer and can cost the accuracy mark.
• Skipping the substitution line. The method mark is for the correctly substituted formula. Write it out even if the arithmetic then goes wrong — you still bank that mark.
• Using the sine rule with no matching pair. Two sides and the angle between them cannot start the sine rule. That is a cosine-rule setup.
• Forgetting the second angle. On SSA problems, always test the obtuse case before you move on.

Test yourself

1. In triangle PQR, angle P = 40°, side q = 10 cm and side r = 6 cm. Find side p.
2. A triangle has sides 5 cm, 7 cm and 9 cm. Find the largest angle.
3. In triangle ABC, angle A = 35°, side a = 6 and side b = 9. Find the two possible sizes of angle B.

Quick answers: (1) Cosine rule: p^2 = 100 + 36 - 120 × cos 40° = 136 - 91.93 = 44.07, so p = 6.64 cm. (2) The largest angle faces the longest side (9). cos = (25 + 49 - 81) / (2 × 5 × 7) = -7 / 70 = -0.1, so the angle = 95.7°. (3) sin B = 9 × sin 35° / 6 = 0.8604, so B = 59.4° or B = 120.6° — both fit, since 35 + 120.6 = 155.6° is still under 180°.

Check your working line by line with the Math solver, and if the whole topic still feels shaky, ask Explain anything to walk you through it, then lock it in with a quick Quiz.

FAQ

When do I use the sine rule versus the cosine rule?

Sine rule when you have a matching pair (a side and its opposite angle). Cosine rule when you do not — that is, three sides, or two sides with the angle between them. If a matching pair exists, the sine rule is usually the quicker route.

Do I have to memorise the formulae for the exam?

On Cambridge IGCSE and current GCSE papers they are given on the formula sheet, so no. But you must know when to use each and how to substitute — that is what earns the marks. AP and A-Level students should still be able to recall and rearrange them from memory.

Can I use the cosine rule to dodge the ambiguous case?

Often, yes. If you find an unknown side first and then use the cosine rule to get an angle, you sidestep the two-answer problem entirely, because the cosine rule returns obtuse angles correctly on its own.

Why does my answer come out as an error or nonsense?

Nine times out of ten it is degree/radian mode, or a mislabelled triangle where a side has been paired with the angle it touches rather than the angle opposite it. Check both before anything else.

In short: no right angle means no SOHCAHTOA — use the sine rule when you have a matching pair, the cosine rule when you do not, and always check the ambiguous case whenever the sine rule hands you an angle.