Cambridge IGCSE Mathematics 0580

Sine and cosine rules

Use the cosine rule when you know two sides and the angle between them, or all three sides. Use the sine rule when you know a side and the angle opposite it. The area of any triangle is $\frac{1}{2} a b \sin C$ , where $C$ is the angle between sides $a$ and $b$ . All three formulas are printed on the Extended papers.

On Cambridge IGCSE 0580 almost every question on this is a multi-step problem, often with bearings, where the marks go to the student who picks the right rule first time.

Updated 28 September 2026

How much it is worth

Sine and cosine rules: 192 marks across 42 real Cambridge 0580 papers — 4.6% of the total, on 27 of the 42. How that ranks against every other 0580 topic →

Which rule to use

You know	You want	Use
Two sides and the angle between them	The third side	Cosine rule
All three sides	An angle	Cosine rule (rearranged)
A side and its opposite angle, plus one more angle	Another side	Sine rule
A side and its opposite angle, plus one more side	Another angle	Sine rule (check for an obtuse answer)
Two sides and the angle between them	The area	½ab sin C

If the triangle has a right angle, neither rule is needed — use Pythagoras and SOHCAHTOA, which are quicker and not on the formula list. What the 0580 formula list gives you →

The sine rule

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Each side is paired with the angle opposite it. To find an angle, flip it over so the unknown is on top: $\frac{\sin A}{a} = \frac{\sin B}{b}$ . You only ever use two of the three fractions at once.

The cosine rule

a^{2} = b^{2} + c^{2} - 2 b c \cos A \cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}

$A$ is the angle between sides $b$ and $c$ , and $a$ is the side opposite it. The formula list gives only the first form; rearrange it for an angle. If $\cos A$ comes out negative, the angle is obtuse — your calculator handles that automatically with $\cos^{- 1}$ .

Area: ½ab sin C

Area = \frac{1}{2} a b \sin C

Two sides and the angle between them. If the angle you have is not between the two sides you have, find the right one first — usually with the sine rule, or from the angles of a triangle summing to 180°.

Obtuse angles and the ambiguous case

$\sin θ = \sin (180^{\circ} - θ)$ , so when the sine rule gives an angle, there may be a second, obtuse answer: $180^{\circ}$ minus the one the calculator shows. The 0580 syllabus lists this ambiguous case explicitly.

Check whether the obtuse answer is possible: it is only valid if it plus the angle you already know is less than 180°. If the diagram shows the angle is obtuse, or the question says so, take $180^{\circ} - θ$ . The cosine rule never has this problem, which is a reason to prefer it for angles when you have all three sides.

Worked example

A ship sails 8 km from A to B, then 5 km from B to C. Angle ABC = 115°. Find the distance AC and angle BAC, correct to 1 decimal place.

Two sides and the angle between them, so the cosine rule for AC:

A C^{2} = 8^{2} + 5^{2} - 2 (8) (5) \cos 115^{\circ} = 89 - 80 (- 0.4226) = 122.81

$A C = 11.08 \dots = 11.1 km$ . Keep the unrounded value for the next step.

Now a side and its opposite angle are known, so the sine rule for angle BAC:

\frac{\sin A}{5} = \frac{\sin 115^{\circ}}{11.08 \dots} \Rightarrow \sin A = 0.4089 \dots \Rightarrow A = {24.1}^{\circ}

The obtuse alternative, 155.9°, is impossible: the triangle already has a 115° angle.

Common mistakes

1.Using the cosine rule with the wrong angle.
The angle in $a^{2} = b^{2} + c^{2} - 2 b c \cos A$ must be the one between $b$ and $c$ , opposite $a$ . Label the triangle before substituting.
2.Calculating b² + c² − 2bc and then multiplying by cos A.
Work out $2 b c \cos A$ as one term and subtract it. Typing the whole thing into a calculator in one go, with brackets, avoids this.
3.Rounding the side before using it again.
Answers are to 3 significant figures (angles to 1 decimal place), but carry the full value into the next step, or the final answer drifts out of the mark scheme’s range.
4.Missing the obtuse angle.
When the sine rule gives an angle and the diagram looks obtuse, the answer is $180^{\circ} - θ$ .
5.Calculator in radians.
Check the display shows D or DEG. $\cos 115$ in radians is a different number entirely.

Common questions

Are the sine and cosine rules given in IGCSE 0580?

Yes, on the Extended papers (2 and 4), along with the area formula $\frac{1}{2} a b \sin C$ . They are Extended content, so they are not on the Core list.

How do I know whether to use the sine or cosine rule?

If you know a side and the angle opposite it, use the sine rule. Otherwise — two sides and the angle between them, or three sides — use the cosine rule.

How do bearings questions use the sine and cosine rules?

The bearings give the angles inside the triangle, usually via parallel north lines and co-interior angles summing to 180°. Once the triangle is drawn with its angles, it is an ordinary sine or cosine rule question. Bearings guide →

Sine and cosine rules on real papers

The papers that leaned on it hardest, as a share of the paper. Each one has its own page on Quanta with the full topic breakdown:

0580/42/O/N/2514 of 100 marks 0580/43/M/J/2416 of 130 marks 0580/41/M/J/2510 of 100 marks

Practise sine and cosine rules against real mark schemes

Quanta has real Cambridge IGCSE Maths 0580 past-paper questions, broken into skill checkpoints, marked criterion by criterion the way examiners mark — and it tracks which skills you’re missing. Free for individual students.

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