OCR GCSE Mathematics: Sine rule and cosine rule (J560) – Revision Notes

These revision notes condense sections 10.05d (sine rule) and 10.05e (cosine rule) of the OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026), together with the area formula ½ab sin C from 10.03a. All of it is Higher tier only: it appears only in the Higher tier column of the specification. For full explanations and longer worked examples, read the study guide.

Related links: the OCR GCSE Mathematics hub, the printable checklist, the practice set and the free 10-minute diagnostics.

Labels

In triangle ABC, side a is opposite angle A, side b is opposite B and side c is opposite C.

        A
       / \
    c /   \ b
     /     \
    B-------C
         a

A side and the angle opposite it form a pair. The sine rule needs a pair. The cosine rule is for cases with no pair.

Formulae

Ref	Result	Use for
10.05d	a / sin A = b / sin B = c / sin C	lengths and angles (Higher tier only)
10.05d	sin A / a = sin B / b = sin C / c	the same rule, upside down, for angles (Higher tier only)
10.05e	a² = b² + c² - 2bc cos A	lengths (Higher tier only)
10.05e	cos A = (b² + c² - a²) / (2bc)	angles (Higher tier only)
10.03a	Area = ½ab sin C	area from two sides and the angle between (Higher tier only)

In 6.02d, the Higher tier column lists these formulae among those to recall and use, so learn all five lines.

Which rule?

You know	Want	Rule
Two angles and a side	another side	Sine
A pair, plus a side or angle	the missing side or angle in that equation	Sine
Two sides and the angle between	third side	Cosine
Three sides	an angle	Cosine
Two sides and the angle between	area	½ab sin C

If only two angles are given, find the third first: the angles add up to 180°.

Method boxes

Sine rule for a length

1. Label the triangle and mark the pair you know.
2. Write the equation with the unknown side on top: x / sin(its angle) = known side / sin(its angle).
3. Multiply both sides by sin(its angle).
4. Round at the end.

Sine rule for an angle

1. Write sin x / (its side) = sin(known angle) / (known side).
2. Multiply to get sin x on its own.
3. Use sin⁻¹. Ask whether the obtuse angle 180° - x also fits.

Cosine rule for a length

1. Identify the included angle and the two sides around it.
2. Write a² = b² + c² - 2bc cos A with your numbers in.
3. Work out 2bc cos A as one signed number.
4. Subtract it from b² + c², then take the square root.

Cosine rule for an angle

1. Put the side opposite the wanted angle first in the numerator: cos A = (b² + c² - a²) / (2bc).
2. Work the top and the bottom separately, then divide.
3. Use cos⁻¹. A negative cosine gives an obtuse angle.

Small worked reminders

Length by sine rule. A = 52°, B = 38°, a = 8.6 cm. Find b. b = 8.6 × sin 38° / sin 52° = 6.719… So b = 6.72 cm.

Angle by cosine rule. Sides 4, 5, 8 (all in cm). The largest angle faces the 8. cos = (4² + 5² - 8²) / (2 × 4 × 5) = -23/40 = -0.575. The angle is 125.1°.

Area. Sides 10 cm and 14 cm with 150° between them. Area = ½ × 10 × 14 × sin 150° = 70 × 0.5 = 35 cm².

Cosine rule becomes Pythagoras. If A = 90° then cos A = 0, so a² = b² + c².

Must-know distinctions

• Sine rule or cosine rule? Look for a pair. A pair means sine. Two sides with the angle between them, or three sides, means cosine.
• Pair or not? Two sides and an angle are not a pair unless the angle faces one of the sides.
• Two answers or one? The sine rule can give an acute angle and an obtuse one. Cosine gives exactly one angle, so find the largest angle first and then use the sine rule safely for the rest.
• Included angle or not? The angle in ½ab sin C must be between the sides a and b.
• Sign. For an obtuse angle, cos is negative, so -2bc cos A adds to b² + c².

Checks before you move on

• Angle sum. The three angles must add to 180°. After finding two angles, find the third both ways if you have time.
• Big side, big angle. The longest side faces the largest angle, and the shortest side faces the smallest. If your answer breaks this, look for a slip.
• Obtuse check. If the largest side squared is more than the other two squared added together, the angle opposite it is obtuse and its cosine is negative.
• Sensible size. A side found by the cosine rule must be shorter than the sum of the other two sides.

Worked reminder of a full set of angles. Take the 4 cm, 5 cm and 8 cm triangle again. The cosine rule gave the largest angle as 125.1°. For the angle opposite the 5 cm side, the sine rule gives sin B = 5 × sin 125.1° / 8 = 0.5118…, so B = 30.8°. The angle opposite the 4 cm side is then 180° - 125.1° - 30.8° = 24.1°. Cosine rule on the 4 cm side confirms 24.1°. Safe to use the sine rule here because the other two angles must be acute.

Setting out multi-step answers

1. Draw and label one clear triangle. Add any given bearings, then work out the interior angle.
2. Write the rule you are using, then the substitution, then the answer. Showing the substitution lets a reader see your method even if a number slips.
3. Keep each unrounded result and use it in the next step.
4. Write the final answer with units (cm, km, degrees) and the accuracy the question asks for.
5. Re-read the question. Does it ask for a length, an angle, an area or a bearing? Bearings are measured clockwise from north, so you may need to add or subtract an angle from a given bearing as a last step.

Quick self-test

1. In triangle ABC, A = 35°, B = 80° and a = 7.2 cm. Find b to 3 s.f.
2. In triangle ABC, a = 9.1 cm, b = 6.4 cm and A = 72°. Find B to 1 d.p.
3. In triangle ABC, b = 5.5 cm, c = 8.0 cm and A = 40°. Find a to 3 s.f.
4. A triangle has sides 5 cm, 6 cm and 7 cm. Find its largest angle to 1 d.p.
5. Two sides of a triangle are 6.5 cm and 9.2 cm, with an angle of 50° between them. Find the area to 3 s.f.
6. A = 58° and B = 74° in triangle ABC. Find C.
7. State what the cosine rule becomes when A = 90°.
8. In triangle ABC, A = 30°, a = 12 cm and b = 17 cm. Find both possible values of B to 1 d.p. and say whether each fits in the triangle.
9. Which rule do you start with for a triangle where you know all three sides and want an angle?
10. In the formula Area = ½ab sin C, which angle do you use?

Answers

1. b = 7.2 × sin 80° / sin 35° = 12.36… so 12.4 cm.
2. sin B = 6.4 × sin 72° / 9.1 = 0.6688…, so B = 41.98…°, which is 42.0°. The given angle faces the longer side, so the obtuse angle is ruled out.
3. a² = 30.25 + 64 - 88 × cos 40° = 94.25 - 67.41… = 26.84…, so a = 5.18 cm.
4. cos = (5² + 6² - 7²) / (2 × 5 × 6) = 12/60 = 0.2, so the angle is 78.5°.
5. Area = ½ × 6.5 × 9.2 × sin 50° = 22.9 cm².
6. C = 180° - 58° - 74° = 48°.
7. cos 90° = 0, so it becomes a² = b² + c² (Pythagoras).
8. sin B = 17 × sin 30° / 12 = 0.7083…, so B = 45.1° or 134.9°. Both fit, because 30° + 45.1° = 75.1° and 30° + 134.9° = 164.9°, and both are below 180°.
9. The cosine rule, rearranged to make cos of the angle the subject.
10. The angle between sides a and b, which is the angle C opposite the third side.

Where marks are usually lost

• Pairing a side with the sine of an angle that is not opposite it. Always draw the triangle and label it.
• Choosing the sine rule when the given angle sits between two known sides, so no pair exists.
• Computing (b² + c² - 2bc) × cos A instead of b² + c² - (2bc cos A).
• Dropping the negative sign of cos for an obtuse angle.
• Rounding an intermediate length or angle, then using it in the next part. Keep the unrounded value.
• Giving an angle to the wrong accuracy. Read the question for 1 d.p. or 3 s.f., and round only once.
• Forgetting that the sine rule can give an obtuse answer, or giving a second answer that does not fit the triangle.
• Using an angle in ½ab sin C that is not between the two sides.
• Stopping after the first stage of a multi-step problem, such as finding a length but not the bearing or area asked for.

Official syllabus

OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026), for first assessment in 2017, published by OCR. Sections 10.05d, 10.05e, 10.03a and 6.02d. All content on this page is Higher tier only.