The sine rule and cosine rule, explained
SOHCAHTOA only works in right-angled triangles. The sine rule and cosine rule work in any triangle, so you can find missing sides and angles in shapes such as roof trusses, survey plots and navigation routes.
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Labelling the triangle
Both rules depend on consistent labels. Call the angles A, B and C, and call the side opposite each angle by the matching lower-case letter: a is opposite A, b is opposite B and c is opposite C. Take a moment to label any exam triangle this way before choosing a rule; most errors come from mismatched pairs.
The sine rule
a / sin A = b / sin B = c / sin C
or, flipped, which is easier when finding an angle:
sin A / a = sin B / b = sin C / c
Use the sine rule when you know a side and its opposite angle, plus one other side or angle. In other words, you have a complete “pair” and one more piece of information.
The cosine rule
a² = b² + c² − 2bc cos A
or, rearranged to find an angle:
cos A = (b² + c² − a²) / 2bc
Use the cosine rule when you know two sides and the angle between them (to find the third side), or all three sides (to find an angle). Notice that when A = 90°, cos A = 0 and the cosine rule becomes Pythagoras' theorem.
Which rule should I use?
- Do you know a side and its opposite angle? Use the sine rule.
- Do you know two sides and the angle between them, or all three sides? Use the cosine rule.
- Is the triangle right-angled? SOHCAHTOA or Pythagoras is quicker.
Worked examples
Example 1: sine rule for a side
In triangle ABC, A = 40°, B = 75° and a = 8 cm. Find b.
- b/sin 75° = 8/sin 40°.
- b = 8 sin 75°/sin 40° ≈ 12.0 cm.
Example 2: sine rule for an angle
In triangle PQR, p = 9 cm, q = 7 cm and P = 64°. Find Q.
- sin Q/7 = sin 64°/9, so sin Q = 7 sin 64°/9 ≈ 0.699.
- Q = sin−1(0.699) ≈ 44.4°.
Example 3: cosine rule for a side
Two sides of a triangle are 5 cm and 8 cm, with an included angle of 60°. Find the third side.
- a² = 25 + 64 − 2 × 5 × 8 × cos 60° = 89 − 40 = 49.
- a = 7 cm. On a non-calculator paper, cos 60° = ½ makes this exact.
Example 4: cosine rule for an angle
A triangle has sides 6 cm, 7 cm and 10 cm. Find the largest angle.
- The largest angle is opposite the longest side, 10 cm.
- cos A = (36 + 49 − 100)/(2 × 6 × 7) = −15/84 ≈ −0.179.
- A ≈ 100.3°. A negative cosine means the angle is obtuse.
Example 5: area of a triangle
Find the area of the triangle in Example 3.
- Area = ½ab sin C = ½ × 5 × 8 × sin 60° = 20 × √3/2.
- Area = 10√3 ≈ 17.3 cm².
Example 6: a bearings problem
A ship sails 12 km on a bearing of 050°, then 9 km on a bearing of 120°. How far is it from its start?
- The angle between the two legs at the turning point is 180° − (120° − 50°) = 110°.
- d² = 144 + 81 − 2 × 12 × 9 × cos 110° ≈ 225 + 73.9 = 298.9.
- d ≈ 17.3 km. A sketch with north lines at each point is essential for bearings.
Multi-step questions
Harder questions chain the rules together: use the cosine rule to find a side, then the sine rule to find an angle, then the area formula. Keep unrounded values in your calculator memory between steps, and round only the final answer, or small rounding errors can grow enough to lose the accuracy mark.
The ambiguous case (Higher and A level)
When you use the sine rule to find an angle, there can be two possible answers, because sin θ = sin(180° − θ). In Example 2, Q could be 44.4° or 135.6°. The second is only possible if it fits in the triangle with the given angle: 135.6° + 64° = 199.6°, which is more than 180°, so here only 44.4° works. Check the obtuse option whenever the question hints there might be two triangles, or asks for “all possible values”.
Area using sine
Area = ½ ab sin C, where a and b are two sides and C is the angle between them. It may appear on the formulae sheet for your exam year, but it is worth learning. It is often combined with the sine or cosine rule in multi-step questions: first find a missing side or angle, then the area.
Common mistakes
- Pairing a side with the wrong angle in the sine rule.
- Using the cosine rule with an angle that is not between the two known sides.
- Calculating b² + c² − 2bc and then multiplying by cos A, instead of doing the multiplication 2bc cos A first.
- Forgetting to square root at the end when finding a side.
- Having the calculator in radians instead of degrees.
Practice questions
Try each one before opening the answer.
Q1. A = 30°, a = 5, B = 45°. Find b.
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Q2. Sides 4 and 6 with included angle 90°. Find the third side using the cosine rule.
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Q3. A triangle has sides 5, 5 and 8. Find the angle opposite the 8.
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Q4. Find the area of a triangle with sides 10 cm and 12 cm and included angle 150°.
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Frequently asked questions
Are the sine and cosine rules on GCSE Foundation?
No. They are Higher tier GCSE topics. Foundation students use SOHCAHTOA and Pythagoras in right-angled triangles.
Are they given in the GCSE formula sheet?
For GCSE exams in 2025, 2026 and 2027, a formulae sheet is provided with the Maths papers, covering formulae students previously had to memorise. Check the sheet for your exam year. Learn the formula anyway: recalling it instantly saves time and avoids copying errors. A level formula booklets may differ; check yours.
Key takeaways
- Label side a opposite angle A, and so on.
- Sine rule: a/sin A = b/sin B = c/sin C, used with a complete side-angle pair.
- Cosine rule: a² = b² + c² − 2bc cos A, used with two sides and the included angle, or three sides.
- Area = ½ab sin C.
- Watch for the ambiguous case when finding an angle with the sine rule.