Maths
What are the sine rule and the cosine rule, and when do you use each?
Quick answer
The sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A find missing sides and angles in any triangle, not just right-angled ones.
In one sentence
The sine rule and the cosine rule are formulas that let you find missing sides and angles in any triangle, even when there is no right angle.
Explained
SOH CAH TOA and Pythagoras' theorem only work in right-angled triangles. Real triangles, such as the path of two ships, the frame of a roof or a field with three fences, usually have no right angle. The sine and cosine rules fill that gap.
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First, the labelling. Call the angles A, B and C, and call the side opposite each angle by the same small letter: side a is opposite angle A, side b is opposite angle B, side c is opposite angle C. Every formula on this page depends on that pairing.
The sine rule says that in every triangle, each side divided by the sine of its opposite angle gives the same number:
a ÷ sin A = b ÷ sin B = c ÷ sin C
To find an angle, it is easier to flip it: sin A ÷ a = sin B ÷ b = sin C ÷ c. You use only two of the three parts at a time, and you need one complete pair (a side and its opposite angle) to start.
The cosine rule is Pythagoras' theorem with a correction for angles that are not 90°:
a² = b² + c² − 2bc cos A
Rearranged to find an angle:
cos A = (b² + c² − a²) ÷ (2bc)
If A = 90°, then cos A = 0 and you get back a² = b² + c², Pythagoras' theorem. If A is obtuse, cos A is negative, so a is longer than Pythagoras would predict, which makes sense: a wider angle pushes the opposite side further out.
Which rule?
- Two angles and a side, or two sides and a non-included angle (a complete pair plus one more) → sine rule.
- Two sides and the angle between them (SAS) → cosine rule for the third side.
- All three sides (SSS) → cosine rule for an angle.
A related formula gives the area without the height: Area = ½ab sin C, using two sides and the angle between them.
Examples
Example 1 (sine rule, side). A = 40°, B = 60°, a = 10 cm. Find b. b ÷ sin 60° = 10 ÷ sin 40°, so b = 10 × sin 60° ÷ sin 40° = 8.660 ÷ 0.6428 ≈ 13.47 cm (2 d.p.).
Example 2 (sine rule, angle). a = 8 cm, A = 50°, b = 6 cm. Find B. sin B = 6 × sin 50° ÷ 8 ≈ 0.5745, so B ≈ 35.1° (1 d.p.). The other angle with this sine, 144.9°, is impossible here because 50° + 144.9° is more than 180°.
Example 3 (cosine rule, side). b = 5 cm, c = 7 cm, A = 60°. Find a. a² = 25 + 49 − 2 × 5 × 7 × cos 60° = 74 − 35 = 39, so a = √39 ≈ 6.24 cm.
Example 4 (cosine rule, angle). Sides 3, 5 and 7. Find the largest angle (opposite 7). cos C = (3² + 5² − 7²) ÷ (2 × 3 × 5) = −15 ÷ 30 = −0.5, so C = 120°.
Common mistakes
- Mismatched pairs. Side a must go with angle A, the angle opposite it. Fix: label the diagram before writing any formula.
- Calculator in radians. sin 30 in radian mode is about −0.988, not 0.5. Fix: check for a "D" or "DEG" on the screen.
- Working out the cosine rule in the wrong order. b² + c² − 2bc cos A is not (b² + c² − 2bc) × cos A. Fix: work out 2bc cos A first, then subtract.
- Missing the second answer (the ambiguous case). When you find an angle with the sine rule, both θ and 180° − θ have the same sine. Fix: check whether 180° − θ also fits in the triangle.
Step by step
Step by step
- Draw and labelSketch the triangle and label angles A, B, C and the opposite sides a, b, c.
- List what you knowWrite down the given sides and angles, and what you need to find.
- Choose the ruleA complete side–angle pair: sine rule. SAS or SSS: cosine rule.
- Substitute and solvePut the numbers in, rearrange, and use sin⁻¹ or cos⁻¹ for an angle.
- CheckThe longest side should face the largest angle, and the angles must add up to 180°.
The ambiguous case. If you know two sides and an angle that is not between them, there may be two possible triangles. With a = 6, b = 8 and A = 40°, sin B = 8 × sin 40° ÷ 6 ≈ 0.857, so B ≈ 59.0° or B ≈ 121.0°. Both work, because 40° + 121.0° is still less than 180°.
Practice questions
Sine rule and cosine rule
Try each one, then check your answer.
- A = 30°, B = 45°, a = 5 cm. Find b.
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b = 5 × sin 45° ÷ sin 30° ≈ 7.07 cm - A = 35°, B = 75°, c = 12 cm. Find a.
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C = 180° − 35° − 75° = 70°; a = 12 × sin 35° ÷ sin 70° ≈ 7.32 cm - a = 8 cm, A = 50°, b = 6 cm. Find B.
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sin B = 6 × sin 50° ÷ 8 ≈ 0.5745, so B ≈ 35.1° - b = 5 cm, c = 7 cm, A = 60°. Find a.
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a² = 25 + 49 − 35 = 39, so a ≈ 6.24 cm - b = 8 cm, c = 11 cm, A = 120°. Find a.
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cos 120° = −0.5, so a² = 64 + 121 + 88 = 273; a ≈ 16.52 cm - a = 9 cm, b = 12 cm, C = 50°. Find c.
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c² = 81 + 144 − 216 × cos 50° ≈ 86.16, so c ≈ 9.28 cm - A triangle has sides a = 7, b = 5, c = 8. Find angle A.
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cos A = (25 + 64 − 49) ÷ 80 = 0.5, so A = 60° - A triangle has sides 3, 5 and 7. Find its largest angle.
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cos C = (9 + 25 − 49) ÷ 30 = −0.5, so 120° - A triangle has sides 5, 12 and 13. Use the cosine rule to find the angle opposite 13.
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cos C = (25 + 144 − 169) ÷ 120 = 0, so 90° (a right angle) - Find the area of a triangle with a = 6 cm, b = 9 cm and C = 30°.
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½ × 6 × 9 × sin 30° = 13.5 cm² - Find the area of a triangle with a = 10 cm, b = 12 cm and C = 75°.
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½ × 10 × 12 × sin 75° ≈ 57.96 cm² - You know all three sides of a triangle. Which rule finds an angle?
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The cosine rule, rearranged: cos A = (b² + c² − a²) ÷ (2bc) - a = 6 cm, b = 8 cm, A = 40°. Find both possible values of B.
Show answer
sin B ≈ 0.857, so B ≈ 59.0° or 121.0° (two triangles) - In triangle ABC, A = 40° and a = 10 cm. Is it possible that B = 150°?
Show answer
40° + 150° = 190° > 180°, so no
Word problems
Sine and cosine rules in real life
Try each one, then check your answer.
- Two ships leave a port. One sails 12 km, the other 15 km, and the angle between their paths is 70°. How far apart are they?
Show answer
d² = 144 + 225 − 360 × cos 70° ≈ 245.87, so d ≈ 15.68 km - From a point on flat ground, the top of a tree is seen at an angle of elevation of 30°. Walking 20 m closer, the angle is 45°. How tall is the tree?
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The angle at the top is 45° − 30° = 15°; distance from the near point = 20 × sin 30° ÷ sin 15° ≈ 38.64 m; height = 38.64 × sin 45° ≈ 27.32 m - A triangular flower bed has sides of 8 m and 10 m with a 65° angle between them. Find its area.
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½ × 8 × 10 × sin 65° ≈ 36.25 m² - A field has sides of 40 m, 50 m and 60 m. Find the largest angle.
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cos θ = (1600 + 2500 − 3600) ÷ 4000 = 0.125, so θ ≈ 82.8° - Two 5 m roof beams meet at the top and their bottoms are 8 m apart. Find the angle at the top.
Show answer
cos θ = (25 + 25 − 64) ÷ 50 = −0.28, so θ ≈ 106.3°
Questions people ask
What is the sine rule used for?
To find a missing side or angle in any triangle when you know a side and the angle opposite it, plus one more side or angle.
What is the cosine rule used for?
To find the third side when you know two sides and the angle between them, or to find an angle when you know all three sides.
Does the sine rule work in right-angled triangles?
Yes, it works in every triangle. With a right angle, sin 90° = 1 and it reduces to the ordinary SOH CAH TOA ratios, which are usually quicker.
What is the ambiguous case of the sine rule?
When you know two sides and an angle that is not between them, there can be two different triangles, because sin θ = sin(180° − θ). Always check whether the obtuse answer also fits.
How do you remember the cosine rule?
Think "Pythagoras minus a correction": a² = b² + c² − 2bc cos A. The side on the left is opposite the angle on the right; the other two sides appear everywhere else.
Why does the sine rule work?
Drop a perpendicular h from C to side c. In the two right-angled triangles, h = b sin A and h = a sin B. So b sin A = a sin B, which rearranges to a ÷ sin A = b ÷ sin B.
Quick quiz
The sine rule and the cosine rule
Check what you know with 4 quick questions.
Question 1 of 4
You know two sides and the angle between them. Which rule finds the third side?
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Related topics
Trigonometry
Trigonometry links the angles and sides of right-angled triangles using sine, cosine and tangent. It lets you find a missing side or angle. SOH CAH TOA helps you remember the three ratios.
Pythagoras' theorem
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c², where c is the longest side (the hypotenuse). For sides 3 and 4, the hypotenuse is 5.
Angles
An angle measures the amount of turn between two lines that meet at a point. Angles are measured in degrees (°); a right angle is 90° and a full turn is 360°.
Angles in parallel lines
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Area of a trapezium (trapezoid)
Add the two parallel sides, multiply by the height and halve: A = ½(a + b)h. Parallel sides of 4 cm and 6 cm, 3 cm apart, give an area of 15 cm².