Trigonometry
Grade 6-8The Sine Rule
The sine rule works on any triangle, not just right-angled ones, connecting each side to the sine of the angle opposite it. This lesson covers the sine rule formula, choosing between the sine rule and the cosine rule, finding a missing side, and finding a missing angle.
What you need to know
- Use the sine rule when you have a matching side and opposite angle, plus one more piece of information.
- a over sin A equals b over sin B equals c over sin C, with each side opposite its angle.
- Flip it to sin A over a when you are finding an angle.
- Finding an obtuse angle can give two possible answers (the ambiguous case).
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The sine rule formula
In any triangle, each side divided by the sine of its opposite angle gives the same value: a/sin A = b/sin B = c/sin C, where side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Only two of the three ratios are needed at once, matched to whichever side and angle are missing.
Labelling matters: side a must be the side directly across from angle A, not next to it.
Choosing between the sine rule and the cosine rule
Use the sine rule when a matching pair, one angle and the side opposite it, is already known, alongside one more piece of information: either two angles and a side, or two sides and a non-included angle. Use the cosine rule instead when two sides and the angle between them are known (SAS), or when all three sides are known (SSS), since the sine rule cannot be used without a known angle-side pair.
If two sides and the included angle are known, that is SAS, and the cosine rule is needed instead of the sine rule.
Finding a missing side
Set up the sine rule using the known angle-side pair and the side you want to find, then rearrange to make the missing side the subject. If the given side is not opposite one of the known angles, find the third angle first using angles in a triangle summing to 180°.
With angle A = 40°, angle B = 65°, and side a = 10 cm (opposite A), find side b: b/sin 65° = 10/sin 40°, so b = 10 × sin 65° ÷ sin 40° = 14.1 cm (3 s.f.).
Finding a missing angle
To find a missing angle, use the reciprocal form of the sine rule, sin A/a = sin B/b, so the unknown sine value ends up on top. Rearrange to find the sine of the angle, then use sin⁻¹ to find the angle itself.
With angle A = 45°, side a = 9 cm (opposite A), and side b = 11 cm: sin B/11 = sin 45°/9, so sin B = 11 × sin 45° ÷ 9 = 0.864, giving B = sin⁻¹(0.864) = 59.8° (1 d.p.).
Worked examples
Three exam-style questions, fully solved
In triangle ABC, angle A = 40°, angle B = 65°, and side a (opposite A) = 10 cm. Find the length of side b. Give your answer correct to 3 significant figures.
Easy- 1.Set up the sine rule: b/sin 65° = 10/sin 40°
- 2.Rearrange to make b the subject: b = 10 × sin 65° ÷ sin 40°
Answer: 14.1 cm
In triangle ABC, angle A = 50°, angle B = 60°, and side a (opposite A) = 8 cm. Find the length of side c. Give your answer correct to 3 significant figures.
Medium- 1.Find the third angle: angle C = 180° - 50° - 60° = 70°
- 2.Set up the sine rule using the known angle-side pair: c/sin 70° = 8/sin 50°
- 3.Rearrange to make c the subject: c = 8 × sin 70° ÷ sin 50°
Answer: 9.81 cm
In triangle ABC, angle A = 45°, side a (opposite A) = 9 cm, and side b (opposite B) = 11 cm. Find the size of angle B. Give your answer correct to 1 decimal place.
Hard- 1.Set up the sine rule using the reciprocal form: sin B/11 = sin 45°/9
- 2.Rearrange to find sin B: sin B = 11 × sin 45° ÷ 9
- 3.Use the inverse sine function: B = sin⁻¹(0.864)
Answer: 59.8°
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
In triangle ABC, angle A = 40°, angle B = 60°, and side a (opposite A) = 8 cm. Work out the length of side b, to 1 decimal place.
Work out the size of the angle marked x, to 1 decimal place.
In triangle PQR, angle P = 45°, angle Q = 55°, and side r (opposite R) = 12 cm. Work out the length of side p, to 1 decimal place.
Work out the size of the angle marked x, to 1 decimal place.
In triangle ABC, angle A = 100°, angle B = 30°, and side a = 25 cm. Work out the length of side c, to 1 decimal place.
In triangle ABC, a = 10 cm, b = 14 cm, and angle A = 35°. Work out the size of angle B, to 1 decimal place.
Avoid these
The mistakes students make most often
Mislabelling which side is opposite which angle, using a side that is adjacent to the angle rather than directly across from it.
Forgetting to find the third angle first when the given side is not opposite one of the known angles.
When finding a missing angle, forgetting to use the reciprocal form of the sine rule, or forgetting to apply sin⁻¹ at the end.
Using the sine rule when the cosine rule is actually needed, such as when two sides and the included angle (SAS) or all three sides (SSS) are given.
Rounding intermediate values too early, which makes the final answer less accurate than the required degree of precision.
FAQ
Questions students and parents ask
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