Sine Rule, Cosine Rule and Area of a Triangle
The sine rule, cosine rule and the area formula (1/2)ab sin(C) find missing sides and angles in any triangle, not just right-angled ones. The sine rule a/sin(A) = b/sin(B) = c/sin(C) suits problems with a matching angle-side pair, while the cosine rule and area formula work when two sides and the included angle, or all three sides, are known; these combine with bearings problems worth up to 9 marks.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Sketch the triangle and label the sides and angles using consistent letters, with a opposite A, b opposite B and c opposite C.
- If you know an angle and its opposite side, use the sine rule a/sin(A) = b/sin(B) = c/sin(C) to find another side or angle.
- If you know two sides and the included angle, or all three sides, use the cosine rule a^2 = b^2 + c^2 - 2bc cos(A), or its rearrangement for finding an angle, instead of the sine rule.
- For area, use Area = 1/2 x a x b x sin(C), where C is the angle between the two given sides a and b.
- Watch for the ambiguous case: if you are given two sides and a non-included angle, check whether a second valid triangle exists.
- In bearings problems, convert compass directions into angles inside the triangle before applying the sine or cosine rule.
Worked example
In triangle ABC, angle A = 40 degrees, angle B = 65 degrees, and side a (opposite angle A) = 7.5 cm. Calculate the length of side b, giving your answer correct to 3 significant figures.
- Use the sine rule: b/sin(B) = a/sin(A).
- Substitute the known values: b/sin(65) = 7.5/sin(40).
- Rearrange to make b the subject: b = 7.5 x sin(65)/sin(40).
- Calculate: b = 7.5 x 0.9063/0.6428 = 10.575..., so b = 10.6 cm to 3 significant figures
Practice questions
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Q1State the sine rule for triangle ABC, where sides a, b, c are opposite angles A, B, C.Show answer
Answer: a/sin(A) = b/sin(B) = c/sin(C)
Q2A triangle has two sides of length 6 cm and 10 cm with an included angle of 50 degrees. Calculate its area, correct to 3 significant figures.Show answer
Answer: 23.0 cm^2 (Area = 1/2 x 6 x 10 x sin(50))
Q3In triangle ABC, angle A = 48 degrees, angle B = 77 degrees, and side a = 9.2 cm (opposite angle A). Find side c (opposite angle C), correct to 3 significant figures.Show answer
Answer: 10.1 cm (angle C = 55 degrees, then use the sine rule)
Q4A triangle has sides a = 9 cm, b = 11 cm and c = 14 cm. Calculate the size of the largest angle, correct to 1 decimal place.Show answer
Answer: 88.3 degrees (largest angle is opposite the longest side, c)
Q5Two sides of a triangle are 8 cm and 9 cm, with an angle of 40 degrees between them. Calculate the length of the third side, correct to 3 significant figures.Show answer
Answer: 5.89 cm (cosine rule)
Q6In triangle PQR, angle P = 42 degrees, side p = 6.5 cm (opposite P) and side q = 9 cm (opposite Q). Find the two possible values of angle Q, correct to 1 decimal place.Show answer
Answer: Q = 67.9 degrees or Q = 112.1 degrees (ambiguous case)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
In triangle DEF, angle D = 57 degrees, angle E = 64 degrees, and side d (opposite angle D) = 11.4 cm. Calculate the length of side f (opposite angle F), giving your answer correct to 3 significant figures.
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A triangle has sides p = 10 cm, q = 13 cm and r = 17 cm, where angle R is opposite side r. Calculate angle R, then use it to calculate the area of the triangle, giving both answers correct to 3 significant figures where appropriate.
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In triangle ABC, AB = 14 cm, BC = 9 cm, and angle ABC = 110 degrees. Calculate the length of AC, giving your answer correct to 3 significant figures.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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