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Non-right-angled triangles

Apply the sine rule and cosine rule to find unknown sides and angles in non-right-angled triangles, calculate triangle area using A=12absin⁡CA = \frac{1}{2}ab\sin C, and identify and solve the ambiguous case.

Worked examples

Finding an unknown side using the sine rule

Straightforward

Problem

In triangle ABCABC, ∠A=50°\angle A = 50°, ∠B=70°\angle B = 70° and a=8a = 8 cm. Find the length of side bb to 2 decimal places.

Applying the cosine rule and finding triangle area

Moderate

Problem

In triangle ABCABC, a=7a = 7, b=10b = 10 and ∠C=60°\angle C = 60°. Find side cc and the area of the triangle.

Solving the ambiguous case

Challenging

Problem

In triangle ABCABC, a=10a = 10, b=12b = 12 and ∠A=40°\angle A = 40°. Determine how many triangles satisfy these conditions and find both possible values of ∠B\angle B.

Practise

Q1·Straightforward
In triangle ABCABC, ∠A=50°\angle A = 50°, ∠B=70°\angle B = 70° and a=8a = 8 cm. Find the length of side bb to 2 decimal places.
Q2·Straightforward
In triangle PQRPQR, ∠P=35°\angle P = 35°, ∠Q=85°\angle Q = 85° and p=12p = 12. Find side qq to 2 decimal places.
Q3·Straightforward
In triangle ABCABC, ∠A=42°\angle A = 42°, ∠C=67°\angle C = 67° and c=15c = 15. Find side aa to 2 decimal places.
Q4·Straightforward
In triangle XYZXYZ, ∠Y=110°\angle Y = 110°, ∠X=25°\angle X = 25° and y=20y = 20. Find side xx to 2 decimal places.
Q5·Moderate
In triangle ABCABC, a=7a = 7, b=10b = 10 and ∠C=60°\angle C = 60°. Find side cc to 2 decimal places.
Q6·Moderate
In triangle PQRPQR, p=5p = 5, q=8q = 8 and ∠R=45°\angle R = 45°. Find side rr to 2 decimal places.
Q7·Moderate
Find the area of triangle ABCABC where a=6a = 6 cm, b=9b = 9 cm and ∠C=55°\angle C = 55°. Give your answer to 2 decimal places.
Q8·Moderate
Find the area of triangle DEFDEF where d=11d = 11, e=7e = 7 and ∠F=72°\angle F = 72°. Give your answer to 2 decimal places.
Q9·Challenging
In triangle ABCABC, a=10a = 10, b=12b = 12 and ∠A=40°\angle A = 40°. How many triangles satisfy these conditions? Enter the number.
Q10·Challenging
In triangle ABCABC, a=8a = 8, b=6b = 6 and ∠A=30°\angle A = 30°. How many triangles satisfy these conditions? Enter the number.
Q11·Challenging
In triangle ABCABC, a=5a = 5, b=10b = 10 and ∠A=25°\angle A = 25°. Find all possible values of ∠B\angle B, to the nearest degree.
Q12·Challenging
In triangle PQRPQR, p=9p = 9, q=14q = 14 and ∠P=35°\angle P = 35°. How many triangles are possible, and what are the valid values of ∠Q\angle Q to the nearest degree?