Find a missing side or angle in a non-right-angled triangle when you know a side and its opposite angle. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
The sine rule is sinAa=sinBb=sinCc. Use it when you have a side and its opposite angle, plus one more side or angle. To find a side: sinAa=sinBb, so a=sinA×sinBb. To find an angle: sinA=a×bsinB, then sin−1, watching the ambiguous case (SSA) on Higher — two possible angles adding to 180∘ if both could fit.
Label sides with the lowercase of the opposite vertex. Sketch, mark what you know, and check you are not in cosine-rule territory (three sides, or two sides and the included angle).
sinAa=sinBb. You need an opposite pair (a side and its opposite angle). Ambiguous case: if you find an acute A and the triangle could be obtuse, check 180−A.
Objectives
Find a side opposite 45 degrees from a side opposite 30.
Find the side opposite 30 from a side opposite 45.
Find an acute angle, and the third angle of a 50 and 60 triangle.
Key points
Label a triangle with side a opposite angle A, b opposite B, c opposite C.
Recognise when the sine rule applies: you have a matching pair (a side and the angle opposite it) plus one more piece.
Use the side version to find a side and the angle version to find an angle.
Find the third angle from 180∘ when two angles are known.
Worked examples
Example 1
Use the sine rule. The side opposite 30 degrees is 6. Find the side opposite 45 degrees.
2 marks
Set up sine rule ratio: a / sin A = b / sin B.
Substitute known values: 6 / sin 30 = b / sin 45.
Solve for b: b = 6 × (sin 45 / sin 30) = 6 × √2 = 6√2.
Answer: 6√2
Common slips
Slip: Multiply sides directly. The sine rule equates the ratio of side to sine of opposite angle.
Slip: Put 45 in the numerator. Side 6 corresponds to angle 30°.
Slip: Leave b = 6. The side opposite 45° is 6√2.
Example 2
Use the sine rule. The side opposite 45 degrees is 7√2. Find the side opposite 30 degrees.
2 marks
Set up ratio: a / sin 30 = 7√2 / sin 45.
Solve for a: a = 7√2 × (sin 30 / sin 45) = 7√2 / √2 = 7.
State the side opposite 30 degrees: 7.
Answer: 7
Common slips
Slip: Multiply 7√2 by √2. Going from 45° down to 30° means dividing by √2. Multiplying gives 14, which is too big.
Slip: Write 14. The side opposite 30° is 7√2 ÷ √2 = 7, not 14.
Slip: Write 7√2. The side is 7.
Example 3
Use the sine rule. The side opposite 30 degrees is 5 and the side opposite the unknown acute angle is 5√2. Find that angle.
3 marks
Set up sine rule: sin θ / (5√2) = sin 30 / 5.
Solve for sin θ: sin θ = (5√2 / 5) × sin 30 = √2 × (1/2) = √2/2.
Find acute angle with sin θ = √2/2: θ = 45 degrees.
Answer: 45
Common slips
Slip: Assume θ = 30°. The opposite side is longer, so the angle is larger.
Slip: Calculate sin θ = 1. sin θ = √2/2.
Slip: Write 60. The acute angle is 45.
Questions and answers
GCSE Maths the sine rule questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Use the sine rule. The side opposite 30 degrees is 7. Find the side opposite 45 degrees.
Show answer
Answer: 7√2
sin 45 / sin 30 = √2.
7 × √2 = 7√2.
M1 — sin 45 / sin 30 = √2.
A1 — Answer 7√2
Question 2 Easy2 marks
In triangle ABC, angle A is 42 degrees, angle B is 63 degrees and the side opposite A is 8.4 cm. Find the side opposite B to 3 significant figures.
Show answer
Answer: 11.2
Write b / sin 63 = 8.4 / sin 42.
The side is 11.2 cm.
M1 — Write b / sin 63 = 8.4 / sin 42.
A1 — Answer 11.2
Question 3 Medium2 marks
Use the sine rule. The side opposite 45 degrees is 9√2. Find the side opposite 30 degrees.
Show answer
Answer: 9
Divide the 45 degree side by √2.
9√2 / √2 = 9.
M1 — Divide the 45 degree side by √2.
A1 — Answer 9
Question 4 Medium2 marks
In triangle ABC, angle A is 40 degrees and angle B is 75 degrees. Side AB is 12 cm. Find side BC to 3 significant figures.
Show answer
Answer: 8.51
The third angle is 65 degrees.
BC is opposite 40 degrees, so BC = 12 sin 40 / sin 65.
M1 — The third angle is 65 degrees.
A1 — Answer 8.51
Question 5 Hard2 marks
In triangle ABC, angle A is 35 degrees and angle B is 80 degrees. Side AB is 15 cm. Find side BC to 3 significant figures.
Show answer
Answer: 9.49
Angle C is 65 degrees.
BC = 15 sin 35 / sin 65.
M1 — Angle C is 65 degrees.
A1 — Answer 9.49
Question 6 Hard2 marks
In triangle ABC, angle A is 30 degrees, angle B is 80 degrees and the side opposite A is 6.5 cm. Find the side opposite B to 3 significant figures.
Show answer
Answer: 12.8
Write b / sin 80 = 6.5 / sin 30.
Calculate the side opposite B.
M1 — Write b / sin 80 = 6.5 / sin 30.
A1 — Answer 12.8
Question 7 Exam-style4 marks
In triangle ABC, angle A is 28 degrees and angle B is 70 degrees. Side AB is 16 cm. Find side BC to 3 significant figures.
Show answer
Answer: 7.59
Angle C is 82 degrees.
BC = 16 sin 28 / sin 82.
B1 — Angle C is 82 degrees.
M1 — BC = 16 sin 28 / sin 82.
M1 — BC = 16 sin 28 / sin 82.
A1 — Answer 7.59
Common mistakes
Most lost marks on this topic are one slip.
Pairing a side with an adjacent angle
Use the opposite angle
Leaving out √2 when the angle changes from 30 to 45
Multiply by sin 45 / sin 30
Finding a third angle from 90 instead of 180
Angles in a triangle add to 180
Spot the slip
The sine rule links a side to the angle opposite it. A student pairs a side with an adjacent angle. What should they pair?
The opposite angle
The adjacent angle
The largest side always
Show the slip
The opposite angle Each side is paired with the angle facing it.
Exam tips
A calculator is useful here. Show the calculation you typed, not only the final line.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Label the sides with lower-case letters and the opposite angles with capitals on the diagram before you start.
Write the sine rule with the numbers in as one full line — the substituted equation is a method mark.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.