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The sine rule

Free Printable The sine rule PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 7)

Download free the sine rule worksheets (PDF), GCSE maths past paper questions and worked solutions for The sine rule (7M). This is a Grade 7 geometry topic from the GCSE questions by topic collection for Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Maths4Everyone, Corbett Maths, Maths Made Easy, Dr Frost, 1st Class Maths, Metatutor, Maths Takeaway, Maths DIY, Dr Austin, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our the sine rule revision notes and worked examples with diagrams, try real GCSE past paper the sine rule questions with step-by-step solutions, or download a PDF of all the sine rule worksheet links.

Free The sine rule PDF Worksheets & Answers

Grade 7 • Topic 7M
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Grade 7 topics
7A - Surds7B - Bounds
7C - Proportion (harder)7D - Algebraic fractions
7E - The quadratic formula7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs7H - Rearranging (harder)
7J - Functions (harder)7K - Iteration
7L - Area of any triangle7M - The sine rule
7N - The cosine rule7P - Congruent triangles
7Q - 3D pythagoras / trig.7R - Histograms
7S - Conditional probability7T - Exact trig. values (harder)
7U - Area of a segment
GCSE Maths Grade 7 Revision Notes & Method

Guide to The sine rule: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

Use the Sine Rule when a non-right-angled triangle involves two opposite side-angle pairs. To find a missing side, put sides on top: asin A = bsin B. To find a missing angle, flip the fractions to put sines on top: sin Aa = sin Bb.

Always label each angle with a capital letter (A, B, C) and the side directly opposite it with the matching lowercase letter (a, b, c). Watch out for the ambiguous case if asked for an obtuse angle (180° − θ).

Example 1: In ΔABC, A = 42°, B = 75°, and b = 13.4 cm. Find side a (3 s.f.).

A B C 42° 75° 13.4 cm a

Solution:

Use asin A = bsin B:
asin 42° = 13.4sin 75°
a = 13.4 × sin 42°sin 75° = 9.2827... cm
Answer: a = 9.28 cm (3 s.f.)

Example 2: In ΔPQR, p = 8.2 cm, q = 11.5 cm, and Q = 68°. Find acute angle P (1 d.p.).

P Q R P 68° 11.5 cm 8.2 cm

Solution:

Use sin Pp = sin Qq ⇒ sin P8.2 = sin 68°11.5
sin P = 8.2 × sin 68°11.5 = 0.66112...
P = sin−1(0.66112...) = 41.385...°
Answer: P = 41.4° (1 d.p.)
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper The sine rule Questions & Answers

Past Paper Example 1 AQA Jun 2025 3H Q17 2 marks
Full Paper (PDF) • Mark Scheme (PDF)
Not drawnaccurately31°119°6.3 cmx
Triangle with angles 119° and 31°, side 6.3 cm opposite 31°. Use the sine rule to work out x. [2 marks]
Use the sine rule to work out x.
Worked Solution & Mark Scheme (2 marks)
x / sin(119°) = 6.3 / sin(31°) M1
⇒ x = [6.3 × sin(119°)] / sin(31°) ≈ 10.7 cm. A1
Answer: x = 10.7 cm
Past Paper Example 2 GCSE Jun 2023 2H Q13 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
ABC is a triangle.
70°15 cm18 cmABC
Calculate the size of angle BAC.
Give your answer correct to 1 decimal place.
Worked Solution & Mark Scheme (4 marks)
(Angles in a triangle add up to 180°)
sin(BCA) / 15 = sin 70° / 18 ⇒ sin(BCA) = 0.78307... ⇒ ∠BCA = 51.54° M3
∠BAC = 180 – (70 + 51.54) = 58.5° A1
Answer: 58.5
Past Paper Example 3 AQA Nov 2024 3H Q19 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
35°117°54 cmxNot drawnaccurately
Triangle with angles 35° and 117°, side 54 cm opposite 117°. Use the sine rule to work out x. [3 marks]
Use the sine rule to work out length x.
You **must** show your working.
Worked Solution & Mark Scheme (3 marks)
x / sin 35° = 54 / sin 117°. M2
x = (54 × sin 35°) ÷ sin 117° ≈ 34.8 cm. A1
Answer: x = 34.8 cm

← 7L Area of any triangle

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