Sine & Cosine Rules
The sine and cosine rules extend trigonometry beyond right-angled triangles, enabling students to solve any triangle given sufficient information. These rules form the cornerstone of GCSE Higher tier trigonometry, appearing in 15-20% of Paper 2 questions.
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Why it matters
Surveying relies heavily on these rules when measuring inaccessible distances. A surveyor calculating the height of a building might measure a baseline of 50 metres and angles of 32° and 78°, using the sine rule to find the building's distance. Engineers designing bridges use the cosine rule to calculate structural supports—given two sides of 15m and 22m with a 65° angle between them, they determine the third side spans 18.7m. Navigation systems in ships and aircraft apply these principles continuously. Emergency services use triangulation with the sine rule to locate distress signals, whilst architects employ the cosine rule to verify roof truss measurements. Students encounter these applications directly in GCSE coursework worth 30 marks across both foundation and higher papers, making mastery essential for grades 6-9.
How to solve sine & cosine rules
Sine & Cosine Rules
- Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Use for AAS or SSA.
- Law of cosines: c² = a² + b² − 2ab·cos(C). Use for SAS (find third side).
- Rearranged: cos(C) = (a² + b² − c²)/(2ab). Use for SSS (find an angle).
- Each side is paired with the sine of the angle opposite it.
Example: a=5, b=7, C=60° → c² = 25 + 49 − 70·(12) = 39, so c ≈ 6.24.
Worked examples
You are given two sides and a non-included angle of a triangle. Which rule applies, and what is its formula?
Answer: Law of sines: a/sin(A) = b/sin(B) = c/sin(C)
- Recognise the SSA configuration → Scenario: SSA — AAS / SSA → sine rule. SAS / SSS → cosine rule.
- Write the formula → a/sin(A) = b/sin(B) = c/sin(C) — Use the law of sines when this configuration is given.
In a triangle, side a = 10, angle A = 45°, angle B = 60°. Find side b.
Answer: b ≈ 12.25
- Identify the rule → AAS → law of sines — With two angles and a non-included side (AAS), the law of sines applies.
- Write the formula with given values → 10/sin(45°) = b/sin(60°) — Pair each side with the sine of its opposite angle.
- Solve for b → b = 10 · sin(60°) / sin(45°) = 10 · 0.866 / 0.7071 — Multiply both sides by sin(B) to isolate b.
- Approximate to 2 decimals → b ≈ 12.25 — Evaluate numerically to the requested precision.
In a triangle, side a = 10, side b = 12, and the included angle C = 45°. Find side c.
Answer: c ≈ 8.62
- Identify the rule → SAS → law of cosines — Two sides and the included angle → use the law of cosines.
- Write the formula with given values → c² = 10² + 12² − 2·10·12·cos(45°) — c² = a² + b² − 2ab·cos(C).
- Solve algebraically → c² = 100 + 144 − 240·0.7071 = 74.29 — Compute each term, then combine.
- Take square root and round → c = √74.29 ≈ 8.62 — Side lengths are positive; round to 2 decimals.
Common mistakes
- Confusing which rule applies to given information. Students use sine rule for SAS (two sides, included angle) problems, writing 8/sin(45°) = 10/sin(B) when they should apply c² = 8² + 10² - 2(8)(10)cos(45°) = 75.15, giving c = 8.67.
- Using degrees instead of radians in calculator mode. Students calculate sin(60) as 0.0175 instead of 0.866, making 12/sin(60°) = 8/sin(45°) yield b = 0.24 rather than the correct b = 9.79.
- Mixing up opposite sides and angles in sine rule. Students write a/sin(B) = b/sin(A) instead of a/sin(A) = b/sin(B), calculating 5/sin(70°) = 7/sin(40°) as 7.42 instead of the correct 3.41.
- Forgetting the cosine rule rearrangement for finding angles. Students use c² = a² + b² - 2ab cos(C) when given three sides, instead of cos(C) = (a² + b² - c²)/(2ab), leading to nonsensical answers like c = 13 when finding angle C = 60°.