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§ Trigonometry

Sine & Cosine Rules

§ Trigonometry

Sine & Cosine Rules

CCSS.HSG.SRT3 min read

The sine and cosine rules are fundamental trigonometric relationships that solve any triangle when sufficient information is given. The sine rule states that a/sin(A) = b/sin(B) = c/sin(C), whilst the cosine rule expresses c² = a² + b² − 2ab·cos(C). These formulae appear in Year 10 of the UK National Curriculum and form essential components of GCSE mathematics.

§ 01

Why it matters

Engineering applications rely on these rules for calculating forces in structural frameworks, where components meet at non-right angles. Surveyors use triangulation to measure distances across rivers or valleys, applying the sine rule when two angles and one side are known. Navigation systems employ cosine rule calculations to determine shortest routes between three points. In physics, vector addition problems require these relationships when forces act at angles other than 90°. Architecture depends on precise angle calculations for roof trusses and bridge supports. The rules also underpin more advanced topics including the unit circle, complex numbers, and Fourier analysis in A-level Further Mathematics.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

You are given two sides and the included angle of a triangle. Which rule applies, and what is its formula?

Answer: Law of cosines: c² = a² + b² − 2ab·cos(C)

  1. Recognise the SAS configuration → Scenario: SAS — AAS / SSA → sine rule. SAS / SSS → cosine rule.
  2. Write the formula → c² = a² + b² − 2ab·cos(C) — Use the law of cosines when this configuration is given.
Easy§ 02

In a triangle, side a = 10, angle A = 30°, angle B = 45°. Find side b.

Answer: b ≈ 14.14

  1. Identify the rule → AAS → law of sines — With two angles and a non-included side (AAS), the law of sines applies.
  2. Write the formula with given values → 10/sin(30°) = b/sin(45°) — Pair each side with the sine of its opposite angle.
  3. Solve for b → b = 10 · sin(45°) / sin(30°) = 10 · 0.70710.5 — Multiply both sides by sin(B) to isolate b.
  4. Approximate to 2 decimals → b ≈ 14.14 — Evaluate numerically to the requested precision.
Medium§ 03

In a triangle, side a = 4, side b = 6, and the included angle C = 60°. Find side c.

Answer: c ≈ 5.29

  1. Identify the rule → SAS → law of cosines — Two sides and the included angle → use the law of cosines.
  2. Write the formula with given values → c² = 4² + 6² − 2·4·6·cos(60°) — c² = a² + b² − 2ab·cos(C).
  3. Solve algebraically → c² = 16 + 36 − 48·0.5 = 28.0 — Compute each term, then combine.
  4. Take square root and round → c = √28.0 ≈ 5.29 — Side lengths are positive; round to 2 decimals.
§ 04

Common mistakes

  • Confusing which rule applies to given information — using sine rule for side-angle-side problems leads to errors like attempting 4/sin(A) = 6/sin(B) when angle C = 60° is the included angle, requiring cosine rule instead
  • Incorrect formula rearrangement when finding angles with cosine rule — writing cos(C) = c²/(a² + b²) instead of cos(C) = (a² + b² − c²)/(2ab), producing wrong results like cos(C) = 0.64 rather than 0.125
  • Using degrees instead of radians in calculator mode — calculating cos(60°) as cos(60 radians) ≈ -0.999 instead of 0.5, making c² = 16 + 36 + 47.952 = 99.952 rather than 28
§ 05

Frequently asked questions

When should I use sine rule versus cosine rule?
Use sine rule for angle-angle-side (AAS) or angle-side-angle (ASA) configurations, where two angles and one side are known. Use cosine rule for side-angle-side (SAS) when finding the third side, or side-side-side (SSS) when finding angles.
How do I know if a triangle is valid before solving?
Check that the sum of angles equals 180° and that any side length is less than the sum of the other two sides. For SSS problems, verify that each side satisfies the triangle inequality before applying cosine rule.
What happens with the ambiguous case in sine rule?
When given two sides and a non-included angle (SSA), two different triangles might be possible. This occurs when the given angle is acute and the opposite side is shorter than the adjacent side but longer than the altitude.
How do I check my answer is reasonable?
Verify that angles sum to 180° and that the longest side lies opposite the largest angle. For cosine rule calculations, ensure that cos(C) lies between -1 and 1, as values outside this range indicate calculation errors.
Can these rules work with obtuse triangles?
Yes, both rules work with obtuse triangles. The cosine of obtuse angles is negative, which the formula handles automatically. For angles greater than 90°, cos(C) becomes negative, correctly increasing the calculated side length in the cosine rule.
§ 06

See also

ModeInequalityRange
§ 06

Where to next?

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