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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 formulas used to solve triangles when not all sides and angles are known. The sine rule states that a/sin(A) = b/sin(B) = c/sin(C), connecting each side to the sine of its opposite angle. The cosine rule, c² = a² + b² - 2ab·cos(C), provides a way to find unknown sides or angles using the Pythagorean theorem with an additional correction term.

§ 01

Why it matters

These rules solve real-world problems in navigation, surveying, and engineering. GPS systems use triangulation based on these principles to determine positions within 3 meters of accuracy. Architects apply the cosine rule when designing roof trusses with specific angles, calculating that a 12-foot beam paired with an 8-foot beam at a 120° angle requires a third beam of approximately 17.44 feet. The sine rule helps engineers determine tower heights by measuring angles from 2 different positions 100 feet apart. These formulas extend beyond basic right-triangle trigonometry, enabling solutions for any triangle configuration. In advanced mathematics, they form the foundation for vectors, complex analysis, and spherical trigonometry used in astronomy and global positioning systems.

§ 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 = 8, angle A = 30°, angle B = 45°. Find side b.

Answer: b ≈ 11.31

  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 → 8/sin(30°) = b/sin(45°) — Pair each side with the sine of its opposite angle.
  3. Solve for b → b = 8 · sin(45°) / sin(30°) = 8 · 0.70710.5 — Multiply both sides by sin(B) to isolate b.
  4. Approximate to 2 decimals → b ≈ 11.31 — 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

  • Applying the sine rule to SAS configurations leads to errors like using a/sin(A) = 6/sin(60°) when given sides a=4, b=6, and included angle C=60°, instead of using the cosine rule c² = 4² + 6² - 2(4)(6)cos(60°) = 28
  • Using degrees instead of radians in calculations produces incorrect results, such as cos(60) ≈ -0.999 instead of cos(60°) = 0.5, leading to c² = 4² + 6² - 48(-0.999) ≈ 100 instead of the correct c² = 28
  • Forgetting the 2ab term in the cosine rule results in c² = a² + b² - cos(C), giving c² = 16 + 36 - 0.5 = 51.5 instead of c² = 16 + 36 - 24 = 28
  • Mixing up which angle corresponds to which side in the sine rule, writing 8/sin(45°) = b/sin(30°) instead of 8/sin(30°) = b/sin(45°), yielding b ≈ 5.66 instead of b ≈ 11.31
§ 05

Frequently asked questions

When should I use the sine rule versus the cosine rule?
Use the sine rule for AAS (two angles, one side) or SSA (two sides, one non-included angle) configurations. Use the cosine rule for SAS (two sides, included angle) or SSS (three sides) scenarios. The sine rule works when you can pair known sides with their opposite angles.
What is the ambiguous case in the sine rule?
The SSA case can produce 0, 1, or 2 valid triangles depending on the given measurements. For example, with sides a=5, b=8, and angle A=30°, the calculation sin(B) = 8sin(30°)/5 = 0.8 yields B ≈ 53.1° or B ≈ 126.9°, creating two possible triangles.
How do I rearrange the cosine rule to find an angle?
Start with c² = a² + b² - 2ab·cos(C), then isolate cos(C): cos(C) = (a² + b² - c²)/(2ab). Finally, take the inverse cosine: C = arccos((a² + b² - c²)/(2ab)). For a triangle with sides 3, 4, 5, angle C = arccos((9+16-25)/24) = arccos(0) = 90°.
Why does the cosine rule include the 2ab·cos(C) term?
This term corrects the Pythagorean theorem for non-right triangles. When C=90°, cos(C)=0, so the term vanishes and the formula becomes c²=a²+b², which is the Pythagorean theorem. For acute angles, the term is positive, making c smaller than the Pythagorean prediction.
Can these rules be used for obtuse triangles?
Yes, both rules work for obtuse triangles. The sine of an obtuse angle is positive, and the cosine is negative. For example, in a triangle with sides 3, 4, 6, the largest angle C satisfies cos(C) = (9+16-36)/24 = -11/24 ≈ -0.458, giving C ≈ 117.3°.
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See also

TheoremTermObtuse angle
§ 06

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