Pre-Calculus 11 · Mathematics · Lesson 7.3
Cosine law
- Level
- Grade 11
- Reading time
- 6 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Solving oblique triangles.
What is the cosine law?
Short answer
The cosine law (law of cosines) relates the three sides of any triangle to one of its angles: c² = a² + b² − 2ab·cos C. It generalizes the Pythagorean theorem to non-right triangles and is used to find a third side from two sides and the included angle, or to find an angle when all three sides are known.
Concept preview
What you will be able to do
- 01State the cosine law and its cyclic forms.
- 02See the cosine law as a generalisation of the Pythagorean theorem.
- 03Find the third side from two sides and the included angle (SAS).
- 04Find an angle from all three sides (SSS) using .
- 05Decide when to use the cosine law versus the sine law, and solve applied problems.
The sine law is powerful, but it has one requirement it cannot do without: a complete side–angle pair to anchor the proportion. Two very common situations refuse to give you one. SAS hands you two sides and the angle wedged between them — but that angle is not opposite either known side. SSS hands you all three sides and no angles at all. In both cases the sine law is stuck at the starting line. The cosine law is built exactly for these two setups, and once you have it, the sine and cosine laws together can crack any triangle.
School Plan
Cosine law
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Related topics
Frequently asked questions
- When do you use the cosine law instead of the sine law?
- Use the cosine law when you know two sides and the included angle, or all three sides; the sine law needs an angle–opposite-side pair.
- How does the cosine law relate to the Pythagorean theorem?
- When the angle is 90°, cos C = 0, so it reduces to c² = a² + b².