Law of Cosines

📊 intermediate

$$c^2 = a^2 + b^2 - 2ab \cos C$$

Generalizes the Pythagorean theorem to any triangle by relating the lengths of sides to the cosine of one of its angles.

Variables & Units

$a, b, c$

Side lengths

Unit: Length units

$C$

Angle opposite to side $c$

Unit: Degrees or Radians

Key Information

📐 Derived From

Euclidean geometry and coordinate geometry.

💡 Example

Finding the distance between two cities given their distances from a third city and the angle between them.

🔧 Applications

Navigation, surveying, and 3D graphics.

⚠️ Constraints

Valid for all Euclidean triangles.

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