Geometry • Topic 14

Law of Cosines

The Law of Cosines is a generalization of the Pythagorean Theorem to non-right triangles. It is the go-to tool for calculating third sides or angles when SSS or SAS is given.

Statement

Theorem (Law of Cosines)

△ A BC

c^{2} = a^{2} + b^{2} - 2 ab cos C

b^{2} = a^{2} + c^{2} - 2 a c cos B

a^{2} = b^{2} + c^{2} - 2 b c cos A

Implications

Pythagorean Theorem: If $∠ C = 9 0^{\circ}$ , then $cos C = 0$ , yielding $c^{2} = a^{2} + b^{2}$ .
Classifying Triangles:

$c^{2} < a^{2} + b^{2} ⟹ ∠ C$ is acute.

$c^{2} > a^{2} + b^{2} ⟹ ∠ C$ is obtuse.

Practice Problems

Exercise (Problem 1)

△ A BC

, sides are 5, 7, and 8. Find the cosine of the largest angle.

Exercise (Problem 2)

Prove that

a = b cos C + c cos B

(Projection Law) using the Law of Cosines or basic trigonometry.

Exercise (Problem 3)

Two ships leave a port at the same time. One travels at 20 km/h on a bearing of

04 5^{\circ}

, the other at 30 km/h on a bearing of

13 5^{\circ}

. How far apart are they after 2 hours?