Law of Cosines / Cosine Rule

The Law of Sines is used to solve triangles when we are given two sides and a non-included angle.

If we are given two sides and an included angle or three sides then we can use the Law of Cosines to solve the triangle.

The Law of Cosines, for any triangle ABC is

a ² = b ² + c ² – 2bc cos A

b ² = a ² + c ² – 2ac cos B

c ² = a ² + b ² – 2ab cos C

The Law of Cosines is also sometimes called the Cosine Rule.

Case 1: Given two sides and an included-angle

Example:

Solve triangle PQR in which p = 6.5 cm, q = 7.4 cm and ∠R = 58°.

Solution:

Using the Cosine rule,

r² = p² + q²– 2pq cos R

r² = (6.5)² + (7.4)²– 2(6.5)(7.4) cos58°
= 46.03

r = 6.78 cm

Using the Sine rule,

∠Q = 180° – 58° – 54.39°
= 67.61°

∠P = 54.39°, ∠Q = 67.61° and r = 6.78 cm

Case 2: Given three sides

Example:

In triangle ABC, a = 9 cm, b = 10 cm and c = 13 cm. Find the size of the largest angle.

Solution:

The largest angle is the one facing the longest side, i.e. C.

c² = a² + b²– 2ab cos C

= 0.067

∠C = 86.2°

Videos

The following video gives more examples of how to use the cosine rule.

The following video shows how to solve an oblique triangle using the Law of Cosines.

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