Law of Cosines / Cosine Rule

In this lesson, we will learn:

• the Law of Cosines
• how to use the Law of Cosines when given two sides and an included angle
• how to use the Law of Cosines when given three sides
• how to proof the Law of Cosines
• how to solve applications or word problems using the Law of Cosine

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle with the cosine of one of its angles.

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 or Cosine Formula.

If we are given two sides and an included angle (SAS) or three sides (SSS) then we can use the Law of Cosines to solve the triangle ie. to find all the unknown sides and angles.

We can use the Law of Sines to solve triangles when we are given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).

Law of Cosines: 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

Law of Cosines: 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 two examples of how to use the cosine rule. One given SAS and the other given SSS.

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

Proof of the Law of Cosines

The following video shows how to prove the law of cosines by using coordinate geometry and the Pythagorean theorem.

Applications using the Law of Cosines

The video shows how to solve two word problems using the Law of Cosines

Example 1: An engineering firm decides to bid on a proposed tunnel through a mountain. Find how long is the tunnel and the bid amount.

Example 2: Find the range of service of a transmission tower.

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