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

Related Topics:

More Lessons on Trigonometry

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}– 2bccosA

b^{2}=a^{2}+c^{2}– 2accosB

c^{2}=a^{2}+b^{2}– 2abcosC

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).

* Example:*

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

* Solution: *

Using the Cosine rule,

*r*^{2} = *p*^{2} + *q*^{2}– 2*pq *cos *R *

*r*^{2} = (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

* 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*.

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

= 0.067

∠*C* = 86.2°

The following video gives two examples of how to use the cosine rule. One given SAS and the other given SSS. Using Law of Cosines to solve triangles given 3 sides (SSS) and 2 sides and the angle in between (SAS).

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

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

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.

Use the Law of Cosines to determine the length across the lake.

Use the Law of Cosines to find the distance a plane has traveled after a change in direction.

Use the Law of Cosines to determine the length of a diagonal of a parallelogram.

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