# Cosine Law

Lesson

## Cosine Rule

In trigonometry the cosine rule relates the lengths of the sides and the cosine of one of its angles.

The Law of Cosines is useful in finding:

• the third side of a triangle when you know two sides and the angle between them
• the angles of a triangle when you know all three sides

ABC is a triangle with side lengths $BC=a$BC=a , $CA=b$CA=b and $AB=c$AB=c and the opposite angles of the sides are respectively angle $A$A, angle $B$B and angle $C$C.

Law of cosines

$a^2=b^2+c^2-2bc\cos A$a2=b2+c22bccosA

$b^2=a^2+c^2-2ac\cos B$b2=a2+c22accosB

$c^2=a^2+b^2-2ab\cos C$c2=a2+b22abcosC

Notice that Pythagoras' Theorem $a^2=b^2+c^2$a2=b2+c2 makes an appearance in the Cosine Rule: $a^2=b^2+c^2-2bc\cos A$a2=b2+c22bccosA

## Finding an angle

Find angle $B$B in the triangle.

Think: All three side lengths are known, so I can apply the cosine rule. The unknown angle B appears opposite side $b=3$b=3.

 $b^2$b2 $=$= $a^2+c^2-2ac\cos B$a2+c2−2accosB $3^2$32 $=$= $5^2+6^2-2\times5\times6\cos B$52+62−2×5×6cosB $9$9 $=$= $25+36-60\cos B$25+36−60cosB $9-61$9−61 $=$= $-60\cos B$−60cosB $\frac{-52}{-60}$−52−60​ $=$= $\cos B$cosB $\cos B$cosB $=$= $0.866667$0.866667 $B$B $=$= $29.9^\circ$29.9°  to $1$1 decimal place

## Finding a side length

Find the value of $x$x in the diagram.

Think: The first thing I always do is identify which side is opposite the given angle. This side is the subject of the formula. To find out which other values we are given I label the sides and angles using $a$a,$b$b and $c$c .

Do

I add the following labels to the triangle:

So I want to find the value of $c$c.

 $c^2$c2 $=$= $a^2+b^2-2ab\cos C$a2+b2−2abcosC $c^2$c2 $=$= $8^2+11^2-2\times8\times11\cos39^\circ$82+112−2×8×11cos39° $c^2$c2 $=$= $64+121-176\cos39^\circ$64+121−176cos39° $c^2$c2 $=$= $48.22$48.22 $c$c $=$= $6.94$6.94

The following interactive demonstrates that the cosine rule holds regardless of the angles or size and shape of the triangle.

#### Examples

##### Question 1

Find the length of $a$a using the cosine rule.

##### Question 2

Find the value of angle $B$B using the cosine rule.

##### Question 3

Find the length of $c$c using the cosine rule.

### Outcomes

#### 10D.T3.02

Explore the development of the cosine law within acute triangles

#### 10D.T3.03

Determine the measures of sides and angles in acute triangles, using the sine law and the cosine law