Cosine Law

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.

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+c2−2bccosA

Example

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

Question 2

Question 3