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+c2−2bccosA

$b^2=a^2+c^2-2ac\cos B$b2=a2+c2−2accosB

$c^2=a^2+b^2-2ab\cos C$c2=a2+b2−2abcosC

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.

$b^2$`b`2	$=$=	$a^2+c^2-2ac\cos B$`a`2+`c`2−2`accosB`
$3^2$32	$=$=	$5^2+6^2-2\times5\times6\cos B$52+62−2×5×6`cosB`
$9$9	$=$=	$25+36-60\cos B$25+36−60`cosB`
$9-61$9−61	$=$=	$-60\cos B$−60`cosB`
$\frac{-52}{-60}$−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.

$c^2$`c`2	$=$=	$a^2+b^2-2ab\cos C$`a`2+`b`2−2`abcosC`
$c^2$`c`2	$=$=	$8^2+11^2-2\times8\times11\cos39^\circ$82+112−2×8×11`cos`39°
$c^2$`c`2	$=$=	$64+121-176\cos39^\circ$64+121−176`cos`39°
$c^2$`c`2	$=$=	$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.

Created with Geogebra

Examples

Question 1

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

Round your answer to two decimal places.

A triangle with its vertices labeled as A, B, and C. The angle at vertex A measures $33^\circ$33°. The side opposite to the $33^\circ$33° angle, side $BC$`BC`, is labeled $a$`a` units, indicating its unknown length. Side $AB$`AB` measures $30$30 units, as shown by the scale line beneath it. Side $AC$`AC` measures $13$13 units. Side $AB$`AB` and Side $AC$`AC` are adjacent to the $33^\circ$33° angle at vertex A.

Question 2

Question 3

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