Law of Cosines

When the Law of Sines Cannot Start

The law of sines needs a matched pair: a side and the angle opposite it. Two very common setups hand you no such pair:

SAS: two sides and the angle between them
SSS: all three sides and no angles at all

For these, we need a different tool. That tool is the law of cosines.

The Law

It relates all three sides to one angle:

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

Here $C$ is the angle, and $c$ is the side directly opposite it. The other two sides, $a$ and $b$ , are the ones that meet at $C$ .

The bar on the right is $a^2 + b^2$ , and it never changes. The green bar is $c^2$ . As the angle $C$ opens, watch $c^2$ climb past that fixed line or fall below it. The shaded gap between them is the correction, $2ab\cos C$ : it shrinks $c^2$ when $C$ is acute, vanishes at $90°$ , and grows $c^2$ when $C$ is obtuse.

It Is Just Pythagoras, Corrected

Look closely at the formula. The first part, $a^2 + b^2$ , is the Pythagorean theorem. The last part, $-2ab\cos C$ , is a correction for the angle not being a right angle.

Watch the correction term in the animation as $C$ passes through $90°$ :

Angle $C$	$\cos C$	Correction	Result
acute ( $< 90°$ )	positive	subtracts	$c^2 < a^2 + b^2$
right ( $= 90°$ )	$0$	vanishes	$c^2 = a^2 + b^2$
obtuse ( $> 90°$ )	negative	adds	$c^2 > a^2 + b^2$

At exactly $90°$ the correction disappears and the law of cosines becomes the Pythagorean theorem.

Example: Two Sides and the Angle Between (SAS)

Given $a = 6$ , $b = 8$ , and the included angle $C = 60°$ . Find $c$ .

\begin{aligned} c^2 &= 6^2 + 8^2 - 2(6)(8)\cos 60° \\[0.5em] &= 36 + 64 - 96(0.5) \\[0.5em] &= 100 - 48 = 52 \end{aligned}

So $c = \sqrt{52} \approx 7.21$ .

Example: Three Sides, Find an Angle (SSS)

Rearranged, the law solves for the angle instead. Move the pieces around:

$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

Given $a = 6$ , $b = 8$ , $c = 7.21$ . Find $C$ .

\begin{aligned} \cos C &= \frac{36 + 64 - 52}{96} \\[0.5em] &= \frac{48}{96} = 0.5 \\[0.5em] C &= \arccos(0.5) = 60° \end{aligned}

Unlike the law of sines, $\arccos$ gives a single answer between $0°$ and $180°$ , so there is no ambiguous case here.

Which Law, When

You know	Use
AAS, ASA, or SSA	Law of Sines
SAS or SSS	Law of Cosines

A simple way to decide: if you have a side paired with its opposite angle, reach for the law of sines. If you are stuck with two sides and the angle between them, or all three sides, reach for the law of cosines.