Law of Cosines Worked Examples Cheat Sheet

Key Facts

• The Law of Cosines is $c^2 = a^2 + b^2 - 2ab\cos C$, where side $c$ is opposite angle $C$.
• To find a missing side with $SAS$, use $c = \sqrt{a^2 + b^2 - 2ab\cos C}$.
• To find a missing angle with $SSS$, use $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$ and then $C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$.
• The included angle is the angle between the two known sides, such as angle $C$ between sides $a$ and $b$.
• If $C = 90^\circ$, then $\cos 90^\circ = 0$, so the Law of Cosines becomes $c^2 = a^2 + b^2$.
• Use the Law of Cosines first for $SAS$ or $SSS$ triangle information.
• After finding one angle in an $SSS$ triangle, the Law of Sines can often find another angle using $\frac{\sin A}{a} = \frac{\sin B}{b}$.
• Always check that the largest angle is opposite the longest side and the smallest angle is opposite the shortest side.

Vocabulary

Law of Cosines

A triangle formula, $c^2 = a^2 + b^2 - 2ab\cos C$, that relates three sides and one included angle.

Included angle

The included angle is the angle formed between two known sides, such as $C$ between sides $a$ and $b$.

Opposite side

An opposite side is the side across from a given angle, so side $c$ is opposite angle $C$.

SAS

$SAS$ means two sides and the included angle are known, which is the standard setup for finding a missing side.

SSS

$SSS$ means all three side lengths are known, which is the standard setup for finding a missing angle.

Inverse cosine

Inverse cosine, written $\cos^{-1}(x)$, gives the angle whose cosine is $x$.