Law of Cosines and Area Formulas

By the end of this lesson, you will be able to apply the Law of Cosines to SAS and SSS cases, choose the correct law for a given triangle, and apply both laws to surveying and navigation problems.

Law of Cosines

$$a^2 = b^2 + c^2 - 2bc\cos A$$ $$b^2 = a^2 + c^2 - 2ac\cos B$$ $$c^2 = a^2 + b^2 - 2ab\cos C$$

Angle from three sides:

$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

Choosing the Right Law

Given	Method
AAS or ASA	Law of Sines
SSA	Law of Sines (ambiguous case)
SAS	Law of Cosines (find opposite side first)
SSS	Law of Cosines (find any angle)

Strategy after Law of Cosines: switch to Law of Sines for remaining parts.

Example 1 — SAS

$b = 12$, $c = 17$, $A = 55°$. Find all parts.

$a^2 = 144 + 289 - 2(12)(17)\cos 55° = 433 - 408(0.574) \approx 433 - 234.2 = 198.8$

$a \approx 14.1$

Find $B$: $\sin B = \dfrac{12\sin 55°}{14.1} \approx 0.697 \implies B \approx 44.1°$

$C = 180 - 55 - 44.1 = 80.9°$

Example 2 — SSS

$a = 5$, $b = 7$, $c = 8$. Find all angles.

$\cos A = \dfrac{49 + 64 - 25}{2(7)(8)} = \dfrac{88}{112} \approx 0.786 \implies A \approx 38.2°$

$\cos B = \dfrac{25 + 64 - 49}{2(5)(8)} = \dfrac{40}{80} = 0.5 \implies B = 60°$

$C = 180 - 38.2 - 60 = 81.8°$

Example 3 — Navigation

A ship sails 30 km at N 40° E, then turns and sails 50 km at S 70° E. How far is the ship from the start?

Find the interior angle of the triangle at the turning point. At the turn, the reverse of the first leg points S 40° W, and the second leg heads S 70° E. As bearings clockwise from north those are $220°$ and $110°$, so the angle between them is $220° - 110° = 110°$.

$d^2 = 30^2 + 50^2 - 2(30)(50)\cos 110° = 900 + 2500 - 3000(-0.342) = 3400 + 1026 = 4426$

$d \approx 66.5$ km

Summary

Law of Cosines: use for SAS (find third side) or SSS (find any angle).