Trigonometry / Unit 8: Oblique Triangles and Vectors
8.3 The Law of Cosines
Notes
For any triangle: $c^2 = a^2 + b^2 - 2ab\cos C$ (and similarly for the other sides).
- Use it for SAS (find the third side) and SSS (find an angle).
- To find an angle: $\displaystyle \cos C = \frac{a^2 + b^2 - c^2}{2ab}$.
- When $C = 90^\circ$ it becomes the Pythagorean theorem.
- Find the largest angle first when solving SSS; then the other two angles are acute.
Practice
In a triangle $a = 7$, $b = 8$ and $C = 60^\circ$. What is $c$?
$c^2 = 49 + 64 - 2(7)(8)\cos 60^\circ = 113 - 56 = 57$.
A triangle has sides 5, 7 and 8. What is the angle opposite the side of length 8, to the nearest tenth of a degree?
$\displaystyle \cos C = \frac{25 + 49 - 64}{2(5)(7)} = \frac{10}{70} = \frac{1}{7}$, so $C \approx 81.8^\circ$.
You know two sides of a triangle and the angle between them. What is the best first step?
With SAS you have no side-angle opposite pair, so the Law of Sines cannot start. The Law of Cosines gives the third side.
Ready for more? Take the Unit 8 test.