Integrated Math 3 / Unit 6: Trigonometric Identities and Laws
6.4 Law of Cosines
Notes
In any triangle: $c^2 = a^2 + b^2 - 2ab\cos C$.
- Use it when you know two sides and the included angle (SAS), or all three sides (SSS).
- To find an angle from three sides: $\displaystyle \cos C = \frac{a^2 + b^2 - c^2}{2ab}$.
- If $C = 90^\circ$, then $\cos C = 0$ and it becomes the Pythagorean Theorem.
- Example: $a = 5$, $b = 8$, $C = 60^\circ$ gives $c^2 = 25 + 64 - 80(0.5) = 49$, so $c = 7$.
Practice
A triangle has sides 5 and 8 with an included angle of $60^\circ$. What is the third side?
$c^2 = 25 + 64 - 2(5)(8)\cos 60^\circ = 89 - 40 = 49$.
A triangle has sides 7, 8 and 9. What is the angle opposite the side of length 9 to the nearest tenth of a degree?
$\displaystyle \cos C = \frac{49 + 64 - 81}{2 \cdot 7 \cdot 8} = \frac{32}{112} = \frac{2}{7}$, so $C \approx 73.4^\circ$.
Which information requires the Law of Cosines to start solving a triangle?
Two sides and the included angle (SAS) give no matching side-angle pair, so use the Law of Cosines. Three angles alone cannot fix the size.
Ready for more? Take the Unit 6 test.