Precalculus / Unit 6: Applications of Trigonometry
6.2 The Law of Cosines
Notes
$c^2 = a^2 + b^2 - 2ab\cos C$. It generalizes the Pythagorean theorem (when $C = 90^\circ$ the last term vanishes).
- Use it for SAS (find the third side) and SSS (find an angle): $\displaystyle \cos C = \frac{a^2 + b^2 - c^2}{2ab}$.
- Find the largest angle first when solving SSS triangles, if you want to avoid ambiguity.
- Example: $a = 3$, $b = 5$, $C = 60^\circ$: $\displaystyle c^2 = 9 + 25 - 30 \cdot \frac{1}{2} = 19$, so $\displaystyle c = \sqrt{19}$.
Practice
Which given information requires the Law of Cosines?
SAS (and SSS) cannot be started with the Law of Sines because no side-angle pair is known.
A triangle has sides $a = 3$ and $b = 5$ with included angle $C = 60^\circ$. What is $c$?
$c^2 = 9 + 25 - 2(3)(5)\cos 60^\circ = 34 - 15 = 19$.
A triangle has sides 7, 8 and 9. What is the cosine of the angle opposite the side of length 9?
$\displaystyle \cos C = \frac{49 + 64 - 81}{2 \cdot 7 \cdot 8} = \frac{32}{112} = \frac{2}{7}$.
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