# A typical triangle has sides a=5 and b=1 and angle C= pi/3. Find c ?

May 11, 2018

$c = \sqrt{{5}^{2} + {1}^{2} - 2 \left(5\right) \left(1\right) \left(\frac{1}{2}\right)} = \sqrt{21}$

#### Explanation:

I have a few thousand trig problems under my belt and it's startling and sad just how many use multiples of ${30}^{\circ}$ or ${45}^{\circ} .$ Question writers, the world has more than two triangles!

$\frac{\pi}{3}$ is of course ${60}^{\circ}$ and its cosine is $\frac{1}{2}$.

By the Law of Cosines,

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos C$

$c = \sqrt{{5}^{2} + {1}^{2} - 2 \left(5\right) \left(1\right) \left(\frac{1}{2}\right)} = \sqrt{21}$