# What is the distance between the following polar coordinates?:  (8,(-21pi)/12), (5,(-3pi)/8)

Feb 2, 2018

$10.937$

#### Explanation:

First $\left(8 , - \frac{21 \pi}{12}\right)$ can be simplified a bit $\left(8 , - \frac{7 \pi}{4}\right)$, and since $- \frac{7 \pi}{4}$ is coterminal to $\frac{\pi}{4}$, we'll use $\left(8 , \frac{\pi}{4}\right)$ as an equivalent point. The other point we'll keep as $\left(5 , - \frac{3 \pi}{8}\right)$.

If we plot the points and use them as two vertices of a triangle with the origin as the third vertex, we have sides 8 and 5 with an angle of $\frac{\pi}{4} + \frac{3 \pi}{8} = \frac{5 \pi}{8}$ between them.

For this triangle we can use the Law of Cosines to find the side opposite $\frac{5 \pi}{8}$, which is the distance between the given points:

$c = \sqrt{{a}^{2} + {b}^{2} - 2 a b \cos \left(C\right)}$

$c = \sqrt{{8}^{2} + {5}^{2} - 2 \left(8\right) \left(5\right) \cos \left(\frac{5 \pi}{8}\right)}$

$c \approx 10.937$