# What is the distance between the following polar coordinates?:  (1,(-23pi)/12), (3,(5pi)/8)

Jun 12, 2017

The distance, $c \approx 3.28$

#### Explanation:

When given two polar points $\left({r}_{1} , {\theta}_{1}\right)$ and $\left({r}_{2} , {\theta}_{2}\right)$, please observe that the two radii ${r}_{1}$ and ${r}_{2}$ form a triangle with the line between the two points. Here is a graph of the, two points, the two radii, and the line between them:

We can use the Law of Cosines to find the length, c, of the blue line:

${c}^{2} = {r}_{1}^{2} + {r}_{2}^{2} - 2 \left({r}_{1}\right) \left({r}_{2}\right) \cos \left({\theta}_{2} - {\theta}_{1}\right) \text{ [1]}$

Using the two given points we assign:

${r}_{1} = 1$, ${r}_{2} = 3$, ${\theta}_{1} = - \frac{23 \pi}{12}$ and ${\theta}_{2} = \frac{5 \pi}{8}$

Substituting into equation [1]:

${c}^{2} = {1}^{2} + {3}^{2} - 2 \left(1\right) \left(3\right) \cos \left(\frac{5 \pi}{8} - \left(- \frac{23 \pi}{12}\right)\right)$

$c \approx 3.28$