# What is the distance between (2 , (5 pi)/8 ) and (3 , (1 pi )/3 )?

Feb 9, 2016

The distance between those two coordinates is $\sqrt{13 - 12 \cos \left(\frac{7 \pi}{24}\right)} \approx 2.39$.

#### Explanation:

You can use the law of cosines to do that.

Let me illustrate why:

Polar coordinates $\left(r , \theta\right)$ are defined by the radius $r$ and the angle $\theta$.

Imagine lines leading from the pole to your respective polar coordinates. Those lines represent two sides of a triangle with lengths $A = 3$ and $B = 2$. The distance between those two coordinates being the third side, $C$.

Furthermore, the angle between $A$ and $B$ can be computed as the difference between the two angles of your polar coordinates:

$\gamma = \frac{5 \pi}{8} - \frac{\pi}{3} = \frac{7 \pi}{24}$

Thus, the length of the side $C$ can be found with the help of law of cosines on that triangle:

${C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cos \left(\gamma\right)$

$= {3}^{2} + {2}^{2} - 2 \cdot 3 \cdot 2 \cdot \cos \left(\frac{7 \pi}{24}\right)$

$= 13 - 12 \cos \left(\frac{7 \pi}{24}\right)$

$\implies C = \sqrt{13 - 12 \cos \left(\frac{7 \pi}{24}\right)} \approx 2.39$