# What is the distance between the following polar coordinates?:  (2,(pi)/3), (4,(pi)/6)

Apr 26, 2017

distance $= 2 \sqrt{5 - 2 \sqrt{3}} \approx 2.479$

#### Explanation:

Given 2 polar points: $\left(2 , \frac{\pi}{3}\right) , \left(4 , \frac{\pi}{6}\right)$

$\frac{\pi}{3} = {60}^{\circ} \text{ and } \frac{\pi}{6} = {30}^{\circ}$

Convert the polar coordinates to rectangular coordinates using $\left(x , y\right) = \left(r \cos \theta , r \sin \theta\right)$:

$\left(2 , \frac{\pi}{3}\right) = \left(2 \cos \left(\frac{\pi}{3}\right) , 2 \sin \left(\frac{\pi}{3}\right)\right) = \left(2 \cdot \frac{1}{2} , 2 \cdot \frac{\sqrt{3}}{2}\right)$

$\left(2 , \frac{\pi}{3}\right) = \left(1 , \sqrt{3}\right)$

$\left(4 , \frac{\pi}{6}\right) = \left(4 \cos \left(\frac{\pi}{6}\right) , 4 \sin \left(\frac{\pi}{6}\right)\right) = \left(4 \cdot \frac{\sqrt{3}}{2} , 4 \cdot \frac{1}{2}\right)$

$\left(4 , \frac{\pi}{6}\right) = \left(2 \sqrt{3} , 2\right)$

Find the distance between the two points:

$d = \sqrt{{\left(1 - 2 \sqrt{3}\right)}^{2} + {\left(\sqrt{3} - 2\right)}^{2}}$

$d = \sqrt{1 - 4 \sqrt{3} + 12 + 3 - 4 \sqrt{3} + 4}$

d = sqrt(20 - 8sqrt(3)) = sqrt(4(5-2sqrt(3))

$d = 2 \sqrt{5 - 2 \sqrt{3}} \approx 2.479$