# What is the distance between the following polar coordinates?:  (5,(-5pi)/3), (5,(11pi)/6)

Dec 17, 2016

$5 \sqrt{2}$

#### Explanation:

Before finding the distance between these points, it is necessary to convert them to rectangular coordinates. These points have been plotted as in the given figure and explained as follows:

To plot the point $\left(5 , \frac{- 5 \pi}{3}\right)$,go $\frac{5 \pi}{3}$ radians or ${300}^{o}$ clockwise from x- axis and mark point P at a radial distance of 5 units from the origin O. This is shown in the black pen. Now draw a perpendicular PR from point P to the x -axis. In the right triangle POR, $\angle$POR is $\frac{\pi}{3}$. OR=5 cos$\frac{\pi}{3} = \frac{5}{2}$ and PR= 5 sin$\frac{\pi}{3} = \frac{5 \sqrt{3}}{2}$. Thus rectangular coordinates of P are $\left(\frac{5}{2} , \frac{5 \sqrt{3}}{2}\right)$

To plot the point $\left(5 , \frac{11 \pi}{6}\right)$, go $\frac{11 \pi}{6}$ or ${330}^{o}$ counterclockwise from x -axis and mark point Q at a radial distance of 5 units from the origin O. This is shown in red pen. Now draw a perpendicular QS from point Q to the x -axis. In the right triangle QOS, $\angle$ QOS is $\frac{- \pi}{6}$. OS =$5 \cos \left(- \frac{\pi}{6}\right) = \frac{5 \sqrt{3}}{2}$ and QS= 5sin$\frac{- \pi}{6} = - \frac{5}{2}$ Thus rectangular coordinates of Q are $\left(\frac{5 \sqrt{3}}{2} , - \frac{5}{2}\right)$

Distance between points P and Q= $\sqrt{{\left(\frac{5}{2} - \frac{5 \sqrt{3}}{2}\right)}^{2} + {\left(\frac{5 \sqrt{3}}{2} + \frac{5}{2}\right)}^{2}}$

=sqrt(2(25/4 +75/4)

= 5$\sqrt{2}$