# What is the distance between (4 ,( 7 pi)/6 ) and (3 , ( -3 pi )/2 )?

Oct 3, 2017

$8.44$

#### Explanation:

Distance =sqrt((3-4)^2+(((-3pi)/2)-((7pi)/6))^2
$= \sqrt{1 + {\left(\frac{16 \pi}{6}\right)}^{2}}$
$= \sqrt{1 + 70.18} = \sqrt{71.18} = 8.44$ (as $\pi = \frac{22}{7}$)

Oct 4, 2017

Distance between the two points is $4 \sqrt{3}$

#### Explanation:

The polar coordinates $\left(4 , \frac{7 \pi}{6}\right)$ are equivalent to rectangular or Cartesian coordinates $\left(4 \cos \left(\frac{7 \pi}{6}\right) , 4 \sin \left(\frac{7 \pi}{6}\right)\right)$ i.e. $\left(- \frac{4 \sqrt{3}}{2} , - 2\right)$, and

polar coordinates $\left(3 , - \frac{3 \pi}{2}\right)$ are equivalent to rectangular or Cartesian coordinates $\left(4 \cos \left(- \frac{3 \pi}{2}\right) , 4 \sin \left(- \frac{3 \pi}{2}\right)\right)$ i.e. $\left(0 , 4\right)$

and distance between the two points is

$\sqrt{{\left(0 - \left(- \frac{4 \sqrt{3}}{2}\right)\right)}^{2} + {\left(4 - \left(- 2\right)\right)}^{2}}$

= $\sqrt{{\left(2 \sqrt{3}\right)}^{2} + {\left(6\right)}^{2}}$

= $\sqrt{12 + 36} = \sqrt{48} = 4 \sqrt{3}$