# What is the distance between (4 ,( 7 pi)/6 ) and (-1 ,( 3pi )/2 )?

Mar 23, 2018

The distance between the two points is $\sqrt{3}$ units

#### Explanation:

To find the distance between these two points, first convert them into regular coordinates. Now, if $\left(r , x\right)$ are the coordinates in polar form, then the coordinates in regular form are $\left(r \cos x , r \sin x\right)$.
Take the first point $\left(4 , \frac{7 \pi}{6}\right)$.
This becomes $\left(4 \cos \left(\frac{7 \pi}{6}\right) , 4 \sin \left(\frac{7 \pi}{6}\right)\right)$

=$\left(- 2 \sqrt{3} , - 2\right)$

The second point is $\left(- 1 , \frac{3 \pi}{2}\right)$
This becomes $\left(- 1 \cos \left(\frac{3 \pi}{2}\right) , - 1 \sin \left(\frac{3 \pi}{2}\right)\right)$

=$\left(0 , 1\right)$

So now the two points are $\left(- 2 \sqrt{3} , - 2\right)$ and $\left(0 , 1\right)$. Now we can use the distance formula
$d = \sqrt{{\left(- 2 \sqrt{3} - 0\right)}^{2} - {\left(- 2 - 1\right)}^{2}}$

=$\sqrt{12 - 9}$

=$\sqrt{3}$