# What is the distance between the following polar coordinates?:  (4,(5pi)/12), (4,(3pi)/12)

Nov 29, 2017

$2.06 \textcolor{w h i t e}{88}$units

#### Explanation:

To find the distance we first need to convert the polar coordinates in Cartesian coordinates. We can do this using the fact that:

$x = r \cos \left(\theta\right) \mathmr{and} y = r \sin \left(\theta\right)$

$\therefore$

For $\left(4 , \frac{5 \pi}{12}\right)$

$x = 4 \cos \left(\frac{5 \pi}{12}\right) = 1.04$

$y = 4 \sin \left(\frac{5 \pi}{12}\right) = 3.86$

Cartesian coordinate:

$\left(1.04 , 3.86\right)$

For $\left(4 , \frac{3 \pi}{12}\right)$

$x = 4 \cos \left(\frac{3 \pi}{12}\right) = 2 \sqrt{2}$

$y = 4 \sin \left(\frac{3 \pi}{12}\right) = 2 \sqrt{2}$

Cartesian coordinate:

$\left(2 \sqrt{2} , 2 \sqrt{2}\right)$

Using the distance formula.

$d = \sqrt{{\left(2 \sqrt{2} - 1.04\right)}^{2} + {\left(2 \sqrt{2} - 3.86\right)}^{2}} = 2.06 \textcolor{w h i t e}{88}$units ( 2 .d.p)