# What is the distance between (3 ,( 5 pi)/12 ) and (-2 , ( 3 pi )/2 )?

Jun 20, 2016

The distance between the two points is approximately $1.18$ units.

#### Explanation:

You can find the distance between two points using the Pythagorean theorem ${c}^{2} = {a}^{2} + {b}^{2}$, where $c$ is the distance between the points (this is what you're looking for), $a$ is the distance between the points in the $x$ direction and $b$ is the distance between the points in the $y$ direction.

To find the distance between the points in the $x$ and $y$ directions, first convert the polar co-ordinates you have here, in form $\left(r , \setminus \theta\right)$, to Cartesian co-ordinates.

The equations that transform between polar and Cartesian co-ordinates are:

$x = r \cos \setminus \theta$
$y = r \sin \setminus \theta$

Converting the first point
$x = 3 \cos \left(\setminus \frac{5 \setminus \pi}{12}\right)$
$x = 0.77646$

$y = 3 \sin \left(\setminus \frac{5 \setminus \pi}{12}\right)$
$y = 2.8978$

Cartesian co-ordinate of first point: $\left(0.776 , 2.90\right)$

Converting the second point
$x = - 2 \cos \left(\setminus \frac{3 \setminus \pi}{2}\right)$
$x = 0$

$y = - 2 \sin \left(\setminus \frac{3 \setminus \pi}{2}\right)$
$y = 2$

Cartesian co-ordinate of first point: $\left(0 , 2\right)$

Calculating $a$
Distance in the $x$ direction is therefore $0.776 - 0 = 0.776$

Calculating $b$
Distance in the $y$ direction is therefore $2.90 - 2 = 0.90$

Calculating $c$
Distance between the two points is therefore $c$, where
${c}^{2} = {a}^{2} + {b}^{2}$
${c}^{2} = {0.776}^{2} + {0.9}^{2}$
${c}^{2} = 1.4122$
$c = 1.1884$
$c \setminus \approx 1.18$

The distance between the two points is approximately $1.18$ units.

The diagrams about halfway down this page, in the section 'Vector addition using components' might be useful in understanding the process just performed.