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# Rectangular to Polar Coordinates

Learn in this page on rectangular to polar coordinates. For more help a students can connect to an online tutor anytime and get the required help in the concept.

The location of a point in the rectangular coordinate system uses the distance of the point from the x and y axis while the polar coordinate system the point is located using the distance from the pole and angle. The parameters in the both coordinate system can be converted into each other. In the following article we will see more about the topic rectangular to polar coordinates in detail.

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## Convert Rectangular to Polar Coordinates

Rectangular coordinate system:

In the two dimensional rectangular coordinate system the location of the point in the system is defined by the distance of the point from the x axis and the y axis. And the values used for the representation of the position are called as the coordinates of the point. Any point in the system is represented as $(x,y)$.

Polar coordinate system:

In the polar coordinate system any point is described using the radial distance r of the point measured from the pole and the angle measure between the reference ray from the pole and the point. The points in the polar coordinates are represented as `$(r,\theta)$.

Conversion from rectangular to polar coordinates:

For the conversion of the coordinates of the points in the rectangular coordinates $(x,y)$ to the polar coordinates $(r,\theta)$ the following relations are used,

$r$ = $\sqrt{x^2 + y^2}$ ;   $\theta$ = $tan^{-1}(\frac{y}{x})$

## Examples

Below are some examples on rectangular to polar coordinates

1. Convert the rectangular coordinates (5,1) into its equivalent polar coordinates.

Solution:

$r$ = $\sqrt{x^2 + y^2}$

$\sqrt{5^2 + 1^2}$

$\sqrt{56}$

$5.09$

$\theta$ = $tan^{-1}(\frac{y}{x})$

= $tan^{-1}(\frac{1}{5})$

$= 11.3\ degrees$

2. Convert the rectangular coordinates (2,8) into its equivalent polar coordinates.

Solution:

$r$ = $\sqrt{x^2 + y^2}$

$\sqrt{2^2 + 8^2}$

$\sqrt{68}$

$8.2$

$\theta$ = $tan^{-1}(\frac{y}{x})$

$tan^{-1}(\frac{8}{2})$

$= 76\ degrees$