# Conversion between Cartesian and Polar Coordinates in Plane/Examples/(4, pi over 3)

## Example of Use of Conversion between Cartesian and Polar Coordinates in Plane

The point $P$ defined in polar coordinates as:

$P = \polar {4, \dfrac \pi 3}$

can be expressed in the corresponding Cartesian coordinates as:

$P = \tuple {2, 2 \sqrt 3}$

## Proof

Let $P$ be expressed in Cartesian coordinates as:

$P = \tuple {x, y}$
 $\ds x$ $=$ $\ds r \cos \theta$ $\ds y$ $=$ $\ds r \sin \theta$

where in this case:

 $\ds r$ $=$ $\ds 4$ $\ds \theta$ $=$ $\ds \dfrac \pi 3$

Hence:

 $\ds x$ $=$ $\ds 4 \cos \dfrac \pi 3$ $\ds$ $=$ $\ds 4 \times \dfrac 1 2$ Cosine of $60 \degrees$ $\ds$ $=$ $\ds 2$

and:

 $\ds y$ $=$ $\ds 4 \sin \dfrac \pi 3$ $\ds$ $=$ $\ds 4 \times \dfrac {\sqrt 3} 2$ Sine of $60 \degrees$ $\ds$ $=$ $\ds 2 \sqrt 3$

Hence the result.

$\blacksquare$