Convert Polar to Cartesian Coordinates

This figure illustrates the propperties that a point has in both the polar and cartesian coordinate system.
Figure 1: Point in Polar and Cartesian Coordinates

To convert a point from the polar coordinate system to cartesian coordinate system the trigonometric functions sine and cosine are used to solve for the and coordinate of the point. A point in the polar coordinate system is in the form of and a point in the cartesian coordinate system is in the form of . The formulas for the conversion are shown below:

Note, by convention, radians are used to measure angles in polar coordinates.

Example 1

Example point in polar and cartesian coordinate

This example shows how to convert the point in the polar coordinate system to its cartesian coordinate equivalent.

  1. Start by setting up the formula for conversion.

  2. Substitute the radius and angle of the polar coordinates into the formula.

  3. Calculte the result of the multiplication.

    Then since we can write the coordinates of the point as:

    The cartesian coordinates of the point in polar coordinates are .

Example 2

Example point in polar and cartesian coordinate

This example shows how to convert the point in the polar coordinate system to its cartesian coordinate equivalent.

  1. Start by setting up the formula for conversion.

  2. Substitute the radius and angle of the polar coordinates into the formula.

  3. Calculte the result of the multiplication.

    Then since we can write the coordinates of the point as:

    The cartesian coordinates of the polar point are .

Explanation

The formula uses on this page can be derived using the circle definitions of the trigonometric functions sine and cosine and then solving for the anc component of the point.

This figure illustrates the geometry used to define the circle definitions of the trigonometric functions.
Figure 2: Circle Functions

Links