Coordinates are often used to denote location of a point on a plane. Rectangular (Cartesian) coordinates and polar coordinates are two different ways of locating a point on a plane. Rectangular coordinates take the form (x, y) where x is the distance of the point from the vertical axis while y is the distance from the horizontal axis. In simpler words, the rectangular coordinates tell us how far along and how far up a point is from the origin. On the other hand, the polar coordinates are written in the form (r,(theta)) where r is the distance of the point from the origin and (theta) is the angle formed by the line drawn from the origin to that point and the positive x–axis. Though these coordinates are written in different ways but they can be made equivalent. It means that, rectangular coordinates can actually be converted into polar coordinates and vice versa.

To convert rectangular coordinates to polar coordinates, we must use the tangent trigonometric ratio and the equation of a circle with center at the origin. To be more precise, listed below are the steps of converting rectangular coordinates to polar coordinates:

1. Solve for r using the equation of a circle whose center is at the origin of a plane. Recall that polar coordinates are actually made up of concentric circles. The equation for r is defined by the equation,

2. Draw a right triangle using the given rectangular coordinates. The x-coordinate is the length of the horizontal leg while the y-coordinate is the length of the vertical leg. Mark the angle about the origin. Use the tangent ratio to look for the marked angle. Since tangent involves the ratio of the opposite and adjacent sides. Since the y-coordinate is the length of the opposite side while the x-coordinate is the adjacent, then tan(theta)=y/x so

3. Write the polar coordinates into the form, (r,(theta)).

To convert polar coordinates to rectangular coordinates, let us first recall that on a unit circle, the cosine of an angle equals the ratio between the x-coordinate and the length of the radius while the sine of an angle equals the ratio between the y-coordinate and the length of the radius. In symbols, cos(theta)= x/r and sin(theta)= y/r. By manipulating these equations, we can have the formulas for x and y, respectively. Therefore, to convert from Polar Coordinates (r,(theta)) to Cartesian Coordinates (x,y), use the formulas, x = r cos(theta) and y = r sin(theta), respectively then write the numbers obtained into the form (x, y).

Converting Rectangular Equations to Polar Equations Video Example