Polar and Rectangular Coordinates: Conversion and Distance Formulas

Points of Polar Coordinates

In the Cartesian plane, a point is located using its distances to the two axes x and y. Another way to represent a point is by specifying the distance from that point to a fixed point and the angle formed between the segment that joins the two points and a fixed line. The fixed point is called the pole and the fixed line is known as the polar axis. This point is represented as , known as a polar coordinate.

In the figure above, r represents the distance between the pole and point P. Angle is considered positive when measured counterclockwise and negative when measured clockwise.

Relation Between Rectangular and Polar Coordinates

If the polar plane is placed over a Cartesian plane so that the pole coincides with the origin, then the following relations are obtained:

Example

Determine the rectangular coordinates of the point (5, 60°).

Steps

Procedure

1. Calculate the value for x:

2. Calculate the value for y:

Answer

The rectangular coordinates of the point (5, 60°) are

Example

Determine the polar coordinates for the point P(4, -6).

Steps

Procedure

1. Obtain the value for r from the formula:
   
2. Obtain the value for :

Since the angle is negative, y is negative and x is positive. The point is located in quadrant IV, so you need to subtract the negative angle from 360°.

Answer

The polar coordinates of the point P(4, -6) are

Distance Between Two Points in Polar Coordinates

To calculate the distance between two points given in polar coordinates, find an expression for d in terms of r₁, r₂ and . Observe the following figure:

Considering the figure above, the triangle formed is oblique. By applying trigonometry and the Law of Cosines , the relation can be expressed as:

Solving for d, it can be written as:

Example

Given the points W(4, 30°) and Z(8, 60°), determine the distance between them.

Steps

Procedure

1. Determine the angle between the points.
   
2. Substitute the given data into the distance formula:
   

Answer

The distance between the two points in polar coordinates is