How to Convert Polar Form to Rectangular Form

To convert from polar coordinates (r, θ) to rectangular coordinates (x, y), you use specific formulas that relate the distance from the origin (r) and the angle from the positive x-axis (θ) to the x and y positions in the Cartesian plane.

Understanding Polar and Rectangular Forms

Before diving into the conversion, let's quickly define the two coordinate systems:

• Polar Form (r, θ): Represents a point in a plane by its distance r from a fixed point (the origin) and its angle θ from a fixed direction (the positive x-axis).
• Rectangular Form (x, y): Represents a point in a plane by its signed distances x and y from two fixed perpendicular lines (the x-axis and y-axis).

The Conversion Formulas

As per the reference provided, to convert from polar coordinates to rectangular coordinates, use the formulas x = r cos θ and y = r sin θ.

• x = r cos θ: This formula calculates the x-coordinate (horizontal position) based on the distance r and the cosine of the angle θ.
• y = r sin θ: This formula calculates the y-coordinate (vertical position) based on the distance r and the sine of the angle θ.

Here, r is the distance from the origin, and θ is the angle measured counterclockwise from the positive x-axis.

Steps for Conversion

Follow these simple steps to convert a point from polar coordinates (r, θ) to rectangular coordinates (x, y):

1. Identify r and θ: Note the given values for the radial distance r and the angle θ from the polar coordinates (r, θ). Ensure your angle θ is in the correct units (usually degrees or radians) for your calculation tool (calculator or software).
2. Apply the formulas:
   • Calculate the x-coordinate using the formula: x = r * cos(θ)
   • Calculate the y-coordinate using the formula: y = r * sin(θ)
3. Write the result: The rectangular coordinates are (x, y).

Example: Converting a Polar Point

Let's convert the polar point (r, θ) = (4, 30°) to rectangular coordinates.

1. Identify r and θ:
   • r = 4
   • θ = 30°
2. Apply the formulas:
   • x = r * cos(θ) = 4 * cos(30°)
   • y = r * sin(θ) = 4 * sin(30°)
3. Calculate the values:
   • cos(30°) = √3 / 2 (or approximately 0.866)
   • sin(30°) = 1 / 2 (or 0.5)
   • x = 4 * (√3 / 2) = 2√3 (or approximately 4 * 0.866 = 3.464)
   • y = 4 * (1 / 2) = 2 (or 4 * 0.5 = 2)
4. Write the result: The rectangular coordinates are (2√3, 2) or approximately (3.464, 2).

Here’s a quick summary in a table format:

Polar Coordinate (r, θ)	Formula for x	Formula for y	Rectangular Coordinate (x, y)
(r, θ)	x = r cos θ	y = r sin θ	(r cos θ, r sin θ)

This process allows you to translate the position of any point given in polar form into its equivalent position in the standard rectangular coordinate system.