This instructional aid was prepared by the Tallahassee Community College Learning Commons.
Polar and Rectangular Coordinate Conversions
Polar Coordinate System – Any ordered pair written in the form of (𝑟 , 𝜃) where r is the r radius
from the Origin point O to a fixed point P and θ is the angle between the Polar Axis and the segment OP
.
Rectangular Coordinate System – Any ordered pair that can be written in the form of (𝑥 , 𝑦) where
x is the horizontal component and y is the vertical component of the point.
x = r cos θ and y = r sin θ
Converting from Polar to Rectangular Coordinates:
Example: Find the Rectangular Coordinates for the point that has Polar Coordinates (2 ,60°).
Solution: x = r cos θ and y = r sin θ
x = 2 c os 60° y = 2 sin 60°
= 2 × 1
2 = 2 × √3
2
= 1 = √𝟑
The Rectangular Coordinates for the point that has Polar Coordinates (2 , 60°) is (𝟏 ,√𝟑 )
Converting from Polar Coordinates to Rectangular Coordinates:
Given r2= x2+ y2 and tanθ = y
x
Example: Find the Polar Coordinates for the point that has Rectangular Coordinates (3 , 3).
Solution: r2= x2+ y2
Given: r2= 32+ 32 tan θ = y
x
r2= 9 + 9 tan θ = 3
3
r2=18 tan θ = 1
r = √18 = 3√2 tan−1(1)=45°
The Polar Coordinates for the point that has Rectangular Coordinates (3 , 3) is (𝟑√𝟐,𝟒𝟓°).
