6.4: Polar to Rectangular Conversions
You are hiking one day with friends. When you stop to examine your map, you mark your position on a polar plot with your campsite at the origin, like this
You decide to plot your position on a different map, which has a rectangular grid on it instead of a polar plot. Can you convert your coordinates from the polar representation to the rectangular one?
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James Sousa Example: Convert a point in polar coordinates to rectangular coordinates
Guidance
Just as
The point
Therefore
These equations, also known as coordinate conversion equations, will enable you to convert from polar to rectangular form.
Example A
Given the following polar coordinates, find the corresponding rectangular coordinates of the points:
Solution:
a) For
The rectangular coordinates of
b) For
The rectangular coordinates of
In addition to writing polar coordinates in rectangular form, the coordinate conversion equations can also be used to write polar equations in rectangular form.
Example B
Write the polar equation
Solution:
The equation is now in rectangular form. The
The rectangular form of the polar equation represents a circle with its centre at (2, 0) and a radius of 2 units.
This is the graph represented by the polar equation
Example C
Write the polar equation
Solution:
Vocabulary
Polar Coordinates: A set of polar coordinates are a set of coordinates plotted on a system that uses the distance from the origin and angle from an axis to describe location.
Rectangular Coordinates: A set of rectangular coordinates are a set of coordinates plotted on a system using basis axes at right angles to each other.
Guided Practice
1. Write the polar equation
2. Write the polar equation
3. Write the polar equation
Solutions:
1.
2.
3.
Concept Problem Solution
You can see from the map that your position is represented in polar coordinates as
Practice
Given the following polar coordinates, find the corresponding rectangular coordinates of the points.

(2,π6) 
(4,2π3) 
(5,π3) 
(3,π4) 
(6,3π4)
Write each polar equation in rectangular form.
 \begin{align*}r=3\sin \theta \end{align*}
 \begin{align*}r=2\cos \theta \end{align*}
 \begin{align*}r=5\csc \theta \end{align*}
 \begin{align*}r=3\sec \theta \end{align*}
 \begin{align*}r=6\cos \theta \end{align*}
 \begin{align*}r=8\sin \theta \end{align*}
 \begin{align*}r=2\csc \theta \end{align*}
 \begin{align*}r=4\sec \theta \end{align*}
 \begin{align*}r=3\cos \theta \end{align*}
 \begin{align*}r=5\sin \theta \end{align*}
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Here you'll learn how to convert a position described in polar coordinates to the equivalent position in rectangular coordinates.