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6.4: Polar to Rectangular Conversions

Difficulty Level: At Grade Created by: CK-12
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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


Just as x and y are usually used to designate the rectangular coordinates of a point, r and θ are usually used to designate the polar coordinates of the point. r is the distance of the point to the origin. θ is the angle that the line from the origin to the point makes with the positive xaxis. The diagram below shows both polar and Cartesian coordinates applied to a point P. By applying trigonometry, we can obtain equations that will show the relationship between polar coordinates (r,θ) and the rectangular coordinates (x,y)

The point P has the polar coordinates (r,θ) and the rectangular coordinates (x,y).



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: W(4,200),H(4,π3)


a) For W(4,200),r=4 and θ=200


The rectangular coordinates of W are approximately (3.76,1.37).

b) For H(4,π3),r=4 and θ=π3


The rectangular coordinates of H are (2,23) or approximately (2,3.46).

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 r=4cosθ in rectangular form.


rr2x2+y2=4cosθ=4rcosθ=4xMultiply both sides by r.r2=x2+y2 and x=rcosθ

The equation is now in rectangular form. The r2 and θ have been replaced. However, the equation, as it appears, does not model any shape with which we are familiar. Therefore, we must continue with the conversion.

x24x+y2x24x+4+y2(x2)2+y2=0=4=4Complete the square for x24x.Factor x24x+4.

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 r=4cosθ for 0θ2π or the rectangular form (x2)2+y2=4.

Example C

Write the polar equation r=3cscθ in rectangular form.


rrcscθr1cscθrsinθy=3cscθ=3=3=3=3divide bycscθsinθ=1cscθy=rsinθ


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 r=6cosθ in rectangular form.

2. Write the polar equation rsinθ=3 in rectangular form.

3. Write the polar equation r=2sinθ in rectangular form.








Concept Problem Solution

You can see from the map that your position is represented in polar coordinates as (3,70). Therefore, the radius is equal to 3 and the angle is equal to 70. The rectangular coordinates of this point can be found as follows:



Given the following polar coordinates, find the corresponding rectangular coordinates of the points.

  1. (2,π6)
  2. (4,2π3)
  3. (5,π3)
  4. (3,π4)
  5. (6,3π4)

Write each polar equation in rectangular form.

  1. \begin{align*}r=3\sin \theta \end{align*}
  2. \begin{align*}r=2\cos \theta \end{align*}
  3. \begin{align*}r=5\csc \theta \end{align*}
  4. \begin{align*}r=3\sec \theta \end{align*}
  5. \begin{align*}r=6\cos \theta \end{align*}
  6. \begin{align*}r=8\sin \theta \end{align*}
  7. \begin{align*}r=2\csc \theta \end{align*}
  8. \begin{align*}r=4\sec \theta \end{align*}
  9. \begin{align*}r=3\cos \theta \end{align*}
  10. \begin{align*}r=5\sin \theta \end{align*}

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Difficulty Level:
At Grade
Date Created:
Sep 26, 2012
Last Modified:
Aug 11, 2016
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