6.5: Rectangular to Polar Conversions
You are looking at a map of your state with a rectangular coordinate grid. It looks like this
You realize that if you convert the the coordinates of Yourtown (labelled with a "YT") to polar coordinates, you can more easily see the distance between the Capitol at the origin and Yourtown. Can you make this conversion from rectangular to polar coordinates? Keep reading, and at the end of this Concept, you'll be able to perform the conversion.
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James Sousa Example: Convert a Point in Rectangular Coordinates to Polar Coordinates Using Radians
Guidance
When converting rectangular coordinates to polar coordinates, we must remember that there are many possible polar coordinates. We will agree that when converting from rectangular coordinates to polar coordinates, one set of polar coordinates will be sufficient for each set of rectangular coordinates. Most graphing calculators are programmed to complete the conversions and they too provide one set of coordinates for each conversion. The conversion of rectangular coordinates to polar coordinates is done using the Pythagorean Theorem and the Arctangent function. The Arctangent function only calculates angles in the first and fourth quadrants so
In addition to these formulas,
Example A
Convert the following rectangular coordinates to polar form.
Solution: For
The polar coordinates of
Example B
Convert the following rectangular coordinates to polar form.
Solution: For
The polar coordinates of
Example C
Convert the following rectangular coordinates to polar form.
Solution: For
The polar coordinates of
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 following rectangular point in polar form:
2. Write the following rectangular point in polar form:
3. Write the following rectangular point in polar form:
Solutions:
1. For
2. For
3.
Concept Problem Solution
To convert these rectangular coordinates into polar coordinates, first use the Pythagorean Theorem:
and then use the tangent function to find the angle:
The polar coordinates for Yourtown are
Practice
Write the following points, given in rectangular form, in polar form using radians where
 (1,3)
 (2,5)
 (2,3)
 (2,1)
 (3,2)
 (4,5)
 (1,2)
 (3,3)
 (2,5)
 (1,4)
 (5,2)
 (1,6)
For each equation, convert the rectangular equation to polar form.

x=5 
2x−3y=5 
2x+4y=2 
(x−1)2+y2=1 
(x+3)2+(y+3)2=18  \begin{align*}y=7\end{align*}
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Here you'll learn how to convert a position described in a rectangular coordinate system to the equivalent position described in a polar coordinate system.