Section 10.5: Polar Coordinates
Objectives: Upon completion of this lesson, you will be able to:
 convert between polar and rectangular coordinates
 convert between polar and rectangular equations
 sketch the graph of a polar equation.
Required Reading, Video, Tutorial
 Read Swokowski/Cole: Section 10.5
o Disregard Examples 8, 9, 10, and 11
 Watch the Section 10.5 Video
 Complete the Section 10.5 Before-Class WA assignment
Discussion
This section is an introduction to the polar coordinate system. As you will see in this section, there are times
when it is advantageous to working in a system other than the rectangular coordinate system.
Points in the rectangular coordinate system are of the form  x, y  . Points in the polar system are of the form
 r,   .
When plotting these points, first locate the angle  as a ray in standard position, and then mark the r
value on the ray. A positive r value moves out on the ray. A r value moves in the opposite direction from
the ray. A point in the rectangular system has a unique name. This is not true of points in the polar system;
please see Figure 2 on page 763 in the text.
An essential element to remember from this section is the relationship between the polar and rectangular
coordinates. The following equations are the key to converting not only points from one system to the other but
also equations.
Polar to Rectangular
Rectangular to Polar
x  r cos
r
y

y  r sin
x
r  x2  y2
 R  tan 1
y
, x0
x
Converting from polar to rectangular form is very straight forward; see Example 1 in the text. However,
converting from rectangular to polar form can be tricky. Example 2 in the text will explain the process.
Sketching the point will help to remind you of the quadrant in which the point lies. Be sure that the polar form
angle is in the same quadrant as the sketched point with a positive r value.
Below are two examples of converting a point in polar form to a point in rectangular form.
 
Example 1: Convert the polar point  6,  to rectangular form.
 3
Math 130: Section 10.5 – Polar Coordinates
Solution: Use the polar to rectangular equations from the chart above and substitute.
x  r cos   6 cos
y  r sin   6sin

1
 6   3
3
2
 3
 6 
  3 3
3
 2 

 
Therefore,  6,   3, 3 3 . These two forms of a point are at the same location.
 3


Example 2: Convert the polar point  2,   to rectangular form.
Solution: Use the polar to rectangular equations from the chart above and substitute.
x  r cos   2cos   2  1  2
y  r sin   2sin   6  0   0
Therefore,
 2,    (2,0).
These two forms of a point are at the same location.
Below are two examples of converting a point in rectangular form to a point in polar form.
Example 3: Convert the rectangular point  2, 2  to polar form.
Solution: The given point is in Quadrant II. Use the rectangular to polar equations form the chart above and
substitute.
r  x 2  y 2  (2) 2  22  8  2 2
tan  R 
y
2

 1,
x
2
Therefore,
R 
 2, 2    2

2,

4
3
4
and  
3
4

 . These two forms of a point are at the same location.

Example 4: Convert the rectangular point  0, 1 to polar form.
Solution: The given point is on the positive y-axis. Use the rectangular to polar equations form the chart above
and substitute.
©2014;Dr. Brenda Shryock
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Math 130: Section 10.5 – Polar Coordinates
r  x 2  y 2  x 2  y 2  02  12  1
tan  R 
y

which is undefined for the x  0 and y  1, so  
x
2
Therefore,
 0, 1  1,


 . These two forms of a point are at the same location.
2
Also, become proficient at converting quadrantal points. This will save you a lot of time. Below are four
examples of a point in rectangular form and its corresponding point in polar form in each of the quadrantal
locations.
Rectangular Point
 3, 0 

 0, 3

 3, 0 

 0,  3

Polar Point
 3, 0 or  3, 0
 3, 90 or  3,


 2
 3,180 or  3,  
 3, 270 or  3,

3 

2 
Examples 3, 4, and 5 in the text demonstrate the conversion of equations from one form to another. Remember
that when converting equations from one form to another, rectangular-form equations should have no r or 
terms and polar-form equations should have no x or y terms.
Example 5: Find a rectangular equation for r 
4
.
3cos   sin 
Solution:
r
4
3cos   sin 
r (3cos   sin  )  4
3r cos   r sin   4
3x  y  4
multiply by the denominator
distribute
substitute x  r cos  and y  r sin 
This is a linear equation.
Example 6: Find a rectangular equation for r  8cos .
Solution:
©2014;Dr. Brenda Shryock
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Math 130: Section 10.5 – Polar Coordinates
r  8cos 
x
r  8 
r
r 2  8x
x2  y 2  8x
x 2  8 x  ___  y 2  0  ___
cos 
x
r
r 2  x2  y 2
complete the square
x  8 x  16  y  16
2
2
( x  4) 2  y 2  16
factor
This is a circle centered at  4, 0  with a radius of 4.
Example 7: Find a polar equation for y  5.
Solution: Begin by using the substitution y  r sin  and then solve for r.
y  5
r sin   5
5
r
 5 csc
sin 
Polar Equations and Graphs
The equation of a cardioid is of the form: r  a  a sin or r  a  a cos .
To graph cardioids, either construct an extensive T-Chart or use the technique of sweeping through the
quadrants. To construct a T-Chart, plug in values for  and computing r. Begin with 0 radians and work your
way counterclockwise around the circle with “special” and quadrantal angles. Examples 6 and 7 in the text
illustrate this technique.
Even though limaçons, roses, lemniscates, and spirals are fascinating graphs, you are not responsible for them.
If you have a graphics calculator, you are encouraged to use this tool to examine all of the graphs of this section.
However, keep in mind, that on Test 3 and the Final Exam, you may be expected not only to graph a polar
equation or two, but you may be asked to fill in a T-Chart with specific values. Do practice graphing these by
hand and then use the calculator to verify your graph.
Please print the polar graphs file on WA for your use. If necessary, this is the grid that will be used on Test 3
and the final exam.
Practice Problems
Answers to odd-numbered problems can be found at the end of your text.
Section Exercises on WA
1, 4, 6, 10, 12, 14, 18, 20, 23, 25, 27, 31, 38, 42, 45, 47, 51, 54, 55, 61, 63, 65, 72
10.5
©2014;Dr. Brenda Shryock
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