Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
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Polar Equations & Graphs
Converting an equation in rectangular coordinates to polar coordinates
1) Convert the following equation from rectangular coordinates to polar coordinates. Simplify as
much as possible (so that you end up with the fewest terms, and the simplest terms)
4xy  9
2) Convert the following equation from rectangular coordinates to polar coordinates. Simplify as
much as possible (so that you end up with the fewest terms, and the simplest terms).
x2  y2  4
3) Convert the following equation from rectangular coordinates to polar coordinates.
y 2  2x
Completing the Square (Review)
4) Review: Completing the square (find more examples on p. 115, 4th ed. [p. 120, 3rd ed.])
x 2  8x  y 2  0
5) Review: Completing the square
x 2  y 2  2  10x  6y
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
Page 2 / 9
Converting an equation in rectangular coordinates to polar coordinates: circles
6) Convert the following equation from polar coordinates to rectangular coordinates. Simplify as
much as possible (so that you end up with the fewest terms, and the simplest terms).
Once you've done that, graph the resulting rectangular equation on the provided graph
r  4 sin 
7) Convert the following equation from polar coordinates to rectangular coordinates. Simplify as
much as possible (so that you end up with the fewest terms, and the simplest terms).
Once you've done that, graph the resulting rectangular equation on the provided graph
r  4 cos 
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
Page 3 / 9
Graphing polar equations by converting to rectangular equations
Your goal for the next several sections is to both gain practice in converting equations from polar to
rectangular form, and also in graphing the resulting rectangular equations. At the end of this section,
you should be able to explain what each of the following equations look like on a graph, as well as
being able to get the equation, given a graph with some key points outlined.
8) Based on the work you do in the next several sections, make sure that you're able to graph these
equations. For each one, be able to describe the center point, and radius, of the circle that is
produced.
r  2  a  sin  
r  2  a  sin  
r  2  a  cos 
r  2  a  cos 
9) First convert the equation to rectangular form, and then graphing the rectangular equation. Once
you've figured out where the center is, and what the radius will be, you can stop.
r  4 sin 
r  6 sin 
r  8 sin 
etc
10) Same activity as the previous exercise.
r  4 sin 
r  6 sin 
r  8 sin 
etc
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
11) Same activity as the previous exercises
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r  2 cos
r  4 cos
r  6 cos
r  2 cos
r  4 cos
r  6 cos
etc
Figuring out the equation (in polar coordinates) of a given polar equation
Given the following pictures, figure out the equation that generated it. Make sure that you're clearly
able to identify the center point, and the radius, along with the equation!
12)
Center Point:
Equation:
13)
Radius:
Center Point:
Equation:
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Radius:
Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
14)
Center Point:
Equation:
§ 9.2
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15)
Radius:
Center Point:
Equation:
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Radius:
Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
Page 6 / 9
Converting an equation in rectangular coordinates to polar coordinates: circles & lines
16) Convert the following equation from polar coordinates to rectangular coordinates. Simplify as
much as possible (so that you end up with the fewest terms, and the simplest terms)
Once you've done that, graph the resulting rectangular equation on the provided graph
r2
17) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular
form, and then graphing the rectangular equation on the provided graph (using the X & Y axes as a
rectangular grid)
r 4
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
Page 7 / 9
18) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular
form, and then graphing the rectangular equation.
 

4
19) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular
form, and then graphing the rectangular equation.
r sin   2
20) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular
form, and then graphing the rectangular equation.
r cos   3
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
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Graphing polar equations using a graphing utility
Your goal for this section is to get an understanding of how several types of polar equations look when graphed, using
your calculator. The first thing you'll need to do is properly configure your calculator.
1. If you're using a TI-83, push the
key, and make sure that
your calculator is in Radian mode, and POLar mode (move the
cursor using the arrow keys, and select a mode by pushing
) When you're done, your screen should look like this:
2. Next, push the
button to bring you to the normal screen for
entering equations to be graphed. You'll notice that much in the
same way that rectangular equations define y in terms of x (such
1
as y  x  1 ), these polar equations define r in terms of θ. So if
2
you want to graph an equation, you'll need to solve it in terms of
r, if it isn't already. Enter the formula listed in the picture to the
right (note that the
button now enters a θ. when pressed).
3. Next, push the
button, and you'll see a picture of the equation
appear. It should look like the one to the right
4. If the screen doesn't look quite right (for example, it's stretched
or compressed), then you should press the
button, and try
selecting the ZTrig setting.
5. If you're interested in looking at a table of values for a given
function (much like the book does), you can first press
to access the "TBLSET" (or "TaBLe SETup") menu),
which should look like the picture to the right. Typically, you'll


want to start your table at 0, and change by , or
, etc
12
4
6. When you're done, push the
buttons to actually see the
table. The calculator then shows you the values that it plugged in for θ,
and the resulting values that it got out of the first equation in the r1
column.
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.2
Page 9 / 9
Graphing polar equations using a graphing utility
Graph each of the following using your graphing utility. For each one, you should be able to identify which type of
equations you're looking at, given a graph. You do not need to know how to go from the graph back to the equation.
A.
(a and b are positive constants)
Cardioid: r  a  a cos  , r  a  a sin  ; try r  2  2 sin 
B.
Limaçon without an inner loop: r  a  b cos  , r  a  b sin  , a > b ; try r  2  1.8 sin 
C.
Limaçon with an inner loop: r  a  b cos  , r  a  b sin  , b > a ; try r  1.5  2.5 sin 
D.
Lemniscate: r  a 2 cos  2  , r  a 2 sin  2  ; try r  4 cos 2 
E.
Rose: r  a cos n  , r  a sin n  ; try r  3 sin 2 
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