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When trying to figure out the graphs of polar equations we
can convert them to rectangular equations particularly if
we recognize the graph in rectangular coordinates.
7!r We could square both sides
492 !r
Now use our conversion:
222yxr !
4922 ! yx
We recognize this as a circle
with center at (0, 0) and a
radius of 7.
On polar graph paper it will centered at the origin and out 7
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Let's try another:
3
TU ! Take the tangent of both sides
!
3tantan TU
Now use our conversion:
3!x
y
We recognize this as a line with slope square root of 3.
3!
x
y!Utan
Multiply both
sides byx
xy 3!
To graph on a polar plot
we'd go to whereand make a line. 3
TU !
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Let's try another: 5sin !r
Now use our conversion:
We recognize this as a
horizontal line 5 units belowthe origin (or on a polar plot
below the pole)
Usinry !
5!y
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Sometimes converting to rectangular equations doesn't
help us figure out what the graph would look like or it is
not necessary.
Usin1!rThe only way we know how toconvert is if there is an rin front of
the sin U term so we'll multiply both
sides by r.Usin2 rrr !
Now use our conversions:
222yxr ! 22 yxr ! Usinry !
yyxyx ! 2222 I still don't
know what the
graph looks
like!In these cases we'll plot points, choosing
a U from the polar form and finding a
corresponding rvalue.
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Before we do this, if we knew something about the
symmetry of the graph we may not have to find as many
points. TESTS FOR SYMMETRY
Symmetry with
Respect to the Polar
Axis (xaxis)
Replace U by - U
and if you get
original equation
back
Symmetry with
Respect to the Line
U = T/2 (yaxis)
Replace U by T - U
and if you get
original equation
back
(r,U)
(r, -U)
Symmetry with
Respect to the Pole
(Origin)
Replace rby - r
and if you get
original equation
back
(r,U)(r,T-U)
(r,U)
(- r,U)
These tests are sufficient but not necessary so if test fails you don't know anything.
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Usin1!rLet's test for symmetry
Polar Axis: U! sin1r Usin1!rFAILS
Line U = T/2: UT! sin1r
UTUT sincoscossin1 !r
Use the difference formula
0 -1
Usin1!r
This IS the original equation!
Pole: Usin1! r
Not the original equation.
So this graph is symmetric
with respect to the line U = T/2
(yaxis). We will only need tochoose U's on the right side of
the graph then and we can use
symmetry to get the other half.
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U Usin1!r
2
T 211 !
3
T 87.1
2
31 }
Let's let each unit be 1/4.
6
T
5.1211 !
0 101 !
6
T
2
1
2
11 !
3
T
13.02
31 }
2
T
011 !Let's plot the symmetric points
This type of graph is called a
cardioid.
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Equations of carioids would look like one of the following:
r= a(1 + cosU) r= a(1 + sinU)
r= a(1 - cosU) r= a(1 - sinU)
wherea > 0
All graphs of cardioids pass through the pole.
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Let's test for symmetry
Polar Axis: U! cos23r
YES!
Line U = T/2: UT! cos23r
UTUT sinsincoscos23 !r
Use the difference formula
-1 0
Ucos23!r
Not the original equation
Pole: Ucos23! r
Not the original equation.
So this graph is symmetric
with respect to the polar axis
(xaxis). We will only need to
choose U's on the top half of
the graph then and we can use
symmetry to get the other half.
Let's try another: Ucos23!r
Ucos23!r
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Equations of limacons without inner loops would look like
one of the following:
r= a +b cosU r= a +b sinU
r= a - b cosU r= a - b sinU
wherea > 0, b > 0, and a >b
These graphs DO NOT pass through the pole.
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Let's test for symmetry
Polar Axis: U! cos21r
YES!
Line U = T/2: UT ! cos21r
UTUT sinsincoscos21 !r
Use the difference formula
-1 0
Ucos21!r
Not the original equation
Pole: Ucos21! r
Not the original equation.
So this graph is symmetric
with respect to the polar axis
(xaxis). We will only need to
choose U's on the top half of
the graph then and we can use
symmetry to get the other half.
Let's try another: Ucos21!r
Ucos21!r
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U
0 3121 !
6
T
73.22
321 }
Let's let each unit be 1/2.
3
T
2
2
121 !
2
T 1021 !
3
2T0
2
121 !
6
5T
73.02
321 }
T 1121 !
Let's plot the symmetric points
This type of graph is called a
limacon with an innerloop.Ucos21!r
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Equations of limacons with inner loops would look like
one of the following:
r= a +b cosU r= a +b sinU
r= a - b cosU r= a - b sinU
wherea > 0, b > 0, and a < b
These graphs will pass through the pole twice.
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Let's test for symmetry
Polar Axis: U! 2cos2rYES!
Line U = T/2: UT! 2cos2r
UU 2cos22cos2 !!rYES!
Pole:
Since graph is symmetric
to both the polar axis and
the line U =T/2 it will also
be with respect to the pole.
So this graph is symmetric
with respect to the pole, the
polar axis and the line U =T/2.
We will only need to choose
U's between 0 and T/2.
Let's try another: U2cos2!r
U2cos2!r
UT 22cos2 !
cos is periodic so can drop the 2T
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U
0 212 !
6
T1
2
12 !
Let's let each unit be 1/2.
4
T
002 !
3
T1
2
12 !
2
T
212 !
Let's plot the symmetric points
This type of graph is called a rose
with 4 petals. U2cos2!r
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Equations of rose curves would look like one of the
following:
r= a cos(nU) r= a sin(nU)
Wheren even has 2n petals and
n odd has n petals
(n {
0 ors
1)
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Let's test for symmetry
Polar Axis: U! 2sin42rFAILS
Line U = T/2: UT! 2sin42r
UU 2sin22sin42 !!r FAILS
Pole:
So this graph is symmetric
with respect to the pole.
Let's try another: U2sin42 !r
U2sin42 !r
UT 22sin4 !
sin is periodic so can drop the 2T
U2sin42
! r
U2sin42 !r
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U
0 004 !
6
T
322
34 !
Let's let each unit be 1/4.
4
T
4
14 !
3
T32
2
34 !
2
T
004 !
This type of graph is called a
lemniscate U2sin42 !r r
0
9.1s
2s
9.1s
0
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Equations of lemniscates would look like one of the
following:
r2 = a2 cos(2U) r2 = a2 sin(2U)
These graphs will pass through the pole and are
propeller shaped.
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Table 7 in your book on page 732
summarizes all of the polar graphs.
You can graph these on your calculator. You'll need tochange to polar mode and also you must be in radians.
If you are in polar function mode when you hit your
button to enter a graph you should see r1 instead ofy1.
Your variable button should now put in U on TI-83's and
it should be a menu choice in 85's & 86's.
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Have fun plotting pretty pictures!
Limacon With InnerLoop
made with TI Calculator
Rose with 7 petals made
with graphing program on
computer
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Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloadedfrom www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephens School Carramar
www.ststephens.wa.edu.au
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