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Using Polar Coordinates
Graphing and
converting polar and
rectangular coordinates
Butterflies are among the most celebrated of all insects.
It’s hard not to notice their beautiful colors and graceful
flight. Their symmetry can be explored with trigonometric
functions and a system for plotting points called the polar
coordinate system. In many cases, polar coordinates are
simpler and easier to use than rectangular coordinates.
You are familiar with
plotting with a rectangular
coordinate system.
We are going to look at a
new coordinate system
called the polar
coordinate system.
The center of the graph is
called the pole.
Angles are measured from
the positive x axis.
Points are
represented by a
radius and an angle
(r, )
radius angle
To plot the point
4,5
First find the angle
Then move out along
the terminal side 5
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The polar coordinate system is formed by fixing a point, O,
which is the pole (or origin).
= directed angle Polar
axis
OPole (Origin)
The polar axis is the ray constructed from O.
Each point P in the plane can be assigned polar coordinates (r, ).
P = (r, )
r is the directed distance from O to P.
is the directed angle (counterclockwise) from the polar axis
to OP.
Graphing Polar Coordinates
The grid at the left is a polar grid. The
typical angles of 30o, 45o, 90o, … are
shown on the graph along with circles of
radius 1, 2, 3, 4, and 5 units.
Points in polar form are given as (r, )
where r is the radius to the point and is
the angle of the point.
On one of your polar graphs, plot the
point (3, 90o)?
A
The point on the graph labeled A is correct.
A negative angle would be measured clockwise like usual.
To plot a point with
a negative radius,
find the terminal
side of the angle
but then measure
from the pole in
the negative
direction of the
terminal side.
4
3,3
3
2,4
330
315
300
270240
225
210
180
150
135
120
0
9060
30
45
Polar coordinates can also be given with the angle in
degrees.
(8, 210°)
(6, -120°)
(-5, 300°)
(-3, 540°)
Graphing Polar Coordinates
Now, try graphing .
Did you get point B?
Polar points have a new aspect. A radius
can be negative! A negative radius
means to go in the exact opposite
direction of the angle.
A
To graph (-4, 240o), find 240o and move 4
units in the opposite direction. The opposite
direction is always a 180o difference.
4
3,2
B
C
Point C is at (-4, 240o). This point could
also be labeled as (4, 60o).
Graphing Polar Coordinates
How would you write point A with a
negative radius?
A correct answer would be (-3, 270o) or
(-3, -90o).
In fact, there are an infinite number of
ways to label a single polar point.
Is (3, 450o) the same point?
A
Don’t forget, you can also use radian angles
as well as angles in degrees.
B
C
On your own, find at least 4 different polar
coordinates for point B.
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The point lies two units from the pole on the
terminal side of the angle
( , ) 2,3
r
.3
3
2,3
33,4
34
2
32
1 2 3
0
3 units from
the pole
Plotting Points
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There are many ways to represent the point 2, .3
2
32
1 2 3
0
2,3
52,3
2 42,2 2,,3 3 3
52,3
additional ways
to represent the
point 2,3
( , ) , 2r r n
( , ) , (2 1)r r n
Let's plot the following points:
2,7
2,7
2
5,7
2
3,7
Notice unlike in the
rectangular
coordinate system,
there are many
ways to list the
same point.
Converting from Rectangular to Polar
Find the polar form for the rectangular point (4, 3).
To find the polar coordinate, we must
calculate the radius and angle to the
given point.(4, 3)
We can use our knowledge of right
triangle trigonometry to find the radius
and angle.
3
4
r
r2 = 32 + 42
r2 = 25
r = 5
tan = ¾
= tan-1(¾)
= 36.87o or 0.64 rad
The polar form of the rectangular point
(4, 3) is (5, 36.87o)
Converting from Rectangular to Polar
In general, the rectangular point (x, y) is converted to polar form (r, θ) by:
1. Finding the radius
(x, y)r2 = x2 + y2
y
x
r 2. Finding the angle
tan = y/x or = tan-1(y/x)
Recall that some angles require
the angle to be converted to the
appropriate quadrant.
However, the angle must be in the second
quadrant, so we add 180o to the answer
and get an angle of 123.70o.
The polar form is ( , 123.70o)
r2 = (-2)2 + 32
r2 = 4 + 9
r2 = 13
r =
Converting from Rectangular to Polar
o3156
2
3
2
3
1
.
tan
tan
13
On your own, find polar form for the point (-2, 3).
(-2, 3)
13
Converting from Polar to Rectanglar
322
34
304
x
x
rx
ocos
cos
2214
304
y
oy
ry
sin
sin
Convert the polar point (4, 30o) to rectangular coordinates.
4
30o
We are given the radius of 4 and angle of 30o.
Find the values of x and y.
Using trig to find the values of x and y, we know
that cos = x/r or x = r cos . Also, sin = y/r or
y = r sin .
x
y
The point in rectangular form is: 2,32
Converting from Polar to Rectanglar
2
3
2
13
3003
x
x
rx
ocos
cos
2
33
233
3003
y
oy
ry
sin
sin
On your own, convert (3, 5π/3) to rectangular coordinates.
-60o
We are given the radius of 3 and angle of 5π/3 or
300o. Find the values of x and y.
The point in rectangular form is:
2
33,
2
3
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(r, )
(x, y)
Polex
y
(Origin)
y
r
x
The relationship between rectangular and polar
coordinates is as follows.
The point (x, y) lies on a
circle of radius r, therefore,
r2 = x2 + y2.
tanyx
cos xr
sinyr
Definitions of
trigonometric functions
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20
Coordinate Conversion
cosx r cos xr
siny r sinyr
2 2 2r x y tanyx
(Pythagorean Identity)
Example:
Convert the point into rectangular coordinates. 4,3
1cos co3
24 s 42
x r
3sin sin 4 23 2
4 3y r
, 2, 2 3x y
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Example:
Convert the point (1,1) into polar coordinates.
, 1,1x y
1tan 11
yx
4
2 2 2 21 1 2r x y
set of polar coordinates is ( , ) 2, .4
One r
5Another set is ( , ) 2, .4
r
Let's take a point in the rectangular coordinate system
and convert it to the polar coordinate system.
(3, 4)
r
Based on the trig you
know can you see
how to find r and ?4
3
r = 5
222 43 r
3
4tan
93.03
4tan 1
We'll find in radians
(5, 0.93)polar coordinates are:
Let's generalize this to find formulas for converting from
rectangular to polar coordinates.
(x, y)
r y
x
222 ryx
x
ytan
22 yxr
x
y1tan
Now let's go the other way, from polar to rectangular
coordinates.Based on the trig you
know can you see
how to find x and y?
44cos
x
rectangular coordinates are:
4,4
4y
x4
222
24
x
44sin
y
222
24
y 22,22
Let's generalize the conversion from polar to rectangular
coordinates.
r
xcos
,r
ry
x
r
ysin
cosrx
sinry
922 yx
Convert the rectangular coordinate system equation to a
polar coordinate system equation.
22 yxr 3r
r must be 3 but there is no
restriction on so consider
all values.
Here each r
unit is 1/2 and
we went out 3
and did all
angles.
? and torelated was
how s,conversion From
22 yxr
Before we do the conversion
let's look at the graph.
Convert the rectangular coordinate system equation to a
polar coordinate system equation.yx 42
cosrx
sinry
sin4cos2
rr
sin4cos22 rr
substitute in for
x and y
We wouldn't recognize what this equation looked like
in polar coordinates but looking at the rectangular
equation we'd know it was a parabola.
What are the polar conversions
we found for x and y?
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.
7r We could square both sides
492 rNow use our conversion:
222 yxr
4922 yxWe 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
Let's try another:
3
Take the tangent of both sides
3tantan
Now use our conversion:
3x
y
We recognize this as a line with slope square root of 3.
3
x
ytan
Multiply both
sides by x
xy 3
To graph on a polar plot
we'd go to where
and make a line. 3
Let's try another: 5sin r
Now use our conversion:
We recognize this as a
horizontal line 5 units below
the origin (or on a polar plot
below the pole)
sinry
5y
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Example:
Convert the polar equation into a rectangular
equation.
4sinr
4sinr
2 4 sinr r Multiply each side by r.
2 2 4x y y Substitute rectangular
coordinates.
22 2 4x y Equation of a circle with
center (0, 2) and radius of 2
Polar form
2 2 4 0x y y
Rectangular and Polar Equations
Equations in rectangular form use variables (x, y), while
equations in polar form use variables (r, ) where is an angle.
Converting from one form to another involves changing the
variables from one form to the other.
We have already used all of the conversions which are necessary.
Converting Polar to Rectangular
cos = x/r
sin = y/r
tan = y/x
r2 = x2 + y2
Converting Rectanglar to Polar
x = r cos
y = r sin
x2 + y2 = r2
Convert Rectangular Equationsto Polar Equations
The goal is to change all x’s and y’s to r’s and ’s.
When possible, solve for r.
Example 1: Convert x2 + y2 = 16 to polar form.
Since x2 + y2 = r2, substitute into the equation.
r2 = 16
Simplify.
r = 4
r = 4 is the equivalent polar equation to x2 + y2 = 16
Convert Rectangular Equationsto Polar Equations
Example 2: Convert y = 3 to polar form.
Since y = r sin , substitute into the equation.
r sin = 3
Solve for r when possible.
r = 3 / sin
r = 3 csc is the equivalent polar equation.
Convert Rectangular Equationsto Polar Equations
Example 3: Convert (x - 3)2 + (y + 3)2 = 18 to polar form.
Square each binomial.
x2 – 6x + 9 + y2 + 6y + 9 = 18
Since x2 + y2 = r2, re-write and simplify by combining like terms.
x2 + y2 – 6x + 6y = 0
Substitute r2 for x2 + y2, r cos for x and r sin for y.
r2 – 6rcos + 6rsin = 0
Factor r as a common factor.
r(r – 6cos + 6sin ) = 0
r = 0 or r – 6cos + 6sin = 0
Solve for r: r = 0 or r = 6cos – 6sin
Convert Polar Equationsto Rectangular Equations
The goal is to change all r’s and ’s to x’s and y’s.
Example 1: Convert r = 4 to rectangular form.
Since r2 = x2 + y2, square both sides to get r2.
r2 = 16
Substitute.
x2 + y2 = 16
x2 + y2 = 16 is the equivalent polar equation to r = 4
Convert Polar Equationsto Rectangular Equations
Example 2: Convert r = 5 cos to rectangular form.
Multiply both sides by r
r2 = 5x
Substitute for r2.
x2 + y2 = 5x is rectangular form.
r2 = 5 r cos
Substitute x for r cos
Convert Polar Equationsto Rectangular Equations
sin*
12r
Example 3: Convert r = 2 csc to rectangular form.
Since csc ß = 1/sin, substitute for csc .
Multiply both sides by 1/sin.
Simplify
y = 2 is rectangular form.
1
12
sin*
sin*sin r
You will notice that polar equations have graphs like the following:
Hit the MODE key.
Arrow down to where it says Func (short for "function" which is a bit misleading since they are all functions).
Now, use the right arrow to choose Pol.
Hit ENTER. (*It's easy to forget this step, but it's crucial: until you hit ENTER you have not actually selected Pol, even though it looks like you have!)
The calculator is now in polar coordinates mode. To see what that means, try this.
Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on.
In the r1= slot, type 5-5sin(θ)
Now hit the familiar X,T,θ,n key, and you get an unfamiliar result. In polar coordinates mode, this key gives you a θ instead of an X.
Finally, close off the parentheses and hit GRAPH.
If you did everything right, you just asked the calculator to graph the polar equation r=5-5sin(θ). The result looks a bit like a valentine.
The WINDOW options are a little different in this mode too. You can still specify X and Y ranges, which define the viewing screen. But you can also specify the θ values that the calculator begins and ends with.
Graph r = 3 sin 2θ
Enter the following window values:
Θmin = 0 Xmin = -6 Ymin = -4
θmax = 2π Xmax = 6 Ymax = 4
Θstep = π/24 Xscl = 1 Yscl = 1
Graph:
a. r = 2 cos θ
b. r = -2 cos θ
c. r = 1 – 2 cos θ
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Each polar graph below is called a Limaçon.
1 2cosr 1 2sinr
–3
–5 5
3
–5 5
3
–3
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Each polar graph below is called a Lemniscate.
2 22 sin 2r 2 23 cos2r
–5 5
3
–3
–5 5
3
–3
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Each polar graph below is called a Rose curve.
2cos3r 3sin 4r
The graph will have n petals if n is odd, and 2n
petals if n is even.
–5 5
3
–3
–5 5
3
–3
a
a
Function Gallery in your book on page
352 summarizes all of the polar graphs.
You can graph these on your calculator. You'll need to
change 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 of y1.
Your variable button should now put in on TI-83's and
it should be a menu choice in 85's & 86's.
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