POLAR COORDINATES
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DEFINITION OF POLAR COORDINATES
Before we can start working with polar coordinates, we must define what we will be
talking about. So let us first set us a diagram that will help us understand what we are
talking about.
First, fix an origin (called the pole) and an
initial ray from O.
Each point P can be located by assigning
to it a polar coordinate pair (r,  ) where r
is the directed distance from O to P and 
is the directed angle from the initial ray to
ray OP.
When  is positive, then the angle was
measured counterclockwise, and when 
is negative, the angle was measured
clockwise. By this fact, a given polar
coordinate is not unique.
EXAMPLE 1:
(see diagram)
Sometimes, there are occasions when we would like to allow r to be negative. For
example, force on an object in a certain direction. This is why we would say that r is a
directed distance.
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EXAMPLE 2:
see diagram
EXAMPLE 3:
Find all the polar coordinates of the point (-3, -  / 4).
SOLUTION:
First of all, let us plot the original point.
(Remember we can go around the circle an infinite number of
times in either direction, which is why I use +/- and multiples of 2
 .)
Now, if r is positive, what is the  that gives the same point? What
is the  / 4 angle in quadrant II? The answer is 3  / 4. Here are
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those related points.
ELEMENTARY COORDINATE EQUATIONS AND INEQUALITIES
Some forms of basic polar equations that occur most often are the equations for a
circle centered at the origin and the equation for the line through the origin.
FACT:
The equation of a circle with radius | a | centered at the origin is r = a.
Why did I say that the radius is | a |? Remember r is a directed distance, so the circle r
= a and r = -a will be the same circle, but start in different places.
EXAMPLE 4:
FACT:
To illustrate the above fact I will graph the half circles r = - 2 and
r = 2 on the interval [0,  ]. r = - 2 is in the blue. r = 2 is in the
red. Notice that r = - 2 starts at  and goes to 2  , whereas r = 2
starts at 0 and goes to  .
The equation of a line through the origin making angle  o with the
initial ray is  =  o.
In my opinion, it is kind of a waste using a polar equation to represent the equation of
a line, but it is possible to do.
Now, let us graph some polar equations and inequalities.
EXAMPLE 5:
r1
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SOLUTION:
The graph of this polar
inequality will be the shaded
region outside the circle of
radius 1.
EXAMPLE 6:
SOLUTION:
This is the graph of the line that
makes the angle 2 / 3 with the
positive x-axis, but goes in the
opposite direction starting at 2.
EXAMPLE 7:
0     , r = -1
SOLUTION:
This is the half circle that starts
at  and goes to 2  . I know
that the interval starts at 0, but r
is negative. Therefore it goes in
the opposite directions.
EXAMPLE 8:
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SOLUTION:
The graph of this set of
inequalities is two wedges cut
out of the circle with radii of -1
and 1, and all circles that are
between those two values by the
lines   / 4.
CONVERTING FROM POLAR TO CARTESIAN AND VICE VERSA
Here are the basic equations that relate polar coordinates to Cartesian coordinates.
x = r cos 
y = r sin 
x2+y2=r2
tan  = y/ x
Here is a diagram to help us understand
where these equations came from.
CONVERTING FROM POLAR COORDINATES TO CARTESIAN COORDINATES
EXAMPLE 9:
Convert (0,  / 2) to Cartesian coordinates.
SOLUTION:
x = 0 cos ( / 2) = 0
y = 0 sin ( / 2) = 0
So (0,  / 2) is equivalent to (0, 0) in Cartesian coordinates.
EXAMPLE 10:
Convert the following polar coordinate to its equivalent Cartesian
coordinate.
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SOLUTION:
So the equivalent Cartesian coordinates for the given polar
coordinate is (-1, -1).
CONVERTING FROM A CARTESIAN EQUATION TO A POLAR EQUATION
EXAMPLE 11:
Convert y = 10 into a polar equation.
SOLUTION:
This is a graph of a horizontal line with y-intercept at (0, 10).
EXAMPLE 12:
Convert x 2 - y 2 = 4 into a polar equation.
SOLUTION:
This is an equation of a hyperbola, and here is its graph.
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EXAMPLE 13:
Convert y 2 = 4x into a polar equation.
SOLUTION:
This is an equation of a parabola, and here is its graph.
EXAMPLE 14:
Convert (x + 2) 2 + (y - 4) 2 = 16 into a polar equation.
SOLUTION:
First of all, I am going to multiply out the original equation.
(x + 2) 2 + (y - 4) 2 = 16  x 2 + 4x + 4 + y 2 - 8y + 16 = 16
 x 2 + y 2 + 4x - 8y = -4
Now convert this equation into its corresponding polar form.
r 2 + 4r cos  - 8r sin  = -4
This is an equation of a circle with center at (-2, 4) and radius 4.
CONVERTING A POLAR EQUATION TO A CARTESIAN EQUATION
EXAMPLE 15:
Convert r sin  = 4 into its equivalent Cartesian equation.
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SOLUTION:
r sin  = 4  y = 4
This is an equation of a horizontal line through the point (0, 4).
EXAMPLE 16:
Convert r sin  = r cos  + 4 into its equivalent Cartesian
equation.
SOLUTION:
r sin  = r cos  + 4  y = x + 4
This is an equation of a line with slope of 1 and y-intercept (0, 4).
EXAMPLE 17:
Convert r = csc  e r cos  into its equivalent Cartesian equation.
SOLUTION:
EXAMPLE 18:
Convert r = 4tan  sec  into its equivalent Cartesian equation.
SOLUTION:
In this set of supplemental notes, I defined what makes up a polar coordinate and that a polar
coordinate for a point is not unique. Then I talked about the polar equations for circles centered
at the origin and lines going through the origin. Finally, I discussed how we could convert from a
Cartesian equation to a polar equation by using some formulas. Work through these examples
taking note how each conversion was done. Polar coordinates are the first type of coordinates
that we will learn in this course and in calculus III. Polar coordinates allow us to graph certain
types of curves easily and simplify integrals. In the next three sets of supplemental notes, we will
investigate applications of polar coordinates, so make sure that you understand what is
happening in this set.
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