Polar Coordinate
December 25, 2016
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Introduction
Throughout this course, we have been familiar with reading and plotting graph on
rectangular coordinates or Cartesian coordinates, with x and y axes. But sometimes
it is easier to write the function and plotting the graph on Polar Coordinates of r and
θ. Before we continue on this topic, it is necessary to learn about parameterisation
first, since Polar Coordinates is just a special type of parameterisation.
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Parameterisation
Parametrics is a technique to plot a graph by specifying the x, y and z values separately. We do so using a parameter, a variable that we chosed to link the equations
together. For example, we frequently use the parameter t.
For example, if we take x = cos t and y = sin t. Notice that, although we have two
separate equations, the x and y are linked by the value of t. In this example, we
take a value for the parameter t and plug that value into both equations, to get a
corresponding point (x, y). When t = 0, we get (1, 0). So, the point (1, 0) corresponds
to the value t = 0.
One of the advantages of using parametric equations is that we can describe many
more graphs than we could when we had only x and y.
Another piece of information we get from parametric equation is direction. The
use of the variable t as a parameter is not random. Often, we assign a meaning to the
parameter and sometimes that meaning is time. When you graph a set of parametric
equations, the graph is swept out in a certain direction. This is an inherent feature
of the parametric equations. We will often start at t = 0 and increase t, giving the
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idea that time is passing. By adjusting the parametric equations, we can reverse the
direction that the graph is swept.
2.1
Graphing/Sketching Parametric Equations
To plot graph, make a table of the parameter t (usually consists of negative, 0 , and
positive value), x value and y value. It would be easier if you know how to input this
into calculator efficiently.
Example 1 Sketch the graph for the parametric equations
1. x = 2t + 1, y = t − 1
2. x = 2t2 , y = 4t
3. x = 3 + 2t2 , y = −4t
4. x = 3 cos θ, y = 3 sin θ
2.2
Eliminating the Parameter
Sometimes we are given the set of parametric equations and we are asked to write
the equation without the parameter by eliminating the parameter. This not always
possible but with some equations there are ways to do it.
The easiest technique to try is to solve one of the equations for the parameter and
then substitute the result in the other equation. Here is an example. x = 2t and
y = t2 . Solve the first equation for t giving t = x/2 and substitute into the second
equation. y = (x/2)2 = x2 /4.
A second technique involves the use of trig identities. For example, given the
parametric equations x = cos t and y = sin t, we know that cos2 (x) + sin2 (x) = 1. So
we can write x2 + y 2 = 1 which eliminates the parameter.
A third way is by inspection. Sometimes it is obvious what a substitution might
be. For example, if we have x = et and y = e3t + 1 then we can rewrite y = (et )3 + 1.
Then we can replace et with x in the last equation (because our first parametric
equation was x = et ) to get y = x3 + 1. Of course, the first technique would have
worked too by solving x = et for t and then substituting and simplifying but by
standing back and looking at the equations more carefully, the solution was much
easier.
Example 2 Obtain the Cartesian equation by eliminating the parameter from the
following paramteric equations.
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1. x = 2t, y = 4t2 − 1
2. x = 4t, y =
4
t
3. x = 2 − m, y = m2 + 4
4. x = 2 − 5 cos θ, y = 1 − 3 sin θ
5. x = 3t2 , y = 6t
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Polar Coordinates System
From basic trigonometri, you know that a point in the plane can be described as
(x, y) or as (r cos θ, r sin θ). Comparing these two forms gives you the equations
x = r cos θ,
y = r sin θ.
These equations are used to convert between polar coordinates and rectangular coordinates. Remember from trig that angles can be described in an infinite number of
Figure 1: Polar Coordinate System in Cartesian
ways, since θ = θ + 2π and θ = θ − 2π. It is always best to use the smallest possible
angle in the interval (−π, π] or [0, 2π] or whatever is required by the context. Note
that these angles are measured in degrees. However, in calculus we almost always
specify angles in radians. There are no restriction of the application, you should
carefully choose when you input the value in your calculator. Most of the time we
use it in degrees since it is more easier to imagine and later plot it in the graph.
One of the biggest differences you will find between trigonometri and polar coordinates is that in trig, r in the above equation is usually 1. Trigonometri focuses on the
unit circle (when r = 1). However, in polar coordinates we generalize the equations
so that r is usually not 1.
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Figure 2: Polar Coordinate plot
Figure 3: Trigonometric Quadrants
The positive x-axis is called the polar axis, labeled L or sometimes x in Figure 2
and the point O is called the pole. All angles are measured from the polar axis with
positive angles in a counter-clockwise direction.
In conclusion, polar coordinates are just parametric equations where the parameter is the angle θ and r is a function of θ.
Example 3 Plot the following points
1. (3, 225◦ )
2. (2, −60◦ )
3. (3, 150◦ )
π
4. (4, )
2
5. (1, −120◦ )
6. (2, −
7π
)
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7. (2, π)
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Converting Polar Coordinates ⇔ Cartesian Coordinates
In order to convert polar coordinates to Cartesian coordinates, we just need the
formula
y
θ = tan-1 ,
x
Example 4 Express the following in polar equations
x = r cos θ,
y = r sin θ,
x2 + y 2 = r2 .
1. y = x2
2. x2 + y 2 = 5
3. x + y = 1
4. xy = a2
5. x = 4
6. y = 2
7. x2 + y 2 = 2x
8. x2 = y
Example 5 Express the points in Example 3 in Cartesian coordinates.
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Graph Sketching of Polar Equation
The method to sketch the graph of polar equation is similar to that of Cartesian
equations. We start with θ usually from 0 to 2π, or in degrees, with distance r and
marked with point in the form (r, θ).
Example 6 Sketch the graph for the polar equations given
1. r = 3
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2. r = π
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3. r = 1 − cos θ
4. r = 2 + cos θ
5. r = 2 sin θ
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6. r = 3 sin 2θ
Knowing if the equation is symmetrical or not will make it easier to draw the
graph since repeating calculations can be avoided. Testing the symmetrical is done
as below
Symmetry
Symmetrical Conditions
About the x-axis
If (r, θ) satisfy the equation, so does (r, −θ) .
About the y-axis
If (r, θ) satisfy the equation, so does (r, π − θ) .
About the pole(origin)
If (r, θ) satisfy the equation, so does (r, π + θ).
Identities that will come in handy :
sin(−θ) = − sin θ,
cos(−θ) = cos θ,
tan(−θ) = − tan θ.
IMPORTANT : It is compulsory for you to test for all axes. It is always the case that
it would symmetry on one axis only, OR symmetry on all three axes.
Usually, if the polar equation is symmetry on x-axis, it is convenient for you to only
look for the value 0◦ − 180◦ when plotting the graph, then reflect the graph on x-axis.
Also, if the polar equation is symmetry on y-axis, it is convenient for you to only
look for the value 0◦ − 90◦ AND 270◦ − 360◦ when plotting the graph, then reflect
the graph on y-axis.
If the polar equation is symmetry on origin(pole), then you can choose either domain
mentioned above, and then rotate 180◦ of every point that you plot. Coincidentally,
if the polar equation is symmetry on the pole, it has the same effect of having to plot
the polar coordinate with negative r, for which we plot the positive (r, θ) first then
reflect it on origin to get (−r, θ).
You might not convince sometimes with your plot after performing the symmetrical
test. Therefore, it is okay, for you to tabulate all the value of r from 0◦ −360◦ provided
you must do the symmetrical test first as required.
Example 7 Test the symmetries of the following equations. Hence sketch the graphs.
1. r = 2 + cos θ
2. r = 3 cos 2θ
3. r = 1 − sin θ
4. r = 2 − 3 sin θ
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5. r = a sin θ
6. r2 = 2a2 cos 2θ
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Intersection of Curves in Polar Coordinates
If both equation are given in r and θ, just equate the r. But sometimes you need to
change it into Cartesian first to find the points of intersection.
Example 8 Find the intersection between each of the following pairs of curves.
1. r = 2 cos θ and r = 2 sin θ
2. r = 1 and r2 = 2 cos 2θ
3. r = a and r = 2a sin θ
4. r = 1 + cos θ and r = 3 cos θ
IMPORTANT : It should be noted that, the process of finding the intersection by
equating the two polar functions, will not detect certain intersection points due to the
polar and angle properties. We usually missed the intersection points that occured
on origin or on axes. The best way to rectify this problem is by plotting both of the
polar coordinate function in a graph.
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Conclusion
For this topic you are expected to know how to
1. Convert Cartesian coordinates to polar coordinates and vice versa
2. Sketch the graph for polar equations and use symmetrical test
3. Find the intersection points of curves
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