Math 144
Activity #11
Polar Coordinates
We use ordered pairs to identify and distinguish points from one another in the Cartesian plane.
When doing so, we need two components, one to give the horizontal distance from the origin and
a second to give a vertical distance from the origin. Is there another way to represent points in a
plane? In this activity, we will explore the possibility of using other means to locate and identify
points in a plane.
Rectangular Plane:
Plot the points A 1,4 , B  3,5 and C 2,5 on the graph below.
Why are the points A and B on opposite sides of the y-axis?
Why are the points B and C on different sides of the x-axis?
Polar Plane:
Below, you will find a graph of the unit circle with grid lines and the Cartesian plane. Locate the
point in quadrant I that is marked on the circle. What is the ordered pair that is associated with
the point marked?
The ordered pair uses two distances, a horizontal distance from the y-axis and a vertical distance
from the x-axis. Instead of using two distances, could you get there by using a distance and an
angle? If so, what would they be for the point marked on the graph?
Find the distance associated with the point (3, 2); then find the angle.
Find the distance associated with the point (2, 2); then find the angle.
Find the distance associated with the point (3, 1); then find the angle.
On the previous page, you were given several ordered pairs that contained a horizontal
component, an x, and a vertical component, a y, and asked to find a distance from the origin and
an angle of rotation for each. With this in mind, could the points (3, 2) and 13 ,33.7 o represent
the same point?

What about the points (3, 1) and

 10 ,18.4  represent the same point?
o
You should have answered yes to both of those. This means that any point in the rectangular
plane can be represented using, a distance from the origin and an angle in standard position.
Describe how you could represent any Cartesian coordinate using an angle and a distance. You
must be careful, the three points you did were all in quadrant I, what if you had the point (–3, 2)?
Look at the figure below; it is a map of Earth as viewed from the North Pole. On the map, the
North Pole acts as the origin, the longitude lines mimic the concentric circles of the polar graph.
How would you define polar coordinates?
Graphing polar and rectangular coordinates together:
Below you will see a polar graph superimposed with a Cartesian system. Use the graph below to
plot each of the following coordinates.
Rectangular:
 4 
 5 
3  
Polar:
 3,
,
 2,
,
 , 
 3 
 6 
2 4
 3 3 3
,
Rectangular:   ,

2
2




3 ,1 ,
What do you notice about the points that you plotted?
3 2 3 3 


 4 , 4 


Extension Question:
Can the radius (distance) value of polar coordinate be negative? If not, why? If so, explain how
to plot the point.
Converting between the systems:
It is often important for us to convert lengths of objects from one unit to another, as in feet to
inches. It is also important to do this with coordinates in a plane. In the Cartesian coordinate
system, we us an x, y  ordered pair to identify a point and in the polar coordinate system, we
use an r ,   ordered pair. Using what you discovered earlier in the activity, you will now find a
way to convert between the two systems.
Locate the point in quadrant I on the graph below. If this is a rectangular plane, the point is x, y  .
If this a polar plane, the point is identified as r ,   .
●
Draw a perpendicular line from the point to the x-axis and then draw a second line from the
origin to the point. You should have a right triangle in quadrant I, where one side is on the
positive x-axis.
Label the sides of the triangle x, y and r appropriately. Label the acute angel in the corner next to
the origin  . Using what you know about right triangles and trigonometry, find four equations
that relate the variables x and y to the variables r and  .
1. Use the Pythagorean Theorem:
2. Find the value of tangent for  and then isolate the polar coordinate:
3. Find the value of cosine for  and then isolate the rectangular coordinate:
4. Find the value of sine for  and then isolate the rectangular coordinate:
The above equations that you created will allow you to convert any polar coordinate to a
rectangular point and any rectangular to polar.
 7 
Convert the point  2,
 to a rectangular coordinate.
 6 
Convert the point (5, 4) to a polar coordinate.
Defining functions in two variables:
In your own words, write a definition for what is a function?
You might have mentioned something about a rule that relates two variables to each other or
maybe you said that each input has exactly one output. So is it possible to be a function if there
are multiple inputs and one output? To answer this, let’s think about something that you do every
time you go through a checkout line in a store. After the item is rung up and totaled by the clerk,
the amount you owe is given to you; we will call this amount cost. Discuss with your group how
the cost is calculated. The cost of an item you want to purchase includes the price being charged
by the retailer and any sales tax required by law. The price is some dollar amount set by the
retailer and the tax is some percent set by the state or city, together these to quantities will
combine to give you the cost you pay for the item. Think about the previous sentence, does this
fit the definition of a function? In order to determine the cost, you need a price and a tax rate.
There is one price and usually one set tax rate; each of these represents an input and both are
needed to get the cost as an output. If we let price = p, tax = t, then in math symbols this looks
like the following: f  p, t   1  t  p = cost. If we add an input, is it still a function? Explain your
answer.
The definition of a function in two variables is a relation that assigns to every ordered pair of
input values (x, y) in a set called the domain a unique output value denoted by f(x, y). The set of
output values is called the range.
What this definition means for us now, is that we are describing a function as a single output
with multiple input values, but it is still a function. With the example on the previous page, we
are describing a single or unique output, called cost that is generated by using two input values
called price and tax. So yes, it is a function.
Give at least one more example of a function related to your major that requires at least two
variable inputs. Discuss these with your group.
Polar functions:
The graphs of polar functions are quite fascinating. Below you can see a few examples of such
graphs along with the equation that makes the graph and their name.
Equation: r  2  2 cos
This graph is called a cardioid. It can be rotated half a
pie in any direction in the polar plane.
Equation: r  3 cos( 2 )
This graph is called a rose. The number of pedals can
will vary as you change the multiplier on theta.
Equation: r  4 sin(  )
This graph is of course a circle.
These are the graphs of functions. They are polar functions and if we were to convert them to
rectangular coordinates, they would not be functions.
Extension Question:
Convert the equation r  4 sin(  ) to rectangular by using the relationships that you found earlier.