Mentor: You are probably familiar with the Cartesian coordinate system which allows you to plot
points and functions on the coordinate plane. If you are given a set of points in the form (x,
y), you should be able to plot them on the x- and y-axes of a graph.
Student: Yes. If I were given the point (2,3), I would start at the origin and go two units to the
right on the x-axis and then move up 3 units in the y direction.
Mentor: That is correct. And if you were given a function, say y=2x+3, you could probably graph that
Student: Of course, I would choose values of x which I would input into the function to get the values
of y for that function. Then I would graph the points on the graph.
Mentor: Yes! Did you know that you can also plot points on another type of graph based on the Polar
Student: What is the Polar Coordinate system?
Mentor: The polar coordinate system is a circular rather than a rectangular coordinate system. You
plot points given in terms of a value, r which is the distance from the center; and an angle
measure which is given in degrees, or more usually radians. Do you remember what radians are?
Student: Sure, they are another unit of measure for measuring angles. A circle contains 360 degrees
which is the same as 2π radians. But, what would you use this system for?
Mentor: Well sometimes it is more appropriate to describe a location in terms of a circular motion
rather than a straight line. This system is well suited to trigonometric functions and is used
in navigation because values can be plotted in terms of distance (r) and direction (angle,
also referred to as the Greek letter theta shown as the symbol θ).
Student: How would you plot points on a polar graph?
Mentor: The polar coordinate system has axes similar to the Cartesian system and a point like the
origin except that it is called the
pole. It also has a ray that extends horizontally through the pole called the
polar axis. Instead of a square grid, there are concentric circles centered at the pole that get
uniformly bigger and radiate out from the pole. Polar coordinate plane looks like this:
Mentor: Like rectangular coordinates, polar coordinates are also named with an ordered pair, but the
meaning of each of the numbers is different. The first number in the coordinate pair, r,
represents the distance from the pole and is like measuring the radius of a circle. The second
number of the pair, θ, represents the angle measure, measured from the horizontal axis.
The polar axis represents an angle measure of 0 radians or 0 degrees. Moving counterclockwise
from this axis represents a positive angle measure and clockwise is negative.
Student: I think I'm getting it. Angles are defined on the polar coordinate system the way they are on
the unit circle, usually in radians. So, if you are given the polar coordinate (3, π/4),
the point is three units from the center with an angle of π/4 radians measured
counterclockwise from the polar axis:
Student: So it sounds like coordinates given can be negative too?
Mentor: That is an excellent question and the answer is yes. Just as on the unit circle, negative
angles go in a clockwise direction from the polar axis. Do you remember -3π/4 is the same
Student: Yes and -4π/3 is the same as 2π/3?
Mentor: That's right. And, since you can add as many rotations in the form of multiples of 2Ï as you
want, any angle can have an infinite amount of negative and positive values. In addition, r
can be negative. A negative r indicates that the point is in the opposite
quadrant from the angle given in the coordinate pair. For example π/4 is located in the 1st
quadrant, so positive r values are plotted r units from the pole in the 1st quadrant and
negative values are plotted in the opposite quadrant, the 3rd quadrant, on the angle
identified as 5π/4. Where do you think the point (-2, π/3) is plotted?
Student: Well, since (2, π/4) was measured 2 units to the right along the polar axis, then -2 would
indicate that the point is measured 2 units to the left along the polar axis and then π/4
degrees in the counterclockwise direction. Since the angle π/3 is in the first quadrant,
then to plot the point (-2, π/3) you would plot a point two units from the pole along the
angle 4π/3. Right?
Mentor: That's right. It would look like this:
Student: So a single point could have many different values?
Mentor: Correct! The values for r can be given as positive and negative values and θ can be
given not only in positive and negative values, but also as any value θ + any multiple
of 2π. So, unlike the Cartesian system where each point has a unique set of coordinates, in
the polar system any point can have an infinite number of coordinates!
Student: That means that the point given as (2,π/4) could also be given as (2,-7π/4) or (2,
9π/4) or (-2,5π/4)!
Mentor: Exactly. Try plotting those points using the Polar Coordinates activity and verify they are
the same point!