What are polar equations?
First, we need to understand that polar equations are graphed on the polar coordinate system, which is a two-dimensional coordinate system wherein a point on the plane (r, θ) is determined by the distance r from the origin and the angle θ from the positive x-axis, measured counter-clockwise. An example of a point z on the polar coordinate system is shown below.
Clearly, polar coordinates (r, θ) of a point are related to the x- and y-coordinates of a point. Considering the picture above, we can see the red lines form a right triangle. Using basic trigonometry, we see that the length of the leg of the right triangle on the x-axis would be x = r cos θ and the length of the leg parallel to the y-axis would be y = r sin θ.
Polar equations are algebraic curves expressed in polar coordinates. For polar equations in this exploration we will define r as a function of θ. The curves (graphs of the polar functions r) will consist of points in the form (r(θ), θ).
First, letÕs consider the graph of r = 1. Based on our definition, this is the set of points a constant distance r from the origin (r does not depend on θ). So, this would be the unit circle.
Similarly, all graphs of r = a where a is a constant will be circles centered at the origin with radius a.
What does the graph of r = a θ look like?
LetÕs consider the graph of the polar equation r = a θ when a = 1 as θ ranges from 0 to 2¹.
It is interesting to note that the curve crosses the x-axis at (-¹, 0) and (2 ¹, 0). This makes sense since r will be a distance of ¹ from the origin (since r = θ ) when θ is ¹ radians from the x-axis, and r will equal 2¹ when r is 2¹ radians from the origin. Here are some other curves of the form r = a θ.
Notice that when a is negative, the curve is a reflection in the x-axis and the y-axis of the curve when a is positive. Also, the curve of r = 0.5 θ crosses the x-axis at (-¹/2, 0) and (¹,0) since r will be half the length of the value of θ. So when θ is ¹ radians from the origin, r will be a distance of ¹/2 from the origin and when θ is 2¹ radians from the origin (back on the positive x-axis), then r will be a distance of ¹ from the origin.
Let us consider the curves of a few other polar equations.
This is a line. What if we substituted 2θ for each θ in the equation above?
What if we substituted θ/2 for 2θ in this equation?
All of these curves have been very different. Watch a movie of the curves of the following form as a ranges from -10 to 10.
What does the graph of r = a cos θ look like?
When a = 1, and θ ranges from 0 to 2 , we get the following graph.
The graph is a circle with radius a/2 centered at (a/2, 0). What if a = 2?
This graph is also a circle with radius a/2 with a center of (a/2, 0). Now letÕs look at the algebraic curve of r = 2 cos θ – 1.
This is an interesting curve. We can watch a movie of the curve r = a cos θ - 1 to see to see the curve as a ranges from -10 to 10.
What if we were to graph the curve of r = 2 cos (θ – 1)?
It looks like the same size circle as r = 2 cos θ, but shifted up and to the left.
What does the graph of r = a/ θ look like?
First, we observe the graph when a = 1.
This graph is a curve whose maximum y-value is a (when we extend out the x-axis, the curve never gets above 1). Will this same thing occur when a = 2?
Yes. The graph of r = 2/ θ is a curve with a maximum y-value of a. Watch this movie that shows the curve of r = a/ θ as a ranges from -10 to 10.
Let us explore a few more graphs of polar equations.
These two curves are very similar.
Now letÕs examine the graph from above in a movie by changing the value of 10 to n and allowing it to range from -10 to 10.