Points, Vectors, and Functions
Bounds on Theta
Polar function plots may leave your head spinning like you got off the Tilt-a-Whirl ride at the amusement park. But using them, it's possible to model that Tilt-a-Whirl ride, and they make some of the sweetest looking plots.
We need to place some bounds on the number of times the plot goes around the origin. These things are complicated, we will still need a calculator to help us plot. Think of it like were placing a polar bear on a leash. You can try to tame it, but it may take you for a walk.
Sometimes wel need to know which bounds on θ give a particular piece of a polar graph. The easiest way to find these bounds is to graph the function on the calculator and play with the bounds until we find the right piece of the graph.
In the example, we need to be careful to find the petal shown in the graph. Other choices for bounds might give us a single petal, but the wrong petal. If we take
π / 4 ≤ θ ≤ 3π/4
we find one petal, but not the one asked for:
Depending on the problem, it might be important to find the specific petal mentioned.
It can also be important to find bounds on θ that trace out the graph exactly once.
The graph of r = cos θ$ for 0 ≤ θ ≤ 2π looks like this:
However, the graph of r = cos θ for 0 ≤ θ ≤ π looks the same.
If asked what values of θ are needed to describe the whole graph of r = cos θ, we would take the narrower bounds:
0 ≤ θ ≤ π.
When asked what bounds on θ give a particular portion of the graph, there are multiple correct answers. We know that r = cos(2θ) for - π/4 ≤ θ ≤ π/4 looks like this:
If instead we take 7π/4 ≤ θ ≤ 9π/4, we find the same piece of the graph:
How do we know if we've found correct bounds for θ? Put them in the calculator and draw the graph. As long as we find the correct portion of the graph, and the calculator traces it only once, then we've found correct bounds for θ.