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We're ready to practice using polar coordinates.
Polar functions are the perfect opportunity to practice. We can think of polar functions like a bear itself—a polar bear in this case. We need to wrangle the polar bear for our confrontation with the calculus bear.
First, a few basics. A polar function is a function described by an equation of the form
r = f (θ).
The values of θ to be used will often be specified by an inequality α ≤ θ ≤ β.
The polar function r = f (θ) can also be described by the parametric equations
It's time to get friendly with our graphing calculators. First, we need to understand how to graph polar functions by hand.
This is one of the many instances in calculus where it's helpful to use a calculator as a tool, but it's important to know what it's output means.
When checking the graph on the graphing calculator it can be helpful to spot-check points, especially at the boundaries of θ and simple angles like 0, , and to make sure r has the right values. Then look at what r is doing between those points to see if it makes sense.
Calculator Tip: If the calculator graph looks like jagged lines instead of looking curvy, try making the θ step size smaller (this may be called Δ θ on the calculator). This affects how carefully the calculator draws the graph.
The best way to graph polar functions is by using a graphing calculator or a computer program. We can wave our hands and pull a rabbit out of a hat. That's because there aren't as many rules about graphing polar functions. Those few rules that we do have can be much more complex.
With a rectangular function
y = f (x)
there are certain rules about how the function stretches or translates if we look at variations such as:
c + f (x)
f (c + x)
where c is a constant.
We have rules like this when dealing with polar functions too, but not as many.
As far as nice rules for graphing go, that's all we get.
We can verify that the function r = f (cθ) is weird by trying different values in the graphing calculator.
The function r = c + f(θ) is also weird. Adding a constant can change whether your r values are positive or negative, which can totally change the shape of the graph. It may also change the bounds we need for θ if we want to find the whole graph.
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 we were placing a polar bear on a leash. You can try to tame it, but it may take you for a walk.
Sometimes we 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
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 looks like this:
If instead we take , 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.
We need one more tool to solve for intersections of polar functions. We need to know how to solve a system of two equations.
We can find where two rectangular functions y = f (x) and y = g(x) intersect by setting them equal to each other and solving the equation f (x) = g(x) for x. This is the place (or places) where both equations are true at the same time.
Like the intersection of two rectangular functions, we can find most of the places two polar equations r = f (θ) and r = g(θ) intersect by setting them equal to each other and solving the equation f (θ) = g(θ) for θ.
It's useful to graph the functions before finding where they intersect. We want to know how many intersection points we need to find. Also, setting equations equal to each other and solving might not catch an intersection point at the origin.
This sort of thing can happen because with polar coordinates there are infinitely many ways to write any point. Pick your favorite and stop there.
When finding where two polar graphs intersect, graph the functions first. Then look at how many intersection points there are and which quadrants they're in. Then we'll know how many points we need to find and roughly where they are.
We can estimate the intersection points from the graph. The graphs of r = cos θ and r = sin θ do look like they hit each other around θ = π⁄4. However, we still need to set the functions equal to each other and solve for θ.