Textbook Reference: Section 8.3 Polar Coordinates
There are lots of ways to give directions.
You might be the kind of person who knows the names of all the roads and how far to go in which direction. Or you might be like me and do everything based on counting how many traffic lights it is to the next turn and giving a bunch of landmarks along the way (usually restaurants).
The point is, the way you give directions depends on who you're giving them to and what purpose you're using them for.
The same applies in math. You know that one way to describe the location of a point in the plane is to say how far left or right it is from the origin (the \(x\)coordinate) and how far up or down it is (the \(y\)coordinate). (These are called rectangular coordinates.) Now we're going to look at another useful way to "give directions" to a point in the plane.
Polar coordinates
The idea behind polar coordinates is that we specify a point using two quantities:
 \(r\), the displacement from the origin
 \(\theta\), the angle from the positive \(x\)axis (which often gets called the polar axis)
Essentially, we're specifying how many steps to take and which direction to face.
Most textbooks write polar coordinates using an ordered pair \((r,\theta)\). This can be a bit confusing because it's not immediately obvious without context whether \((3,\pi)\) means "move \(3\) units to the right and \(\pi\) units upward" or "move \(3\) units in the direction of the angle \(\pi\).
So, we're going to take a page from electrical engineering's book and write it in a more evocative format: \[(r\,\angle\,\theta)\] You can read this as "move \(r\) steps in the direction of \(\theta\)." So, \((3\,\angle\,\pi)\) is the point \(3\) units from the origin in the direction of \(\pi\), which brings us to the point \((3,0)\).
We often plot polar coordinates on a special grid:
Each circle contains all the points with the same \(r\), and each line through the origin contains all the points with the same \(\theta\).
Converting between rectangular and polar coordinates
We can visualize the relationship between \(x\), \(y\), \(r\), and \(\theta\) with the following picture:
Wait a minute! We have a right triangle, and a marked angle ... that means we can use circular functions to describe the relationship! Basic SOHCAHTOA gives us: \[\sin\theta=\dfrac yr\quad\cos\theta=\dfrac xr\quad\tan\theta=\dfrac yx\] If we solve the first two equations for \(x\) and \(y\), we get: \[\boxed{x=r\cos\theta}\qquad \boxed{y=r\sin\theta}\] Using these equations, we can take any polar coordinates \((r\,\angle\,\theta)\) and find the corresponding set of rectangular coordinates \((x,y)\).
For example, if our polar coordinates are \((3\,\angle\,\pi)\), then: \[ \begin{align*} x &= 3\cos\pi=3\\ y &= 3\sin\pi=0 \end{align*} \] Thus the rectangular coordinates are \((3,0)\), just as we'd expect.
The Pythagorean Theorem and the tangent function let us go in the other direction: \[\boxed{r^2=x^2+y^2}\qquad\boxed{\tan\theta=\dfrac yx}\] Using these equations, we can take any rectangualr coordinates \((x,y)\) and find a corresponding set of polar coordinates \((r\,\angle\,\theta)\).
By the way, did you notice that I said "the" the first time, but "a" the second time?
That's because, unlike in rectangular coordinates where there's only one way to describe a point, there are many ways (infinitely many ways in fact) to describe a point in polar coordinates. This should be no surprise, considering that the circular functions are involved! You'll get a feel for this in the activity.
Preview Activity 25
Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

Convert from polar coordinates to rectangular coordinates:
 \((5\,\angle\,\frac{\pi}2)\)
 \((4\,\angle\,\frac{4\pi}3)\)
 \((4\,\angle\,\frac{3\pi}4)\)

For each of the following rectangular coordinates, find two different ways to represent them in polar coordinates — one with a positive value of \(\theta\), and another with a negative value of \(\theta\).
 \((0,2)\)
 \((4\sqrt{2},4\sqrt{2})\)
 \((3,3\sqrt{3})\)

Notice that with the way we've set things up, there's technically nothing stopping us from letting \(r\) be negative. Let's see if we can make sense of what that would mean.
 Use the formulas given earlier to rewrite \((3\,\angle\,\frac{3\pi}{2})\) in rectangular coordinates.
 Find another way to write this same point in polar coordinates, but with a positive \(r\).
 A negative value of \(r\) can be thought of as "walking backwards." Explain what this means more specifically with the given point from parts a and b.

Answer AT LEAST one of the following questions:
 What was something you found interesting about this activity?
 What was an "aha" moment you had while doing this activity?
 What was the muddiest point of this activity for you?
 What question(s) do you have about anything you've done?
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