\( \definecolor{flatred}{RGB}{192, 57, 43} \definecolor{flatblue}{RGB}{41, 128, 185} \definecolor{flatgreen}{RGB}{39, 174, 96} \definecolor{flatpurple}{RGB}{142, 68, 173} \definecolor{flatlightblue}{RGB}{52, 152, 219} \definecolor{flatlightred}{RGB}{231, 76, 60} \definecolor{flatlightgreen}{RGB}{46, 204, 113} \definecolor{flatteal}{RGB}{22, 160, 133} \definecolor{flatpink}{RGB}{243, 104, 224}\)

February 6, 2021

The Unit Circle

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Textbook Reference: Section 5.1 Angles

Behold, the unit circle:

It's pretty unassuming – it's just a circle whose center is the origin \((0,0)\) and whose radius is \(1\). Simple, right? But this circle is going to be the main character in our story throughout the semester. You'll see what I mean as we go on, trust me. You and this unit circle are going to become the best of friends.


Often when you think of angles, you usually think of the corners of polygons. These angles can be anywhere, which is why you usually bring a protractor with you when you need to measure them. They're also pretty static – they just kind of stay where they are, as if they're holding the polygon together and keeping it from collapsing.

In this class, on the other hand, we're often going to be thinking of angles as being dynamic – they can open and close, like a hinge. To keep things from getting too crazy, we'll often put our angles into standard position, which means that one side of the angle (the initial side) is locked down on the x-axis, and the other side (the terminal side) is free to move around. We mark the initial side as \(0^\circ\), and as the angle opens counterclockwise, the angle gets bigger, and we can mark the number of degrees at the corresponding point on the circle.

Drag the blue point to change the angle.

Important Angles

Some angles on the unit circle are so common that they're worth marking off: the multiples of \(30^\circ\) and \(45^\circ\).

Make sure you familiarize yourself with these angles. Ideally, you want to get to a point where you can visualize where they are at any time. If someone walks up to you on the street and says, "Hey, where's \(120^\circ\) on the unit circle?", you should be able to point upward and a little to the left. (You might look at them a little funny afterward, but your friend the unit circle will be pleased.)

Coterminal Angles

Now, you might be thinking:

"Do angles have to be between \(0^\circ\) and \(360^\circ\)?"

Okay, well maybe you weren't thinking that, but I bet you are now. It seems pretty limiting to say that angles are only allowed to be within that certain range. Well, you know what? Let's just see what happens if we allow angles to be bigger than \(360^\circ\):

They wrap back around on themselves.

You could imagine the angle revolving around and around forever, and it could get as big as you want.

Okay, what about negative angles?

If counterclockwise movement is positive, then it should be no surprise that clockwise movement is negative:

But that's cheating, you might think! Is \(450^\circ\) really different from \(90^\circ\)?

Well … yes and no.

On one hand, if this were a circular track, and you ran from \(0^\circ\) to \(450^\circ\), you'd feel a lot more tired than if you just ran from \(0^\circ\) to \(90^\circ\). But, you have to admit that the angles end in the same place. We have a name for that – we say that \(90^\circ\) and \(450^\circ\) are coterminal angles.

Starting at any angle, you can always go around a full circle (\(360^\circ\)) in either direction, and when you stop, you'll be at an angle that's coterminal to the one you were at before.

In other words, an angle can be ANY real number.

This will be really useful.


In mathematics, you always have to be asking why something is true. Sometimes doing that leads you to some pretty amazing insights, especially when you realize that some things you always took for granted don't always have to be set in stone.

You probably didn't even blink when I said that there were \(360^\circ\) in a full circle. But did you ever think about why?

Well, one reason why is that it's actually what the Babylonians used. But when it really comes down to it, \(360^\circ\) is just a convenient-yet-arbitrary number for the degrees in a circle.

We could have divided a circle into, say, \(400\) pieces instead. Then a right angle would have been \(100\) pieces, and so on. (France actually tried to do this during the French Revolution, when they turned everything into the decimal system. They called these divisions of the circle "grads." Unlike the meter and the kilogram, the grad didn't catch on.)

To really understand circles, we need to come up with a measure that doesn't depend on some arbitrary number. We need a more natural measure.

And you might have thought of one already, something that brings back memories of drawing little pizzas in elementary school:


Whether we think of a right angle as being \(90\) degrees or \(100\) "grads", it's still \(1/4\) of a circle. Thinking of our angles actively again, we can imagine that the angle turned \(1/4\) of the way around the circle. We can use the Greek letter \(\tau\) (tau) to stand for a "turn" and write it as follows:\[90^\circ=\tfrac14\tau\]

This arguably makes it even easier to work with angles than when we had degrees! If you want to go \(5/12\) of the way around the circle, then your angle is just \(\tfrac{5}{12}\tau\): five-twelfths of a turn. Simple and clean.


We're definitely closer to having a natural unit of angle measurement, but we haven't really linked it to the circle yet. We could just as easily apply our fraction concept to a square, or a hexagon:

So what makes the circle special?

As the angle opens, the arc of the circle grows proportionally with it. With any other shape, the amount marked off on the perimeter will grow faster or slower depending on how close the shape is to the origin, but with a circle, double the angle always means double the arc.

You could imagine wrapping a string around the circle, starting at the initial side and ending at the terminal side, and then measuring how long it is. The problem is that would depend on how big the circle's radius is. But we can fix that too, by always basing our measurements on the simplest possible circle: the unit circle, with a radius of \(1\).

If we wrapped our string all the way around the unit circle, you'd notice it would be about \(6.28\) units long. That number is actually a really important number in mathematics – and, in fact, it's actually called tau!\[\tau=6.2831853071795864769228\ldots\]

Tau is the circumference of a unit circle, and it's one of those deep and fundamental constants that "makes math tick." Wherever there's a circle, you can be sure that \(\tau\) is lurking somewhere behind the scenes.

This means that we can still use our measurements we said before, though we do it in a funny way. We say that a string 1 unit long would be one radian, since it's exactly as long as the radius (which we also said was 1 unit long on the unit circle). That means that \(1\) radian is about \(57.3^\circ\), which is a little inconvenient.

By Lucas Vieira - Own work, Public Domain,

But we almost never use whole-number measurements for radians – instead we use fractions of \(\tau\). So, the whole circle is \(\tau\) radians, halfway around the circle is \(\tfrac12\tau\) radians, a right angle is \(\tfrac14\tau\) radians, an eighth of a circle is \(\tfrac18\tau\), and so on. If you keep things in terms of \(\tau\), you get to keep those intuitive fractions.

(Click the figure to enlarge it.)

There's just one catch.

The unfortunate thing about \(\tau\) is that, due to how history has gone, it has a much more famous little brother: \(\pi\) (pi), which is half of \(\tau\). Or, put another way, \(\tau=2\pi\).

So, the way mathematicians usually measure radians is in terms of \(\pi\) instead of \(\tau\). A full circle has a total of \(2\pi\) radians, since (using the formula \(C=2\pi r\) that you learned back in elementary school) the circumference of the unit circle is \(2\pi\). That means that halfway around the circle is \(\pi\), a right angle is \(\tfrac12\pi\) (or often written \(\tfrac\pi2\)), an eighth of a circle is \(\tfrac14\pi\) (or \(\tfrac\pi4\)), and so on. There's a lot of multiplying by \(2\) and dividing by \(2\), and it can get very confusing if you're not careful.

(Click the figure to enlarge it.)

It would be much nicer if we could just always do things in terms of \(\tau\). And there are some people who are proposing that we do exactly that. However, until that day comes, the rest of the world still uses \(\pi\). But you'll be okay if you just remember one thing:

Radians are angles measured as fractions of \(2\pi\).

If you're looking at \(\tfrac18\) of a circle, that's \(\tfrac18\) of \(2\pi\):\[\dfrac18\cdot 2\pi=\dfrac{2\pi}{8}=\dfrac\pi4\]

In other words, as long as you remember the intuition that made radians work in the first place, you'll always have something to fall back on.

Preview Activity 1

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

  1. List out as many divisors (that is, factors) of \(360\) as you can think of. Why would this have made it convenient for the Babylonians to split up the circle into \(360^\circ\)?
  2. Find at least three angles that are coterminal to \(20^\circ\).
  3. Remember that the four quadrants are laid out as follows.
    In which quadrants would each of the following angles be found? (It might help you to sketch these out.)
    1. \(110^\circ\)
    2. \(215^\circ\)
    3. \(-250^\circ\)
    4. \(950^\circ\)
  4. Each of the following is the radian measure of an angle. Multiply each one out, writing your answers in lowest terms. Then try to figure out the corresponding angle in degrees based on visualizing it on the unit circle.
    1. \(\tfrac{1}{2}(2\pi)\)
    2. \(\tfrac{2}{3}(2\pi)\)
    3. \(\tfrac{3}{4}(2\pi)\)
    4. \(\tfrac{5}{6}(2\pi)\)
    5. \(\tfrac{3}{8}(2\pi)\)
    6. \(\tfrac{5}{12}(2\pi)\)
  5. Answer AT LEAST one of the following questions:
    1. What was something you found interesting about this reading?
    2. What was an "a-ha" moment you had while doing this reading?
    3. What was the muddiest point of this reading for you?
    4. What question(s) do you have about anything you've read?


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