Buckle up readers, it's Trig time!

Trigonometry can be scary to many students, and in my opinion, a lot

of that is because one of the most confusing concepts in trigonometry

occurs right at the very beginning, in the form of the Unit Circle

and Radians.

Let's start at the beginning. Give yourself a circle with a radius of 1. Now center that circle on the origin of a coordinate plane, so that the line of the circle itself passes through the points (1,0) (0,1) (-1,0) and (0, -1). Got that?

Now, this circle is referred to as the Unit Circle, because the radius is one unit and it is therefore easier for us to do various manipulations and calculations with it.

Now choose any point on the circle (we'll call the coordinates of

that point (x,y)), draw the radius to it (which will still be a

length of 1), and drop a line back perpendicular to one of the axes.

Do that and you'll have a right triangle with the radius as the

hypotenuse, meaning it has a hypotenuse of 1. This becomes very

useful later on for side length calculations involving the

Pythagorean Theorem. But for right now, let's talk angles.

So in basic geometry you should have learned that there are 360 degrees in a circle. If we want to find out the measure of the angle located at the origin in our right

triangle above (what is usually referred to as a central angle), it

makes sense that we could figure it out based on what portion of the

circle the angle is taking up. If the last slice of pie was taking

up 1/8 of the full pan, we'd know that that angle was 1/8 of 360

degrees, or 45 degrees. That's why we bother with the circle in the

first place, rather than simply dropping the right triangle randomly

in the middle of a blank piece of paper. It gives us some context to

work with, so that we can think of our angles as parts of a whole.

So what is a Radian?

Well, a Radian is a unit of angle measurement, like degrees. Only this unit is written in terms of the distance around the circumference. So whereas degree measurements

are based on a portion of 360 (360 being the “whole” benchmark),

Radians are based on a portion of the circumference. They become

very handy for finding the measure of an angle when all you know is

the length of the arc that the angle cuts out of the circle. Say you

didn't know what portion of the pie you had left, but you had a tape

measure and could measure the length of its crust. You'd be left with

what we call an Arc Length, or the distance around the circle that

the angle in question cuts off, and with the power of radians, you

could figure out how much of the pie you had left.

So let's talk radians, shall we?

The circumference of a circle, you'll remember from geometry, is 2πr, or 2 pi times the radius. Well, we know the radius of the Unit Circle, it's 1, remember? That's the definition of a Unit Circle. That means that the circumference
of the Unit Circle (and of our tasty dessert) is simply
2π.

So if we wanted to know the measure of that mystery angle at the center of our last piece of pie, we could certainly find it in terms of what portion of
2π the length of that crust was. If it was half of
2π, we'd have half the circle, and so on. In fact, before we come back to our tasty treat, let's look at some simple examples to give you an idea of what's going on here.

I just mentioned half the circle, so let's look at that first. Now, we

know that that angle would be 180 degrees, since it's a straight line

and half of the full 360, but what portion of the circumference would

it be? What would the length of the curved portion of that

hemisphere be? Well, it'd be half of 2π,

wouldn't it? So it would be 2π/2, or just a single
π.

Let's look at one-quarter of the circle. We know the angle would be 90

degrees, and the circumference would be one-quarter of 2π,

which is 2π/4,

or π/2. If we look at three-quarters of the circle, similarly we

find that the circumference is ¾ of 2π,

which is 6π/4, or 3π/2.

What we've just described are the measures of those angles in radians.

Don't be confused into thinking that one is arc lengths and the other

is angle measurements; radians are a unit for describing angles in

terms of circumference, so it's actually still the same thing. It's

like describing the volume of a container in both cups and liters;

the actual volume hasn't changed, but the metric by which you're

measuring it has.

These are all good values to remember, by the way, as they make the

conversion from degrees to radians much easier. Commit these four to

memory – it will come in handy later, I promise!

90 degrees = π/2 radians

180 degrees = π radians

270 degrees = 3π/2 radians

360 degrees = 2π radians

Incidentally, you'll now see why I harp so much on leaving values as fractions for

as long as possible. Many students like converting to decimals

because the decimal version often looks less intimidating, but a lot

of higher math disciplines (like this one) virtually require you to

be able to work with fractions. Take 3π/2, for example. Plug that

into your calculator, and you'll get 4.712388980385, but if I handed

you that number on a test, you'd have absolutely no way of knowing

that it's actually 3π/2 and therefore a 270-degree angle or

three-quarters of the circle, either of which are much more useful

things to know than that random string of numbers. This is

especially important wherever π or other irrational numbers come in,

since they are often non-repeating, non-terminating decimals which

would have to be rounded off and therefore render you an estimation

rather than an exact answer. 3π/2 is exact; 4.712388980385 is not.

But back to our tasty dessert conundrum. Say you measure the crust on

your remaining piece of pie, and it comes out to be π/4. (Don't ask

me what kind of tape measure would give you that; maybe it was a

hand-me-down from your mathematician grandfather. Problems get

really funky if you try to take the π out of the picture here.) How

do you figure out what the angle is in degrees?

Well, by now you've probably figured out that π/4 is the measure of the

angle in radians. So to convert to degrees, what do you do?

The easiest solution is to set it up as a proportion. You remember

those, right?

(π/4) (part) =
x (part)

2π (whole) 360 (whole)

And solve from there. Cross-multiply:

360π = 2πx

4

Don't worry, the π's cancel each other out. Now do you see why I like

leaving things as fractions?

360/4 = 2x

90 = 2x

x = 45

So that π/4 radians converts to a simple 45 degree angle. And, if

you're feeling industrious, that means that the piece of pie you have

left is 45/360 or 1/8th of the whole pie. Nifty, huh?

Although, to be fair, you could have figured out how much of the pie you had

without ever converting to degrees in the first place. You simply

know that you have:

(π/4)(part) out of 2π (whole), so

(π/4) =
π * 1 = 1 = 1

2π 4 2π 4*2 8

So 1/8th the total pie. The π's cancel each other out again. Starting to

make sense to you? Always leave things as fractions!

It's beneficial to get used to working with radians right off the bat, as

most higher math levels will work almost exclusively in radians

rather than degrees. Why? Well, radians are generally more useful

in wider contexts than degrees, partly because they are based on a

distance measurement rather than a somewhat arbitrary 360. You just

have to learn to take a deep breath and not be intimidated by working

with π...it'll work itself out in the end, I promise. And if you

come up against a problem where the π's don't cancel themselves out,

it's always permissible (and in fact preferable) to leave your answer

in terms of π, where it's easier for the next mathematician to pick

it up and work with it further.

I hope you've enjoyed our mathematical journey, and that radians make a

bit more sense to you now. Stay tuned for more exciting journeys in

the future!