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Mathematical Journeys: What's a Radian?

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 .


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  the length of that crust was. If it was half of , 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 ,
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 ,
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 ,
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!

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