Pre-Calculus: Convert between Degrees and Radians

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The Trigonometric Functions
Angles and Radian Measure
Converting Between Degrees and Radians Page [1 of 3]
So now we measure angles, when you actually want to measure an angle, in degrees. Of course you know that, for
example, if I go up to here, that’s 90 degrees. This would be 180. This sort of part of our, sort of life, this, which may
be is a little less a part of your life is 270, and then you make a complete circle, full circle, once around, that’s 360
degrees. In fact, you hear this all the time like when people are talking and they say, “Oh, I did a 180.” That means
they looked right around them. Well, where did that come from? Where did the measure of once around being 360
degrees, where did that come from? I mean why 360? Why not 1,000 degrees? Or why not 100 degrees? Why did
we pick 360? Well the answer actually is pretty pathetic. It’s because, a long time ago, people were thinking, “Well,
that sort of once around is like one complete cycle. So let’s just think of about sort of a year. Once complete cycle of
sort of a year, and so how many days? Well they had trouble counting exactly, so back then, it was, yeah, it’s about
360 days in a year to make a complete cycle. So let’s just call that the measure of an angle, once around, 360. That
means half of it, a straight angle, would be 180, and then this thing, a right angle, would be 90 and so forth and so on,
and all of mathematics took off. But what a lame, ridiculous, foolish way of just arbitrarily giving a measure to angles.
Well, it turns out that mathematicians really don’t like that because it doesn’t make a lot of sense. First of all, there are
365 days in a year, and the leap years, what happens then? Do we change the angle measures during leap years?
Well, what do you do? So, anyway, it’s a big ol’ mess. So what we try to do is think about what makes more sense
and what is more useful in actually doing sort of mathematics and trying to compute things and answering questions.
So, to inspire this, let’s think about what happens if we just sort of look at a circle that goes once around. Let’s take a
look at a circle that goes once around and see if you can come up with a better notion of angle measure than just
degrees. So, if you consider going sort of once around, let’s take a circle and for simplicity, let’s suppose the circle
has radius 1. Sort of once around we may think of as sort of like the length of this circle, just once around. Let’s just
take that to be the length. Now the length of a circle is sometimes called the circumference and actually there’s a
formula for circumference it’s two pi times the radius, 2? r. That’s the length of the boundary of a circle of radius, r.
So the length of this thing sort of once around would be 2? x 1, in this case since it has a radius 1. So in fact this
distance here would be 2? . That’s the distance if I start at this point right here and go once around. That’s 2? .
Now that sort of makes sense. I mean that’s sort of a formula that we understand, sort of independent of all of this
stuff. So why don’t we just call once around 2? and create a whole new measure of angles, not degrees anymore,
because that would be 360, but now a whole new measure of angles, where we just make about once around being
called 2? . Well, this new measure is actually called “measure in radians,” – radians, sort of like radius, in a circle.
So I’d call this angle once around, which is 360°, I’d call that 2? radians. So now I’m defining a new way of actually
measuring angles. These are called radians.
So, once around a complete cycle is going to be 2? radians. So what does that mean? It means that 360°=2? , and
you don’t put a degrees symbol, you just write radians, or rad. Well, now in fact, this allows us to do all sorts of
conversions. Because if you know this fact, which is pretty easy to remember once you sort think about how radians
are made, suppose I want to convert one type of measure to the other. Well, if I divide both sides by 360, I see that
one degree would equal, and if I divide this by 360, I would see 2? ÷ 360, which would be
180
?
radian. So, I see 1°
is
180
?
radian. So if I want to convert from degrees to radians, if I had 60 degrees here, I would have to multiply this
side by 60. I multiply this side by 60 which gives me 60°. I multiply this side by 60, I get radians. So, in fact this is a
conversion from degrees to radians. What if I were given radians and I want to convert back to degrees? Well, you
could just go back to this original fact and divide both sides by 2? . If you divide this by 2? , you would see 360 ÷ 2
is 180, so you’d see
180
?
degrees would equal 1 radian.
So, if someone told you “I have so many radians, how many degrees is it?” you just take that number and multiply it
by
180
?
. And you’d get how many degrees. If someone gave you degrees and they wanted to know how many
radians, you just multiply the degrees by
180
?
.
The Trigonometric Functions
Angles and Radian Measure
Converting Between Degrees and Radians Page [2 of 3]
Now, don’t memorize these two formulas, because it’s not worth it. All you have to remember is this thing right here.
That once around is 360 and if you think about it as a circumference of a circle of radius 1, that’s just 2? . So, just
remember this and then you can always go back and forth, depending upon your needs.
So, let me show you some examples. Let’s make some conversions. For example, 360 we already saw, degrees
equals 2? radian. What about 180°? Well, that’s pretty straightforward, because that’s just half of this, so I take half
of this. That would be ? . So, 180° is also ? radians. What about 90°? Well, we can figure that out, too, because
that’s just half of this and so I take half of this and I would see
2
?
. So,
2
?
radians is the exact same angle as 90°, just
a different type of measuring unit.
What about some more exotic ones? Like, for example, what about 30°? Well, how would I do this? Well, here’s
exactly how I would do it, and I hope you would do it, too. The first thing I’d say is I don’t know what that is, so I’d go
back to square one. I know that 360° = 2? rad. And now I have 30°, so if I divide everything through by 360, that
gives me the 1° =
180
?
rad, and so if I multiply everything through by 30, both sides by 30, on this side I’ll see 30° and
if I multiply both sides here by 30, what do I see? I see ? ÷ 6 rad. So, therefore, I’ve made the conversion. 30° =
? ÷ 6 rad. You see how I did that? I just went back to the basics, divided by the 360 here and then multiplied
everything through by 30. What about 45°? What’s the conversion to radian there? Well, let’s see, I know that 1° =
180
?
rad, so if I multiply through by 45, what would I see? I would see 45°, and here I’d see 45? ÷ 180 and that
could be simplified a little bit and what would I see? I would see
4
?
. And if you take 45 and divide it by 180, you get
one-fourth. So
4
?
is the radian way of saying 45°. It’s sort of like a conversion to a different language.
What about 270°? Well, I just multiply everything here by 270 and then what would you see on this side? 270 ÷
180? would be 3? ÷ 2 rad. And, in fact, that sort of makes sense, 90° is
2
?
, so what’s 270? It’s three times this.
So, three times this
3
2
?
. Makes a lot of sense.
Let’s go back the other way now. Suppose I gave you radians. Like, suppose I said to you, “I’m thinking of an angle
and it’s
9
?
rad. What measure is it in degrees? Well, now what do I do? Well, now I don’t use this formula. But,
instead, what I remember is that, since 360° = 2? rad, if I divide both sides by 2? now, I see
180
?
degrees = 1
radian. So, now what do I do? Well, now I multiply both sides by
9r
?
. I don’t want 1 radian, I want
9r
?
. So, I put
9r
?
here, notice the ? ’s cancel, ? , ? on the bottom and then over 9, well 9 cancels with the 180 and I’m just left
with 20°. So, in fact this is 20°. How would you convert 3? rad? Well you would use the exact same procedure.
You’d come back here and multiply everything through by 3? . And if you multiply everything through by 3? , the
? ’s cancel, and I have 3 x 180°. And so what’s that? Well, 3 x 180° is actually 360 + 180. You can add them up and
see what you get. And you get a 4, so you get a 540°. So, in fact, it’s easy to convert from radians to degrees or from
degrees to radians, by just remembering that 360° = 2? radian.
The Trigonometric Functions
Angles and Radian Measure
Converting Between Degrees and Radians Page [3 of 3]
Now, a great question you should be thinking is, “Okay, fine, there’s this radian thing, we can measure things in
radians, but why bother? Why not just stick with the standard 360°, so I can say, Oops, he did a 180, instead of
saying, Oop, he did a ? .” That sort of sounds funny. Well, it turns out that using radian measure actually allows us,
and it actually empowers us to measure all sorts of important things, especially with respect to trigonometric functions.
All the functions that we’ll take a look at, the trigonometric functions, will be measured, the angles in radians. We’ll
see why right up next. Watch the next lecture, you’ll see the power of radians in action.