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# Intro to radians

Sal explains the definition and motivation for radians and the relationship between radians and degrees. Created by Sal Khan.

Video transcript

You are by now probably
used to the idea of measuring angles in degrees. We use it in everyday language. We've done some examples
on this playlist where if you had an angle
like that, you might call that a
30-degree angle. If you have an
angle like this, you could call that a
90-degree angle. And we'd often use this
symbol, just like that. If you were to go
180 degrees, you'd essentially form
a straight line. Let me make these proper angles. If you go 360 degrees,
you have essentially done one full rotation. And if you watch figure
skating on the Olympics, and someone does a
rotation, they'll say, oh, they did a 360. Or especially in some
skateboarding competitions, and things like that. But the one thing to realize--
and it might not be obvious right from the get-go-- but
this whole notion of degrees, this is a
human-constructed system. This is not the only way
that you can measure angles. And if you think
about it, you'll say, well, why do we call a
full rotation 360 degrees? And there's some
possible theories. And I encourage you
to think about them. Why does 360 degrees show up in
our culture as a full rotation? Well, there's a couple
of theories there. One is ancient calendars. And even our calendar
is close to this, but ancient calendars were
based on 360 days in a year. Some ancient
astronomers observed that things seemed to move
1/360 of the sky per day. Another theory is the
ancient Babylonians liked equilateral
triangles a lot. And they had a base
60 number system. So they had 60 symbols. We only have 10. We have a base 10. They had 60. So in our system, we like
to divide things into 10. They probably liked to
divide things into 60. So if you had a circle,
and you divided it into 6 equilateral
triangles, and each of those equilateral triangles
you divided into 60 sections, because you have a
base 60 number system, then you might end
up with 360 degrees. What I want to think
about in this video is an alternate way
of measuring angles. And that alternate way--
even though it might not seem as intuitive to you from
the get-go-- in some ways is much more mathematically
pure than degrees. It's not based on these cultural
artifacts of base 60 number systems or
astronomical patterns. To some degree, an
alien on another planet would not use degrees,
especially if the degrees are motivated by these
astronomical phenomena. But they might use what we're
going to define as a radian. There's a certain degree
of purity here-- radians. So let's just cut to the chase
and define what a radian is. So let me draw a
circle here, my best attempt at drawing a circle. Not bad. And let me draw the
center of the circle. And now let me draw this radius. And let's say that this
radius-- and you might already notice the word radius is very
close to the word radians. And that's not a coincidence. So let's say that this circle
has a radius of length r. Now let's construct an angle. I'll call that angle theta. So let's construct
an angle theta. So let's call this angle
right over here theta. And let's just say, for
the sake of argument, that this angle is just
the exact right measure so that if you look at the
arc that subtends this angle-- and that seems like
a very fancy word. But let me draw the angle. So if you were to draw
the angle-- so if you look at the arc that subtends
the angle, that's a fancy word. That's really just
talking about the arc along the circle that intersects
the two sides of the angles. So this arc right over here
subtends the angle theta. So let me write that down. Subtends this arc,
subtends angle theta. Let's say theta is the exact
right size so that this arc is also the same length as
the radius of the circle. So this arc is also of length r. So given that, if you were
defining a new type of angle measurement, and you
wanted to call it a radian, which is very close
to a radius, how many radians would you define
this angle to be? Well, the most obvious
one, if you kind of view a radian as another
way of saying radiuses, or I guess radii. Well, you say, look,
this is subtended by an arc of one radius. So why don't we call this right
over here one radian, which is exactly how a
radian is defined. When you have a circle, and you
have an angle of one radian, the arc that subtends it
is exactly one radius long. Which you can imagine might
be a little bit useful as we start to interpret more
and more types of circles. When you give a
degrees, you really have to do a little
bit of math and think about the circumference
and all of that to think about how many radiuses
are subtending that angle. Here, the angle in radians
tells you exactly the arc length that is subtending the angles. So let's do a couple of
thought experiments here. So given that, what would
be the angle in radians if we were to go-- so let
me draw another circle here. So that's the center, and
we'll start right over there. So what would happen if I
had an angle-- if I wanted to measure in radians, what
angle would this be in radians? And you could almost
think of it as radiuses. So what would that angle be? Going one full
revolution in degrees, that would be 360 degrees. Based on this definition,
what would this be in radians? Well, let's think about the
arc that subtends this angle. The arc that subtends this angle
is the entire circumference of this circle. Well, what's the
circumference of a circle in terms of radiuses? So if this has length r,
if the radius is length r, what's the circumference of
the circle in terms of r? Well, we know that. That's going to be 2 pi r. So going back to this
angle, the length of the arc that subtends this
angle is how many radiuses? Well, it's 2 pi radiuses. It's 2 pi times r. So this angle right
over here, I'll call this a different-- well,
let's call this angle x. x in this case is going
to be 2 pi radians. And it is subtended by an
arc length of 2 pi radiuses. If the radius was
one unit, then this would be 2 pi times
1, 2 pi radiuses. So given that,
let's start to think about how we can convert
between radians and degrees, and vice versa. If I were to have-- and we
can just follow up over here. If we do one full
revolution-- that is, 2 pi radians-- how many degrees
is this going to be equal to? Well, we already know this. A full revolution in
degrees is 360 degrees. Well, I could either write
out the word degrees, or I can use this little
degree notation there. Actually, let me write
out the word degrees. It might make things a
little bit clearer that we're kind of using units
in both cases. Now, if we wanted to
simplify this a little bit, we could divide both sides by 2. In which case, on
the left-hand side, we would get pi radians would
be equal to how many degrees? Well, it would be
equal to 180 degrees. And I could write it that way,
or I could write it that way. And you see over here,
this is 180 degrees. And you also see if you were
to draw a circle around here, we've gone halfway
around the circle. So the arc length, or the
arc that subtends the angle, is half the circumference. Half the circumference
is pi radiuses. So we call this pi radians. Pi radians is 180 degrees. And from this, we can
come up with conversions. So one radian would
be how many degrees? Well, to do that,
we would just have to divide both sides by pi. And on the left-hand
side, you'd be left with 1-- I'll just
write it singular now. 1 radian is equal to-- I'm
just dividing both sides. Let me make it clear what I'm
doing here, just to show you this isn't some voodoo. So I'm just dividing
both sides by pi here. On the left-hand side,
you're left with 1. And on the right-hand
side, you're left with 180/pi degrees. So 1 radian is equal to
180/pi degrees, which is starting to make
it an interesting way to convert them. Let's think about
it the other way. If I were to have 1 degree,
how many radians is that? Well, let's start
off with-- let me rewrite this thing over here. We said pi radians is
equal to 180 degrees. So now we want to
think about 1 degree. So let's solve for 1 degree. 1 degree, we can divide
both sides by 180. We are left with pi/180
radians is equal to 1 degree. So pi/180 radians is
equal to 1 degree. This might seem
confusing and daunting. And it was for me
the first time I was exposed to this,
especially because we're not exposed to this in
our everyday life. But what we're going to see
over the next few examples is that as long as we keep
in mind this whole idea that 2 pi radians is
equal to 360 degrees, or that pi radians is equal to
180 degrees, which is the two things that I do
keep in my mind. We can always re-derive
these two things. You might say, hey,
how do I remember if it's pi/180 or 180/pi
to convert the two things? Well, just remember, which
is hopefully intuitive, that 2 pi radians is
equal to 360 degrees. And we'll work through a bunch
of examples in the next video, to just make sure
that we're used to converting one
way or the other.