>> This is your math gal,
Julie Harland.
Please visit my web site
at yourmathgal.com where all
of my videos are organized
by topic.
All right.
In this video we're going
to define what a radian
measure of an angle is.
So first of all,
the most common way
of measuring angles is
in degrees.
Most people have heard
of that.
But another measure
for angles is called
radian measure.
And this measure is used
in trigonometry, calculus
and science.
So a radian is simply a
real number.
It's not, like, 10-degrees.
It's like, well,
this angle measures 5,
or this angle measures
two-thirds, etceteras.
So you could actually say
in angle measures, 2 radians,
or you can just put 2--
this is an abbreviation just
putting RAD,
or you could simply write
that the angle's measure is 2.
You don't have
to write the word radians.
So it's a little bit strange.
You've got a angle,
but you're just saying it's
a number.
So if you don't see the
degrees symbol,
the angle is not in degrees.
So it's very important,
if you're trying
to show a measure of angle
and you mean 20-degrees,
you need to put the degrees
symbol because if you don't it
means 20 radians
and you'll find
out that's totally a
different angle.
So what is a radian?
How do we measure an angle
in radians?
It's kind of a mouthful.
It takes a little thinking
to get used to the idea.
So here I'm going
to explain how we're going
to do it.
You're going to draw a circle
of radius R.
And then you're going
to draw an angle.
I'll call it theta.
So the vertex of that angle is
at the center of the circle.
And you're going
to use an arrow
to show the direction the
angle is going.
And we're going
to have it go
counter-clockwise
to be a positive angle.
The angle cuts off an arc
on the circle and if
that arc has length S--
I don't know why,
but they use the letter S
for the length of an arc--
and the angle is
opening counter-clockwise.
Like I said,
we'll do the radian measure
of theta is defined
as theta equals S
over R. Okay, so let's see.
How-- that's quite a mouthful
so you sort of have to think
about that, maybe read it more
than once.
Let's try it.
So what does theta do?
You're going to draw a circle.
So that's step one.
So I have a circle here.
And it said to do it
of radius R.
So from the center
to any point
on the circle is R.
That might be 2-feet.
It might be 5-inches,
etceteras.
It's just some number.
All right, then what?
Draw an angle so the vertex is
at the center of the circle.
All right.
So let's do that.
So now we're going
to draw an angle.
So I'm going to take--
and how about I just start it
right here?
Whoops. I have to undo that.
And let's say I go
up to here, okay.
And I'm going to call
that theta.
So this is still a length
of R, right,
because from the center
to any point is a length
of R. All right.
So we drawn an angle.
So the vertex is the center
of the circle, using an arrow
to show the direction.
Okay, I want to show
that it's going--
I didn't really show the
little arrow first.
It's going this way, right?
So we'll say that's theta.
The angle cuts off an arc
on the circle.
All right.
So what I'm talking
about here is this right here.
And I'm going to show
that with a arrow as well,
for now-- even though it's
just a length.
And I'm going to call
that S. All right.
So here's the picture.
So hopefully you're okay
so far.
If that arc has length S,
here it is, right,
and the angle is opening
counter-clockwise,
which it is,
the radian measure
of theta is equal to S over R.
So we take this angle measure.
We say we take the length
of this blue line here,
the dark blue, and divide it
by the radius
and that is the measure
of theta.
That is the definition
of the radian measure
of theta.
It's like,
you get out the dictionary
and that's what it says.
All right.
So we have this definition.
If we have this picture then
theta is S over R. I'm going
to go ahead and just get rid
of this R here just
so we can see basically what
it looks like.
That's our central angle.
Central angle just means
that the vertex is
at the center for theta.
All right.
So it's the length
of this arc- arc, right,
like you could measure
that somehow.
And then you could divide it
by the radius.
That's the angle measure
in radians for theta.
All right.
Now, what about if the circle
has a radius of 1?
Then we would have theta
equals S over R, right,
since R is 1,
or we could just say theta
equals S. So some people
like to think of it this way
for the-- how to think
about what radians
really means.
Say, well,
if you have a measure--
if you have a circle that's
always just a radius of 1,
then however long
that arc length is,
that's the same as the measure
of this angle.
So it's kind
of a interesting thing.
The measure
of theta is the same
as the length of S. Okay,
but that only works
in a unit circle so that
if the circle has radius
of one.
Okay, so you still may be
wondering, okay, let's,
you know, what more is there
to this?
I'm not really getting any
real numbers.
I know I'm not getting a
number sense yet.
So let's go ahead and think
about what 1 radian is first.
All right.
So let's say we want
to know what 1 radian is.
Basically, remember that S--
I'm sorry, theta equals S
over R. So the question is,
how do I make that equal to 1?
It means the radius
and the arc length have
to be the same number
to have it be 1.
So that's how big a radian is.
You can go look
at a picture here.
Imagine if you had a string
and you measured this radius
and then you took
that string along the circle
and wherever it ended,
that means this arc length is
the same as the radius.
Then I would get theta.
Here I would have R over R,
which is equal to 1.
Same thing over here.
If I have on--
this is a unit circle.
A unit circle just means the
radius ones.
>> So here we have a unit
circle, which means we have a
radius one 1.
If you're looking at that,
it's the same thing.
You would get 1 over 1,
which is 1.
So some people always think
of, well, I want
to have any circle at all.
I just think of S over R
for people who like to deal
with unit circle,
it's just whatever the length
of the arc length is
because you're always just
putting it over 1.
So those are the two examples.
So if you look
at this picture here.
No, it looks like it's more
than 45-degrees, which it is.
And later, you know,
you're going to find
out more exactly, or closer,
or approximately how many
degrees this is.
But just notice
that 1 radian is a lot bigger
than 1 degree.
One degree is tiny, right,
because, I mean,
you can't even draw it it's
so tiny on this piece
of paper I have anyway.
So it's important
that you say,
this is 1 radian.
If you put 1 degree you're way
off because 1 degree is, like,
so tiny it's like, you know,
tinier than that--
very, very small.
So be careful.
If you want to write something
in radians,
you do not use the
degrees symbol.
So this has just been an
introduction to the definition
of radian measures
so that we have an
understanding
of just what it means.
And now you want
to see my video showing how
to convert between radians
and degrees.
So we'll be doing
that especially
for the special angles
in degrees such as 30-degrees,
45-degrees, 90-degrees,
etceteras.
I know there's lots and lots
and lots of special angles
and we're going
to see how would we write
these in radian measure.
This is your math gal,
Julie Harland.
Please visit my web site
at yourmathgal.com where all
of my videos are organized
by topic.