Radians and degrees

What a radian is. Converting radians to degrees and vice versa.

Welcome to the presentation
on radians and degrees.

So you all are probably already
reasonably familiar with

the concept of degrees.

I think in our angles models
we actually drill you

through a bunch of problems.

You're probably familiar that
a right angle is 90 degrees.

Or half a right angle
-- 45 degrees.

And you're also probably
familiar with the concept that

in a circle -- and that's my
best adept at a circle -- in a

that's my best attempt of a circle

circle, there are 360 degrees.

So today I'm going to introduce
you to another measure or unit

for angles and this
is called a radian.

So what is a radian?

So I'll start with the
definition and I think this

might give you a little
intuition for why it's

even called radian.

Let me use this circle tool and
actually draw a nice circle.

I'm still using the radian
tool, the circle tool.

OK.

This is a radius of length r.

A radian is the angle
that subtends an arc.

And all subtend means is if
this is angle, and this is

the arc, this angle subtends
this arc and this arc

subtends this angle.

So a radian -- one radian -- is
the angle that subtends an arc

that's the length
of the radius.

So the length of
this is also r.

And this angle is one radian.

i think that's messy.

Let me draw a bigger circle.

Here you go.

And I'm going to do this
because I was wondering

why they do radians.

We all know degrees.

But actually when you think
about it it actually makes a

reasonable amount of sense.

So let me use the
line tool now.

And let's say that this radius
is a length r and that this arc

right here is also length r.

Then this angle, what's called
theta, is equal to one radian.

And now it makes sense that
they call it a radian.

It's kind of like a radius.

So let me ask a question:
how many radians are

there in a circle?

Well, if this is r, what is
the whole circumference

of a circle?

It's 2 pi r, right?

You know that from the
basic geometry module.

So if the radian is the angle
that sub tends an arc of r,

then the angle that subtends an
arc of 2 pi r is 2 pi radians.

So this angle is 2 pi radians.

If you're still confused,
think of it this way.

An angle of 2 pi radians going
all the way around subtends

an arc of 2 pi radiuses.

Or radii.

I don't know how to say
the plural of radius.

Maybe it's radians.

And I don't know.

So why am I going through all
of this mess and confusing you?

I just want to one, give you an
intuition for why it's called

a radian and kind of how
it relates to a circle.

And then given that there 2 pi
radians in a circle, we can now

figure out a relationship
between radians and degrees.

Let me delete this.

So we said in a circle,
there are 2 pi radians.

And how many degrees
are there in a circle?

If we went around a whole
circle how many degrees?

Well that's equal
to 360 degrees.

So there.

We have an equation that sets
up a conversion between

radians and degrees.

So one radian is equal to
360 over 2 pi degrees.

I just divided both
sides by 2 pi.

Which equals 180
over pi degrees.

Similarly, we could have
done the other way.

We could have divided both
sides by 360 and we could have

said 1 degree -- I'm just going
to divide both sides but

360 and I'm flipping it.

1 degree is equal to 2
pi over 360 radians.

Which equals pi
over 180 radians.

So then we have a conversion:
1 radian equals 180 over pi

degrees and 1 degree equals
pi over 180 radians.

Amd if you ever forget
these, it doesn't hurt

to to memorize this.

But if you ever forget it,
I always go back to this.

That 2 pi radians is
equal to 360 degrees.

Or another way that actually
makes the algebra a little

simpler is if you just
think of a half circle.

A half circle -- this angle
-- is a 180 degrees, right?

That's a degree sign.

I could also write degrees out.

And that's also equal
to pi radians.

So pi radians equal 180 degrees
and we can get to see the math.

1 radian equals 180 over pi
degrees or 1 degree is equal

to pi over 180 radians.

So let's do a couple of
problems were you'll get

the intuition for this.

If I asked you 45 degrees --
to convert that into radians.

Well, we know that 1 degree
os pi over 180 radians.

So 45 degrees is equal to 45
times pi over 180 radians.

And let's see, 45
divided by 180.

45 goes into 180 four times so
this equals pi over 4 radians.

45 degrees is equal to
pi over 4 radians.

And just keep in mind, these
are just two different units

or two different ways
of measuring angles.

And the reason why I do this is
this is actually the

mathematical standard for
measuring angles, although most

of us are more familiar with
degrees just from

everyday life.

Let's do a couple
of other examples.

Just always remember:
this 1 radian equals

180 over pi degrees.

1 degree equals pi
over 180 radians.

If you ever get confused,
just write this out.

this is what I do because I
always forget whether it's

pi over 180 or 180 over pi.

I just remember pi radians
is equal to 180 degrees.

Let's do another one.

So if I were to say pi
over 2 radians equals

how many degrees?

Well I already forgot what I
had just written so I just

remind myself that pi radians
is equal to 180 degrees.

Oh, my wife just got home, so
I'm just going to have to leave

the presentation like that
and I will continue it later.

Actually, let me just finish
this problem and then I'll

go attend to my wife.

But we know that pi radians is
equal to 180 degrees, right?

So one radian is equal to 180
over -- that's one radian -- is

equal to 180 over pi degrees.

I just figure out the
formula again because

I always forget it.

So let's go back here.

So pi over 2 radians is
equal to pi over 2 times

180 over pi degrees.

And that equals 90 degrees.

I'll do one more example.

Let's say 30 degrees.

Once again, I forgot the
formula so I just remember

that pi radians is
equal to 180 degrees.

So 1 degree is equal to
pi over 180 radians.

So 30 degrees is equal to 30
times pi over 180 radians

which equals -- 30 goes
into 180 six times.

That equals pi over 6 radians.

Hopefully you have a sense of
how to convert between degrees

and radians now and even why
it's called a radian because

it's very closely related to
a radius and you'll feel

comfortable when someone asks
you to, I don't know, deal with

radians as opposed to degrees.

I'll see you in the
next presentation.