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Welcome to the presentation
on radians and degrees.

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So you all are probably already
reasonably familiar with

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the concept of degrees.

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I think in our angles models
we actually drill you

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through a bunch of problems.

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

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Or half a right angle
-- 45 degrees.

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And you're also probably
familiar with the concept that

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

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that's my best attempt of a circle

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circle, there are 360 degrees.

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

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for angles and this
is called a radian.

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So what is a radian?

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

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might give you a little
intuition for why it's

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even called radian.

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

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

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OK.

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This is a radius of length r.

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A radian is the angle
that subtends an arc.

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

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the arc, this angle subtends
this arc and this arc

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subtends this angle.

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

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that's the length
of the radius.

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So the length of
this is also r.

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And this angle is one radian.

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i think that's messy.

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Let me draw a bigger circle.

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Here you go.

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

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why they do radians.

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We all know degrees.

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But actually when you think
about it it actually makes a

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reasonable amount of sense.

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So let me use the
line tool now.

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

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right here is also length r.

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

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

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It's kind of like a radius.

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So let me ask a question:
how many radians are

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there in a circle?

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

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of a circle?

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It's 2 pi r, right?

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You know that from the
basic geometry module.

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

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

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So this angle is 2 pi radians.

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

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

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an arc of 2 pi radiuses.

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Or radii.

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

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Maybe it's radians.

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And I don't know.

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

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

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

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

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figure out a relationship
between radians and degrees.

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Let me delete this.

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

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And how many degrees
are there in a circle?

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If we went around a whole
circle how many degrees?

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Well that's equal
to 360 degrees.

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So there.

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We have an equation that sets
up a conversion between

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radians and degrees.

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

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I just divided both
sides by 2 pi.

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Which equals 180
over pi degrees.

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Similarly, we could have
done the other way.

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

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

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360 and I'm flipping it.

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1 degree is equal to 2
pi over 360 radians.

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Which equals pi
over 180 radians.

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

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degrees and 1 degree equals
pi over 180 radians.

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

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to to memorize this.

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

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That 2 pi radians is
equal to 360 degrees.

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Or another way that actually
makes the algebra a little

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simpler is if you just
think of a half circle.

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

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That's a degree sign.

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I could also write degrees out.

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And that's also equal
to pi radians.

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

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

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to pi over 180 radians.

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

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the intuition for this.

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

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

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

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And let's see, 45
divided by 180.

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

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45 degrees is equal to
pi over 4 radians.

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

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or two different ways
of measuring angles.

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

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mathematical standard for
measuring angles, although most

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of us are more familiar with
degrees just from

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everyday life.

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Let's do a couple
of other examples.

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Just always remember:
this 1 radian equals

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180 over pi degrees.

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1 degree equals pi
over 180 radians.

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If you ever get confused,
just write this out.

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

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pi over 180 or 180 over pi.

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I just remember pi radians
is equal to 180 degrees.

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Let's do another one.

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So if I were to say pi
over 2 radians equals

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how many degrees?

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

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remind myself that pi radians
is equal to 180 degrees.

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

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the presentation like that
and I will continue it later.

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

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go attend to my wife.

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

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

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equal to 180 over pi degrees.

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I just figure out the
formula again because

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I always forget it.

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So let's go back here.

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

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180 over pi degrees.

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And that equals 90 degrees.

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I'll do one more example.

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Let's say 30 degrees.

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Once again, I forgot the
formula so I just remember

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that pi radians is
equal to 180 degrees.

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So 1 degree is equal to
pi over 180 radians.

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

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which equals -- 30 goes
into 180 six times.

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That equals pi over 6 radians.

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Hopefully you have a sense of
how to convert between degrees

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

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

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

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radians as opposed to degrees.

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I'll see you in the
next presentation.