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 module
I mean, lets see
we actually drill you through a bunch of problems.
You're probably familiar that an angle
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 attempt at a circle --
in a 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.
So by definition...
Oh,oh,I'm still using the radian tool, the circle tool.
OK.
So this is --let's say
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 subtends 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.
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.
right, let's say 2 pi raidians
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.
Right , 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 by 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 pi over 180 degrees
(laughs)1 radian equals 180 over pi degrees
and 1 degree equals pi over 180 radians.
And if you ever forget these
it doesn't hurt 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 or the algebric expression
a little simpler is
if you just think of a half circle.
A half circle --
this angle -- is a 180 degrees, right?
180 degrees
That's a degree sign.
I could also write degrees out.
And that's also equal to pi radians,right.
So pi radians equal 180 degrees
and we can get to see the math.
1 radian equals 180 over pi degrees
or 1 degree,that's 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 is 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.