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
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.