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 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.