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Welcome to the presentation

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on radians and degrees

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So you all are probably already reasonably

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familiar with the concept of degrees.

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I think in our angles module

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I mean, lets see

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we actually drill you through a bunch of problems.

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You're probably familiar that an angle

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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

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the concept that in a circle --

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and that's my best attempt at a circle --

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in a circle, there are 360 degrees.

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So today I'm going to introduce you to

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another measure or unit for angles

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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

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and I think this might give you a little intuition

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for why it's even called radian.

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Let me use this circle tool

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and actually draw a nice circle.

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So by definition...

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

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OK.

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So this is --let's say

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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,

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and this is the arc,

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this angle subtends this arc

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and this arc subtends this angle.

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So a radian

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-- 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

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it actually makes a 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

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and that this arc right here is also length r.

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Then this angle, what's called theta,

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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:

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how many radians are there in a circle?

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Well, if this is r,

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what is the whole circumference 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

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that subtends an arc of r,

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then the angle that subtends an arc of 2 pi r

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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

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all the way around subtends 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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I don't know.

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So why am I going through all of this mess

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and confusing you?

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I just want to one, give you an intuition

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for why it's called a radian

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and kind of how it relates to a circle.

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

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we can now figure out

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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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right, let's say 2 pi raidians

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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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Right , 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

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and we could have said 1 degree --

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I'm just going to divide both sides by 360

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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:

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1 radian equals pi over 180 degrees

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(laughs)1 radian equals 180 over pi degrees

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and 1 degree equals pi over 180 radians.

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And if you ever forget these

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it doesn't hurt 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 or the algebric expression

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a little simpler is

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if you just think of a half circle.

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A half circle --

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this angle -- is a 180 degrees, right?

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180 degrees

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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,right.

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So pi radians equal 180 degrees

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and we can get to see the math.

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1 radian equals 180 over pi degrees

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or 1 degree,that's degree, is equal to pi over 180 radians.

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So let's do a couple of problems were

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you'll get the intuition for this.

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If I asked you 45 degrees --

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to convert that into radians.

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

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So 45 degrees is equal to

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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

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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,

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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

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the mathematical standard for measuring angles,

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although most of us are more familiar with degrees

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just from everyday life.

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Let's do a couple of other examples.

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Just always remember:

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this 1 radian equals 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

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whether it's 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

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so I just remind myself that

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pi radians is equal to 180 degrees.

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Oh, my wife just got home,

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so I'm just going to have to leave

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the presentation like that

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and I will continue it later.

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Actually, let me just finish this problem

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and then I'll go attend to my wife.

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

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right?

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So one radian is equal to 180 over --

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that's one radian -- is equal to 180 over pi degrees.

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I just figure out the formula again

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because 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

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pi over 2 times 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

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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

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how to convert between degrees and radians now

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and even why it's called a radian

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because it's very closely related to a radius

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and you'll feel comfortable when someone asks you to,

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I don't know,deal with radians as opposed to degrees.

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I'll see you in the next presentation.