# Converting between Systems of Angle Rotations

Hello. This video is going to address

converting between systems of angle rotation. Okay, one thing I want to make

clear is that in this trigonometry class we’re going to look at several forms of

movement and I want to really just be clear about what rotation is. So we

can have horizontal movements, so you’re moving from left to right. We can have

vertical movement, you’re moving up and down, and then you can have rotation,

right, where essentially you’re pivoting around a point. That’s rotation.

Okay and if you’re in a standardized rotation you start at zero degrees is

due east. Now the basic process of converting

between any two systems is – and it doesn’t matter when I say any two

systems – that could be volume, mass, distance… in this case we’re gonna

address angle rotation, but that process is always the same. You multiply by one

using an equivalency between the two systems. So then you choose the denominator

units to be the unit’s you’re starting with and the numerator units to be the

units you want to end with, all right? So let’s do a really familiar example so we

can really just break it down, something that you already know, and look at the

pieces. So let’s say I tell you this distance is three inches, okay? And you

want to convert to feet. So you know, here’s your equivalency: you know that

there are 12 inches in one foot, okay, so when you multiply by one, that could

be 12 inches divided by one foot (oops) or “1” could also be one foot divided by 12

inches. Because remember when I write 12 inches, I’m representing a certain

distance, and that distance is the same distance as one foot, so the same

distance over itself is “1.” Now, if I start with three inches and my

process is to multiply by “1,” and one of these equivalencies, right, either this

one or this one and I want to choose the denominator units to be the units I’m

starting with and the numerator units to be the units I want to end up with. So

essentially in the denominator I’m gonna have inches, which is what I’m starting

with and feet which is why I want to end up with. So in this case I want this one

right here: one foot is 12 inches and what’s gonna happen is the inches cancel

out the numerator and denominator and then you’re gonna be left with feet and

then you just multiply the numbers three times one divided by 12 is 1/4 or 0.25.

Now I know you do this (laughter), but that’s why I wanted to use it is because you do

know it – it’s simple you can follow through. It’s really no different when we

go and we talk about angle rotations. It’s gonna be the same

process. You just have to learn the equivalencies, commit them to memory, and

then understand that you’re multiplying by “1” and know which

equivalency to use. That’s it. Okay so here are some equivalencies for

systems of measurement. There are others but these are the most common ones. So

you have one revolution around the circle, meaning you start at a point, you

go all around the circle, and you end up exactly the same place. That is 360

degrees. Then you have one degree is 60 minutes and do be careful about these

symbols, right? And then one minute is 60 seconds. So we’re gonna work with these

before we get to this. And then visually I just want to kind of let you see… I’m

not the greatest artist, but we’ll try. So here’s one revolution. I start at zero,

right, and I go all the way around like this, pivoting around that central point

there – boom! I end up at the same place. I went 360 degrees. Well, that’s a lot of

subdivisions, right, so I tried to draw and here’s what one degree might look

like, and it’s already too big, I’m sure. But let’s bear with me, suppose this one

little slice here is this, this rotation of one degree. There. So that’s

going from here to here. Then when I write that one degree is 60 minutes, what we’re

saying is we’re taking this degree, this rotation, and literally dividing it into

60 parts like this, right? I don’t know there’s 60 of them, but humor me.

And then when we say that one minute is 60 seconds, it means that I

take one of these minutes, and again I’m not going to be able to really draw this

well, but here’s maybe a minute. And again, we know that’s too big but

let’s say that’s a minute, and I now transfer that down here as a rotation, I

divide a minute into 60 seconds. So you can really see that there are an awful lot

of seconds in just one degree, all right? So let’s do that conversion. If I want to

know how many seconds are there in one degree – go ahead and pause the video see

if you can go ahead and do that – make that transition. So what you

would start here would actually be a double one (conversion) – you have one degree, and

remember your equivalencies are right here, so we know that one degree is

60 minutes, so I want to say, well in one degree there are 60 minutes, but then

that leaves me with 60 minutes, and now I want to go into seconds. So I know that

they’re here. In one minute there are 60 seconds so in one minute there are 60

seconds. So now these degrees would cancel out, these minutes would cancel

out, you’d multiply one times 60 times 60 to get 3600 divided by 1 divided

by 1, so we have 3600 seconds are in one degree. So obviously there’s a lot of

precision here. Okay so now let’s take another example, and let’s say you were

asked to convert 17 degrees 22 minutes and 46 seconds

into what they call decimal degrees. Now what does that mean? Well remember one

degree is being divided so much into minutes and seconds that you’re now

actually finding a fraction or decimal part of it – of a full degree. So

we’ll see hopefully what that looks like at the end. So what this is really equal

to is 17 degrees plus 22 minutes plus 46 seconds. So we can go ahead and convert

each one. Now we are already in degrees here. Now we need to work with

these to get them into degrees. So if we have here 22 minutes I want to multiply

by “1” but here I would have that one – that there are 60 minutes

in one degree, so now the minutes cancel out, and so I basically have 22 times 1

divided by 60 or 22 divided by 60 and you’re gonna get 0.3667 degrees. So now

we’re in degrees, and now we have to convert still the 46 seconds into degrees.

And again we’re gonna go and say okay, well there’s 60 seconds in one minute

and then now we’re into minutes and now I want to say well there’s 60 minutes in

one degree so now I use different color here these cancel out these cancel out

and I’m gonna be left with degrees, right? I’m gonna be left with degrees and if you

multiply 46 times one times one divided by 60 divided by 60 – and that’s actually

what you’re going to do in your calculator, you’re gonna get 0.0128

degrees. So now if I add all that up: 17 degrees plus 0.3667

degrees plus 0.0128 degrees, you’re gonna get 17.3795 degrees. Now I’m not sure visually I can show you what that

really means, but we’re gonna try it okay? So what that means is you’re

your starting rotation here at zero degrees and you start and you go 17

degrees, right? And then let’s say this would be – again much bigger than it

really is – but it’ll say that’s 18 degrees. Well you’re saying how much more

do I have in degrees? Well 22 minutes… well, we know there’s about –

sorry twenty yeah twenty two minutes – there’s about 60 minutes in a degree so

this is almost a third right? 20 divided by 60 is about a third, and notice we got

0.3, would have been about a third, but you know there’s a little bit more, and

then you still have your seconds. So what you’re saying is you’re going 17 degrees

plus a little bit more, almost let’s say point 4 right? Plus another point 4 of

the way like that so you’re actually rotating from here to about here. All

right thanks so much for listening. Oh wait! There’s something else I wanted to

talk about. Wait hold on this this is a little bit ahead.

I’m jumping a little bit ahead, but it’s the same concept. y]You’re gonna find out

later that 360 degrees or one revolution, right? one revolution is the same as two

pi radians – two pi radians – so radians is gonna be your new system. Another another

way of measuring angle rotation. So you’re gonna do the same thing. So let’s

say for example somebody says convert, let’s say, 28 degrees

into radians. Well remember you basically have that

“1” is 360 degrees divided by what? 2 pi or “1” is also 2 pi divided by 360 degrees.

So go ahead and set that up. Pause and set that up. What would you do? Alright, so hopefully you would say 28

degrees…you’re gonna multiply by “1” and remember you want to get out of degrees and go

into radians, right? So you’d say this equivalency: 360 degrees and then here 2

pi, if you want you can write radians to help you out, and essentially what’s

gonna happen then is you’re gonna cancel out radians –

I’m sorry cancel out degrees and go into radians. And so you’ll actually multiply

28 times 2 times pi and then divided by 360, and you should get 0.489

radians and then of course there’s something you’ll learn later but they

don’t actually write radians, but it’s understood. And let’s say they asked you

to convert 1.7 radians into degrees. Again, you can pause the video and

see how what you would get. So hopefully you’d say one point seven –

if you need to write it down, radians – and then write two pi radians if you need to, and then here, 360 degrees and let me just mention here to do this. This

confuses students sometimes. So if you can see that. I’ll start putting it in there.

One point seven – no let me bring this up to you – so I have one point seven, and I

want to multiply by the 360 and divide by two and pi. So you can either use

parentheses or you could do one point seven times 360 – oops, see if this will focus –

and then you go divided by 2, divided by pi, and your pi is second and then

the carrot key. Okay? That’s what you would do. So you get about ninety seven

point four degrees. All right? Thanks so much for listening!