Everyone. Welcome back. So as we've been talking about angles, we've gotten really used to the idea of working with degrees. If you go around ac a circle, full circle, you'll go a full 360 degrees. And if I asked you to draw me something like 100 and 20 you'd be able to draw me something that looks about like this we're gonna look at in this video is another unit of measuring angles called radiance. It's kind of like how we use inches and centimeters to measure distance. We can use degrees and radiance to measure angles. All right, now, we'll be using radiance a lot in later videos when we talk about functions and graphing. But for now, I just want to under understand the basic difference between degrees and radiance and how we draw them. And more importantly also how we convert between them. Let's go ahead and get started here. All right. So a radiant really is just a different unit for measuring angles based on the idea of going around the circumference of a circle. All right. So what we saw is that if you go full three full circle, that's 360 degrees. So if I cut up the circle into 360 tiny little pieces, each one of those things would be a degree. So this tiny little sliver over here, that's what one degree is. What a radiant is, is. It's basically just based on the idea of going around the circumference of a circle. If you go around the circumference of a circle, that's two pi times the radius. That's a formula from geometry. So the whole idea of a radiant is if you grab the radius of a circle and you take that distance and then start going around the circumference of a circle that same distance. But now is curved. The angle that you're making there, that little pizza slice that you've just made is one radiant and it's about 57 degrees. That's about like approximate, like it's a little bit of a decimal. All right. So if you go round a full circle, that's a full two pi radiance. So a full circle is 360 degrees and two pi radiance. But usually when we work with radiance, we're gonna work with them in multiples or fractions of pi. So just as we cut up 360 into four quarters, we can cut up two pi into four quarters. If you go halfway around half of two pi is one pi radiant. If you go half of that, that's gonna be pi over two radiant. If you go three quarters of the way around the circle, that's gonna be three pi over two radiance. So that's what our sort of axes become in radiant form. So how do we go back and forth between degrees and radiance? Well, it really just comes down to these two formulas over here to convert back and forth between degrees and radiance or ratings and degrees. We're just gonna multiply whatever angle that we have and by pi over 180 or 100 and 80 over pi, all right, we do a couple of examples of this. In fact, our first example is gonna be how do we convert 100 and 20 degrees this angle that we have over here? What does that end up being in terms of radiance? All right. So we're gonna do here is this is an angle that's in terms of degrees, that's stated degrees. So that means we're gonna use this top equation over here. All right. So to convert this to radiance, we're gonna have to do is take our 120 degrees and we're gonna have to multiply it by something. All right. So I don't actually want you to memorize these formulas. All you have to do is just memorize that the two numbers involved are pi and 180. How do you figure out which one goes on top? It really just comes down to what unit you're trying to get rid of. I'm trying to get rid of degrees. So that means I want degrees on the bottom. I want the 180 to be on the bottom because now what happens is you'll see the degrees will cancel and then you'll end up with P radiance being on top. All right. So then how does this simplify? Well, what happens is 1 20/1 80 you can knock a zero off and 12/18 just simplifies to a fraction. So what happens is you're still left with a P over here. But now this simplifies to a fraction of two thirds. So this is two pi over three radiance. All right. So this, this two pi over three is exactly what 100 and 20 degrees is. It's just written in a different unit. These two angles here mean the exact same thing. They both mean an angle that looks like this. It's just in a different unit. All right. So that's how you convert from degrees to radiance. Let's do one more example over here because now we're actually gonna do the opposite. Now we're gonna convert this angle that's given to us in radiance and we're gonna convert it to degrees. All right. So this is an angle that's gonna be theta radiance. So what we're gonna do is we're gonna convert this to the degrees. So how do we do this? Well, we take what the given angle is six pi over five and we know we're gonna have to multiply by pi over 180 or 180. Over pi how do we know which one? Well, it just depends on what we're trying to get rid of. We're trying to get rid of radiance and left, be left with degrees. So that means that radiance, we want to be on the bottom and 100 and 80 degrees on top. Usually you can tell that this is gonna be the case because you want the pie to cancel. So you want to divide by pie to get rid of it. All right. So that's a good way to sort of uh double check that. So we're gonna cancel out the pies over here. And basically when you actually sort of multiply these numbers out which you're gonna get here is you're gonna get 1000 and 80 degrees divided by, divided by five. All right. And if you work the south, what you'll actually end up getting here is you'll end up getting 216 degrees. All right. So that's what six pie or five is gonna become 6, 216. Hopefully, this made sense, folks. Thanks for watching and I'll see you in the next one.

2

Problem

Problem

Convert the angle $540°$ from degrees to radians.

A

$3\pi$ rad

B

$\frac{3\pi}{2}$ rad

C

$\frac{5\pi}{2}$ rad

D

$4\pi$ rad

3

Problem

Problem

Convert the angle $-\frac{5\pi}{6}$ from radians to degrees.