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 a circle, full circle, you'll go full 360 degrees. And if I ask you to draw me something like a 120°, you'd be able to draw me something that looks about like this. What we're going to look at in this video is another unit of measuring angles called radians. It's kind of like how we use inches and centimeters to measure distance. We can use degrees and radians to measure angles. Alright? Now we'll be using radians a lot in later videos when we talk about functions and graphing, but for now, I just want you to understand the basic difference between degrees and radians and how we draw them, and more importantly, also how we convert between them. Let's go ahead and get started here.
So a radian really is just a different unit for measuring angles based on the idea of going around the circumference of a circle. So what we saw is that if you go a 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 radian 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 \(2\pi\) times the radius, that's a formula from geometry. So the whole idea of a radian 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 curved, the angle that you're making there, that little pizza slice that you've just made is 1 radian. And it's about 57 degrees. That's about approximate. Look at a little bit of a decimal. Alright? So if you go around a full circle, that's a full \(2\pi\) radians. So a full circle is 360 degrees \(2\pi\) radians. But usually when we work with radians, we're going to work with them in multiples or fractions of pi. So just as we cut up 360 into 4 quarters, we can cut up \(2\pi\) into 4 quarters. If you go halfway around, half of \(2\pi\) is \(\pi\) radian. If you go half of that, that's going to be \(\frac{\pi}{2}\) radians. If you go 3 quarters of the way around the circle, that's going to be \(\frac{3\pi}{2}\) radians. So that's what our sort of axes become in radian form. So how do we go back and forth between degrees and radians? Well, it really just comes down to these two formulas over here. To convert back and forth between degrees and radians or radians and degrees, we're just going to multiply whatever angle that we have by \(\frac{\pi}{180}\) or \(\frac{180}{\pi}\). Alright? We're going to do a couple of examples of this. In fact, our first example is going to be, how do we convert 120 degrees? This angle that we have over here, what does that end up being in terms of radians? Alright?
So what we're going to do here is this is an angle that's in terms of degrees. That's theta degrees. So that means we're going to use this top equation over here. So to convert this to radians, what we're going to have to do is take our 120 degrees, and we're going to have to multiply it by something. 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 pi radians being on top. So then how does this simplify? Well, what happens is 120 over 180, you can knock a 0 off, and 12 over 18 just simplifies to a fraction. So what happens is you're still left with a pi over here, but now this simplifies to a fraction of \(\frac{2}{3}\). So this is \(\frac{2\pi}{3}\) radians. So this, this \(\frac{2\pi}{3}\), is exactly what 120 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. Alright? So that's how you convert from degrees to radians. Let's do one more example over here because now we're actually going to do the opposite. Now we're going to convert this angle that's given to us in radians, and we're going to convert it to degrees.
So this is an angle that's going to be theta radians. So what we're going to do is we're going to convert this to theta in degrees. So how do we do this? Well, we take what the given angle is, \(\frac{6\pi}{5}\), and we know we're going to have to multiply by \(\frac{\pi}{180}\) or \(\frac{180}{\pi}\). 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 radians and be left with degrees. So that means that radians, we want to be on the bottom and 180 degrees on top. Usually, you can tell that this is going to be the case because you want the pi to cancel, so you want to divide by pi to get rid of it. So we're going to cancel out the pies over here. And, basically, when you actually sort of multiply these numbers out, what you're going to get here is you're going to get 1,080 degrees divided by 5. And if you work this out, what you'll actually end up getting here is you'll end up getting 216 degrees. So that's what \(\frac{6\pi}{5}\) radians is going to become, 216 degrees. Hopefully, this made sense, folks. Thanks for watching, and I'll see you in the next one.
