Topic A: Lessons 8-10: Trigonometric graphs and radians
Current time:0:00Total duration:7:02
- [Instructor] We're asked to convert 150 degrees and negative 45 degrees to radians. Let's think about the relationship between degrees and radians, and to do that, let me just draw a little circle here. So that's the center of the circle, and then do my best shot, best attempt to freehand draw a reasonable-looking circle. That's not, I've done worse than that. Alright, now, if we were to go in degrees, if we were to go one time around the circle like that, how many degrees is that? We know that that would be 360 degrees. If we did the same thing, how many radians is that, if we were to go all the way around the circle? We just have to remember, when we're measuring in terms of radians, we're really talking about the arc that subtends that angle. So if you go all the way around, you're really talking about the arc length of the entire circle, or essentially the circumference of the circle. And you're essentially saying, how many radius's this is, or radii, or how many radii is the circumference of the circle. You know a circumference of a circle is two pi times the radius, or you could say that the length of the circumference of the circle is two pi radii. If you wanna know the exact length, you just have to get the length of the radius and multiply it by two pi. That just comes from the, really, actually the definition of pi, but it comes from what we know as the formula for the circumference of a circle. If we were to go all the way around this, this is also two pi radians. That tells us that two pi radians, as an angle measure, is the exact same thing, and I'm gonna write it out, as 360 360 degrees. And then we can take all of this relationship and manipulate it in different ways. If we wanna simplify a little bit, we can divide both sides of this equation by two, in which case, you are left with, if you divide both sides by two, you are left with pi radians is equal to 180 degrees. How can we use this relationship now to figure out what 150 degrees is? Well, this relationship, we could write it in different ways. We could divide both sides by 180 degrees, and we could get pi radians over 180 degrees is equal to one, which is just another way of saying that there are pi radians for every 180 degrees, or you could say, pi over 180 radians per degree. The other option, you could divide both sides of this by pi radians. You could say, you would get on the left hand side you'd get one, and you would also get, on the right hand side, you would get 180 degrees for every pi radians. Or you could interpret this as 180 over pi degrees per radian. How would we figure out, how would we do what they asked us? Let's convert 150 degrees to radians. Let me write the word out. So, 150 degrees. Well, we wanna convert this to radians, so we really care about how many radians there are per degree, actually, let me do that in that color. (typing) We care about how many radians there are per degree. We'll do that same green color. Per degree. How many radians are there per degree? Well, we already know, there's pi radians for every 180 degrees, or there are pi... Let me do that yellow color. There are pi over 180 radians per degree. And so, if we multiply, and this all works out because you have degrees in the numerator, degrees in the denominator, these cancel out, and so you are left with 150 times pi divided by 180 radians. So what do we get? This becomes, let me just rewrite it. 150 times pi. All of that over 180, so this is equal to, and we get it in radians. And so, if we simplify it, let's see, we can divide the numerator and the denominator both by, looks like, 30. So if you divide the numerator by 30, you get five. You divide the denominator by 30, you get six. So you get five pi over six radians, or 5/6 pi radians, depending how you wanna do it. Now let's do the same thing for negative 45 degrees. What do you get for negative 45 degrees if you were to convert that to radians? Same exact process. You have negative, and I'll do this one a little quicker. Negative 45 degrees. I'll write down the word. Times, times pi radians, pi radians for every 180 degrees. The degrees cancel out, and you're left with negative 45 pi over 180 radians. So this is equal to negative 45 pi over 180, over 180 radians. How can we simplify this? Well it looks like they're both, at minimum, divisible by nine, nine times five is 45, this is nine times 20, so actually it's gonna be divisible by more than just, let's see... Actually, they're both divisible by 45. What am I doing? If you divide the numerator by 45, you get one. You divide the denominator by 45, 45 goes into 180 four times. You're left with negative pi over four radians. This is equal to negative pi over four radians. And we are done.