Angles, arc lengths, and trig functions — Basic example

- [Instructor] In the figure at left, I pasted it up here, point O, point O is the center of a circle of radius 1.5, we see that right over there. The missing sector of the circle has central an, the missing sector of the circle has central angle AOB equal to two pi over three radians, that's this right over here, and that's the central angle for the missing, for the missing sector right over there. What is the length of arc ACB? Now ACB, ACB is this arc, it's kind of the rest of the circle, the part that's not missing, so this is ACB right over here. Well the way that I would tackle this, I would think well what's the circumference of the entire circle, and then what fraction of the entire circle is this arc length? So let's first think about the circumference of the entire circle. Circumference is equal to two pi times r. In this case, our r, our radius is 1.5. It's gonna be two pi times 1.5. Now two times 1.5 is three, and so this is going to be equal to, this is going to be equal to three pi. So the circumference of the entire circle is three pi. And now there's a couple of ways that you could do it. You could figure out well what is the length of this arc right over here and then subtract that from the circumference, and then you'd be left with the magenta part. Or we could figure out the central angle of the magenta part we could figure out this angle, and think about well what fraction is that going to be if we were to go all the way around, and if we're thinking in radians, going all the way around is two pi radians. So what fraction is this angle of two pi, and then that's going to be the same fraction that this arc length is of the entire circumference. Well what's this angle going to be? Well it's going to be, if we want, remember if we went all the way around, if we went all the way around the circle, if we went all the way around the circle, that'd be two pi radians. But if we wanna figure out this magenta central angle, it's going to be two pi minus this two pi over three. So this is going to be two pi, let me do it in that magenta color, so the central angle for this piece of the circle that's kind of the central angle for ACB is going to be two pi minus two pi over three. I'm going all the way around but then I'm subtracting out this part right over here. Now what's two pi minus two pi over three? Let's see, I can find a common denominator, instead of writing it as two pi, I can write that as six pi over three, so let me do that. It's gonna be six pi over three minus two pi over three. Well that's going to be four pi over three, four pi over three. So once again this angle right over here is four pi over three. Four, four pi over three radians. Now what fraction is that of if we were to go all the way around the circle? Well once again this central angle is four pi over three, if you were to go all the way around the circle, that's two pi, so this is the fraction of the entire circle that this arc represents. And so let's just multiply that times the entire circumference, times three pi, and let's try to simplify it. Let's see, we have pi divided by pi, and let's see if we take this three and multiply it times the numerator, this three is gonna cancel with that three, and we're gonna be left with four pi over two, four pi divided by two is equal to, is equal to two pi. So that's the length, that's the length of this arc. It's actually exactly 2/3 of the entire, 2/3 of the entire circumference. So let me just select that, two pi, and we're done.