Angles, arc lengths, and trig functions — Basic example

- [Instructor] What is the length of arc ABC? So arc ABC is this arc right over here and the reason why they have to give us three points is because if they just said arc AC, it would still ambiguous because there's two arc ACs. It could be this arc right over here or it could be this arc over here. And so when they tell us B we know that we're going from A to C through B, so that's why this kind of, by giving us that third letter, it resolves the ambiguity on which arc we're talking about. What is the length of that arc? Well the first thing we could do is well, let's just think about the entire, what's the circumference of this circle? And then we can say well, what fraction of the entire circumference is this arc? And a big hint here is if we were to go all the way around the circle, that would be 360 degrees, but we're only going 135 of those 360 degrees. But let's first think about the circumference. So the circumference of a circle is equal to two, is equal to two pi times the radius of the circle, and they give us the radius as three inches. So the circumference is going to be two pi times three inches, so times three inches, or two times three is six, times pi, so it's going to be six pi, six pi inches. Now, and actually if you look at the choices, if you look at the choices over here, the entire circumference of the circle is six pi inches. Remember, pi is 3.14159, so this is going to be something in the, you know, it's going to be a little bit over 18 or 19 inches. So you could immediately, if the entire circumference is a little over 18 or 19 inches, this is only a fraction of that entire circumference, so actually if you even were looking at the choices, you'd say, well hey, these are way too large. But actually, let's see if we can get to the exact right answer. So the entire circumference is six pi inches, but this arc length isn't the entire circumference. It only goes 135 degrees out of 360 degrees, so we could say, so it is 135/360 of the entire circumference. Remember, if we were to go all the way around, that's 360 degrees, while this angle right over here is 135 degrees, the angle that is subtended by this arc. So this arc length is going to be 135/360 of the entire circumference, so times six pi, six pi inches. So let's see if we can simplify this a little bit. So let's see, we could divide the numerator and the denominator by six. Six divided by six is one. 360 divided by six is 60. Let's see, if we divide 135 by five, that is going to be 27. Yeah, 100 divided by five is 20, and then 35 divided by five is seven, so this is going to be 27, and then this would be 12, if we divide 60 by five. Let's see, we could then divide the numerator and the denominator by three. 27 divided by three is nine, and 12 divided by three is four, so we're left with 9/4 times pi inches, so 9/4 times pi inches, or we could multiply the numerator times the pi. We would say nine pi over four inches. Now when you look at the choices, that is this one, that is this one right over there.