Unit
Trigonometric Functions
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to talk about arc length and the area of sector. I have a circle drawn here and an angle drawn in the circle. The circle has radius r and the angle has measure of theta. Now you remember from our definition of radian measure that theta equals s over r, so if we know arch length and the radius, we can find the measure of angle theta.
Well to get to the formula for arc length, we just multiply both sides by r and we get s equals r times theta, that's our arc length formula. But remember this formula only works if theta is in radians.
Now the formula for the area of a sector is a little trickier, A equals 1/2 r² theta. Let's use these formulas in some problems.
The problem says find the arc length and the area of the sector shown. The first one is a circle with radius 15 meters and the angle is two pi over 3. So arc length remember is s equals r times theta. Now whatever units you're using for the length of your radius, the arc length is going to inherit those. So you have 15 meters times 2 pi over 3. Now the 3 and the 15 will cancel leaving a 5 and you get a product of 10 pi meters, it's about 31 meters.
Now the area is 1/2 r² theta, so 1/2 of 15m² times 2 pi over 3. Now because the 15 meters is getting squared, my units are going to end up being meters squared an notice also the 2s cancel and so we have 15 times 15 divided by 3, that's 15 times 5 or 75, don't forget the pi and that's meters squared.
In this example, our angle is given in degrees, don't forget that the two formula that I've just shown you, they only work if your angle is in radians so you have to convert and let's use that first. We can use our fact 180 degrees equals pi radians. Now in order to get 70, how do we get 70? We can multiply 70 by pi over 180; our conversion factor from degrees to radians. So 70 times pi over 180 is 7 pi over 18. It's not a nice angle, but at least it's in radians.
So now our arc length formula s equals r times theta, the radius is 12 feet times 7 pi over 18, we'll get a little cancellation here with the 12 and the 18 leaving a 2 here and a 3 here. 2 times 7, 14 pi over 3 and this will be in feet.
Now the area is 1/2 r² theta. 1/2, 12ft², so the units of this answer are going to be in feet squared times 7 pi over 18. Now we should do a little cancellation before we multiply through let me write it down here, 1/2 times 12 I'll leave the units out for the moment times 12, 7 pi over oops sorry over 18.
Now 12 and 18 cancel leaving a 3 and a 2, but then the 2 and the 1/2 cancel and you're left with 12 times 7, 84 pi over 3, again that's in feet squared. So that's your arc length formula, area of the sector formula, don't forget and make sure theta is in radians.