I am working on learning trigonometry and having some trouble understanding a piece of algebra involved, so I hope this is posted in the right forum.
So the formula for the area of a segment in Degrees is K=theta/360 times 2piR2 So then to convert to Radians we turn the 360 in the preceding equation to a 2pi. Then the pi in the bottom of the first term cancels, and the pi in the second equation cancels, leaving us with 1/2 r2 theta.
So it seems that when we have 2pi on the top and 2pi on the bottom, the 2pi on the top becomes (1) but on the bottom the pi just goes away leaving us with the two? This doesn't seem to make any sense to me. If we cancel a 2pi out of the top and end up with one, then shouldn't we do the same on the bottom and end up with a 2pi there as well? Where is the 1/2 coming from?
To further complicate things, when we look at examples of unit conversions from radians to degrees, we have a situation where we are doing things like 3pi/4 times 180/pi. Here, the pi terms simply cancel out on both top and bottom, but without the three going down to a two. But it seems that if we were following the rule implied by the previous example, the 3 pi should become a 2 pi.
So my question is: when we are canceling terms with variables (at least, I assume pi should be treated as a variable? Is this my problem, since it's actually a constant? If so I still don't see why it is being treated differently in the two examples I cited) how do I know when the coefficient should be lowered or left the same?
So the formula for the area of a segment in Degrees is K=theta/360 times 2piR2 So then to convert to Radians we turn the 360 in the preceding equation to a 2pi. Then the pi in the bottom of the first term cancels, and the pi in the second equation cancels, leaving us with 1/2 r2 theta.
So it seems that when we have 2pi on the top and 2pi on the bottom, the 2pi on the top becomes (1) but on the bottom the pi just goes away leaving us with the two? This doesn't seem to make any sense to me. If we cancel a 2pi out of the top and end up with one, then shouldn't we do the same on the bottom and end up with a 2pi there as well? Where is the 1/2 coming from?
To further complicate things, when we look at examples of unit conversions from radians to degrees, we have a situation where we are doing things like 3pi/4 times 180/pi. Here, the pi terms simply cancel out on both top and bottom, but without the three going down to a two. But it seems that if we were following the rule implied by the previous example, the 3 pi should become a 2 pi.
So my question is: when we are canceling terms with variables (at least, I assume pi should be treated as a variable? Is this my problem, since it's actually a constant? If so I still don't see why it is being treated differently in the two examples I cited) how do I know when the coefficient should be lowered or left the same?
