Why do radians even exist? Why would you use them instead of degrees?
In: Mathematics
If we know that the ratio of the diameter (2R) and circumference (I) of a circle is a constant and call it pi = I/2R then 2Rpi = I. If we introduce the radius R as a characteristics distance for a circle we can talk about the ratio of I and R. 2pi = I/R. If we don’t fix I to be the circumference but allow some portion of it p we get 2pi p = Ip/R. The number on the left side works how we’d want an angle to work. The total distance covered on the circumference pI = 2pi p R.
Very convenient isn’t it this number which ranges from 0 to 2pi describes a full rotation around a circle and can be turned into distance travelled around the circumference by multiplying with the radius of the specific circle we are looking at. So the length of a circular curve is just the internal angle that traces the curve times the radius. You won’t get a more natural way of introducing some measure of angles than this.
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