Originally posted by joe shmo
After much work trying to involve integration to determine arclength ( which will be my next step) Ive decided to try and simplify the process by using the chord length as an approximation
so Ill start by trying to find a right triangle whose tangent is "x"
this yields a triangle with the following sides
HYP : sqrt(x^2 + 1)
OPP : x
such ...[text shortened]... without introducing more trancendental functions....I cant seem to figure a way to do it?
The arclength of the sector swept out is fairly easy to calculate. All you really need to do is read the following equation out loud:
arctan(x) = 1/2x
"The angle whose tangent is x is 1/2x."
From that, it's easy to see that the angle A = 1/2x, and the corresponding right angle triangle has side lengths 1, x and SQRT(1+x^2) as you stated above. From the definition of a radian, the arc length is therefore:
L = (A/2pi)*(2pi*r) = A*r = (1/2x)*SQRT(1+x^2)
However, I'm not sure this really helps you find an analytical answer to your equation. I'm not even sure it's possible, given that arctan(x) seems to be defined in terms of an infinite series:
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...