TITLE: Bisecting an angle doesn't lead to Trisecting?
QUESTION [5 upvotes]: http://en.wikipedia.org/wiki/Talk:Angle_trisection
If you take the angle, and draw a circle at the corner of the angle. You mark two points along the edges of the angle.
Those two points form the tangent of an isosceles triangle.
If you can trisect a line, trisecting the tangent doesn't equal trisecting the angle?

REPLY [3 votes]: Suppose that we want to "trisect" an angle $3\theta$ by the method suggested.  By using basic trigonometry and then calculating a Taylor series we find that the middle of the three angles constructed is
$$2\tan^{-1}\Bigl(\frac{1}{3}\tan\frac{3\theta}{2}\Bigr)
  =\theta+\frac{2}{3}\theta^3+\frac{1}{2}\theta^5+\cdots\ .$$
This is not equal to $\theta$, so the trisection is not exact.
However, this gives some kind of answer to the comment by David H on Andre Nicolas' answer: the error in the trisection is approximately proportional to $\theta^3$, so for small values of $\theta$, the construction will be very close.