Sorry, no pictures. Explanation here.
To trisect the angle $AOB$ (with $OA=OB$), do:
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOC$, respectively. One may verify that \[ \beta = \arctan\bigg( \frac{ \sin\frac{\alpha}{2} \sin\big(\frac{\pi}{6}+\frac{\alpha}{2}\big) }{ 1 + \sin\frac{\alpha}{2} \cos\big(\frac{\pi}{6}+\frac{\alpha}{2}\big) } \bigg) = \frac{1}{4}\alpha + \frac{\sqrt{3}}{16} \alpha^2 - \frac{1}{16} \alpha^3 + O(\alpha^4). \]
This says that $\ds \beta \approx \frac{1}{4}\alpha$ when $\alpha$ is small, so small angles are quadrisected, rather than trisected! (This is clearly visible in the interactive diagram above.) For not-so-small angles, the method works reasonably well. In fact, it produces exact trisection for angles $\alpha=\pi/2$ and $\alpha=\pi$.
The worst error in the range $0 \le \alpha \le \pi$ is 0.0214 radians = 1.23 degrees. This occurs at $\alpha=2\arctan(\sqrt{3}\pm\sqrt{2})$ which corresponds to angles of approximately 35 degrees and 145 degrees.
This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on June 14, 2010.
Go to list of trisections |