Sorry, no pictures. Explanation here.
We wish to trisect the given angle $AOB$ represented by the circular arc $AB$ centered at $O$, as shown in the diagram above.
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOT$, respectively. It is straightforward to show that \[ \beta = \frac{\alpha}{4} + \arcsin\Big( \frac{1}{3}\sin\frac{1}{4}\alpha \Big) = \frac{1}{3}\alpha - \frac{1}{2^4\cdot3^4} \alpha^3 + O(\alpha^7) = \frac{1}{3}\alpha - \frac{1}{1296} \alpha^3 + O(\alpha^7). \] The term after $\alpha^3$ is $\alpha^7$. That's not a typo.
The error $ \ds e(\alpha) = \frac{\alpha}{3} - \beta $ is monotonically increasing in $\alpha$. The worst error on the interval $0 \le \alpha \le \pi/2$ is $e(\pi/2) =$ 0.003 radians = 0.171 degrees. The worst error on the interval $0 \le \alpha \le \pi$ is $e(\pi)$ = 0.024 radians = 1.367 degrees.
This applet was created by
Rouben Rostamian
using
David Joyce's
Geometry
Applet
on July 26, 2002.
Cosmetic revisions on June 13, 2010.
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