User:Jalexbnbl/sandbox


 * Downsize the 60 ° angle JMK with four angle bisectors. The 40 ° trisector of angle JMK is also a trisector of the 3.75 ° angle NMO. A good approximation to the desired radius is MR, where R is a point on the chord ON lying a third of the way from O to N. Trisecting a chord (as opposed in an arc) of a circle is possible with straightedge and compass. The error in using MR as the trisector of JMK is the measure of the angle between MR and the true trisector, i.e., $$ \arctan((\tan 3.75/2)/3 - 3.75/6) $$ = 0.00020 °, and the measure of RMK is 40.00020 °.

Half-angle and half-side formulae
With $$2s=(a+b+c)$$ and $$2S=(A+B+C)$$,

\begin{align} & \sin{\textstyle\frac{1}{2}}A=\left[\frac{\sin(s{-}b)\sin(s{-}c)}{\sin b\sin c}\right]^{1/2} &\qquad &\sin{\textstyle\frac{1}{2}}a=\left[\frac{-\cos S\cos (S{-}A)}{\sin B\sin C}\right]^{1/2}\\[2ex] & \cos{\textstyle\frac{1}{2}}A=\left[\frac{\sin s\sin(s{-}a)}{\sin b\sin c}\right]^{1/2} &\qquad &\cos{\textstyle\frac{1}{2}}a=\left[\frac{\cos (S{-}B)\cos (S{-}C)}{\sin B\sin C}\right]^{1/2}\\[2ex] & \tan{\textstyle\frac{1}{2}}A=\left[\frac{\sin(s{-}b)\sin(s{-}c)}{\sin s\sin(s{-}a)}\right]^{1/2} &\qquad &\tan{\textstyle\frac{1}{2}}a=\left[\frac{-\cos S\cos (S{-}A)}{\cos (S{-}B)\cos(S{-}C)}\right]^{1/2} \end{align} $$