Say I have the trig identity
$ \tan \frac{\theta}{2} = \frac{ 1 - \cos \theta }{ \sin \theta } $
And you have the 3,4,5 triangle:
taking the angle $\theta$ as marked, why can't we just say
\tan \frac{\theta}{2} = \frac{ 1.5 }{ 4 }
= \frac{ 3 }{ 8 }
(Like, if you half $\theta$, then the opposite side length is seen as half, right?)
But this doesn't work or check out with the identity:
\tan \frac{\theta}{2} = \frac{ 1 - \frac{4}{5} }{ \frac{3}{5} }
= \frac{ \frac{1}{5} }{ \frac{3}{5} }
= \frac{1}{3}