Half angle formulas 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:
$$
\tan \frac{\theta}{2}  = \frac{ 1 - \frac{4}{5} }{ \frac{3}{5}  }
= \frac{ \frac{1}{5} }{ \frac{3}{5}  }
= \frac{1}{3}
$$

Actually, if you half an angle, it will divide the opposite side proportionally to the two other sides. (see http://en.wikipedia.org/wiki/Angle_bisector_theorem )
In your case, 3 would be divided into parts $4/3$ and $5/3$.
So you get $\tan \theta/2=(4/3)/4= 1/3$. Everything works out fine.
