Question

If θ is an angle in standard position whose terminal side passes through (3, 4), evaluate tan(1/2)θ.

Answer 1

The tangent half angle formula, one of several, is

$\tan \dfrac a 2 = \dfrac{1 - \cos a}{\sin a}$

We have θ is opposite 4 in the 3/4/5 right triangle so

$\cos \theta = \dfrac{3}{5}$

$\sin \theta = \dfrac{4}{5}$

$\tan \dfrac{\theta}{2} = \dfrac{1 - 3/5}{4/5} = \dfrac{5-3}{4}=\dfrac{1}{2}$

Answer: 1/2

This is actually pretty deep.  It says half the big acute angle in the 3/4/5 triangle is the small diagonal angle of the 1x2 rectangle.   Similarly, the small acute angle in 3/4/5 triangle is twice the small diagonal angle of the 1x3 rectangle.