New Angle When Opposite Side is Halved

Question

Suppose you have a right triangle with any length sides. The value of one of the angles is $\theta$ and the opposite side is a. If I change the triangle so that the new length of side a is $\frac a2$, does the new value of $\theta$ always become $\frac \theta 2$?

Answer 1

No, but close. $\sin \theta$ becomes $\frac{1}{2}\sin \theta$. In the limit angle goes to zero this becomes the same as $\theta$ becoming $\frac{\theta}{2}$, so it's a good approximation for right triangles with one side disproportionally small compared to others.