In future, it would be helpful to tell us the situation you are looking at (here "find the volume of the smaller of the two regions bounded by the sphere x^2+y^2+z^2=16 and the cone z=sqrt(x^2+y^2)") first, then ask the question! Doing it the other way is confusing.
The "bottom angle" is the angle at the vertex of the cone, correct? In the xz-plane (y= 0) z= sqrt(x^2+ y^2) becomes z= sqrt(x^2)= |x| and so is a pair of lines z= x for x non-negative and z= -x for x negative. Each of those makes an angle of pi/4 with the x-axis and pi/2 with each other.