The harder, more useful question
Inverse kinematics flips the problem: given a target position you want the end effector to reach, what joint angles will get it there? This is what you actually want most of the time in practice, since a task like 'pick up the cup at this location' is naturally expressed as a target position, not as a list of joint angles nobody has calculated yet.
The catch is that inverse kinematics is much harder than forward kinematics, and it doesn't always have a single clean answer. A target position might be reachable by multiple different joint configurations, imagine reaching for a doorknob with your elbow up versus your elbow down, both hand positions can be identical while the arm shape differs. Some target positions are entirely unreachable, sitting outside the arm's physical workspace no matter how the joints are arranged.