Robotic Arm Path Planning (Mechanical & Aerospace Engineering)

In robotics, the forward kinematics function maps the joint angles of a robotic arm to a physical coordinate $(x, y)$ in space. However, to control the robot, engineers must solve the inverse kinematics problem: given a target coordinate $(x, y)$, calculate the joint angle $\theta$ required to reach it.

For a single-link arm of length $L = 5\text{ units}$, the position is given by $x = 5\cos\theta$ and $y = 5\sin\theta$.

1. Reconstruct the trigonometric relationship to show that $\theta = \arctan\left(\frac{y}{x}\right)$.
2. If the arm needs to target the coordinate $(x, y) = (-3, 4)$ in Quadrant II, evaluate the joint angle $\theta$ in radians, accounting for quadrant corrections.