Introduction:
In split phase motors, the torque developed is proportional to the sine of the angle between the main winding current (Im) and the auxiliary winding current (Is). This relationship can be understood by considering the working principle of a split phase motor.

Working Principle of Split Phase Motor:
A split phase motor is a type of single-phase induction motor that has two windings - the main winding and the auxiliary winding. These windings are spatially displaced from each other, resulting in a phase difference between the currents flowing through them.

When the motor is powered, a rotating magnetic field is created by the interaction of the currents in the main and auxiliary windings. This rotating magnetic field induces a torque on the rotor, causing it to rotate.

Torque Development:
The torque developed in a split phase motor is directly influenced by the phase difference between the main and auxiliary windings. This phase difference is determined by the angle between the currents flowing through the windings.

Sine of the Angle between Im and Is:
The sine function relates the angle of a right-angled triangle to the ratio of the lengths of its sides. In the case of a split phase motor, the angle between Im and Is corresponds to the phase difference between the main and auxiliary windings.

The torque developed in a split phase motor is directly proportional to the sine of the angle between Im and Is. This means that as the angle increases, the torque also increases. When the angle is zero, the torque is at its minimum value.

Explanation:
The torque developed in a split phase motor can be calculated using the formula:

T = K * sin(θ)

Where:
T = Torque developed
K = Constant of proportionality
θ = Angle between Im and Is

The sine function ensures that the torque varies sinusoidally with the angle between the currents. This relationship is crucial for the proper functioning of the split phase motor.

Conclusion:
In conclusion, the torque developed by a split phase motor is directly proportional to the sine of the angle between the main winding current (Im) and the auxiliary winding current (Is). This relationship is a result of the working principle of the split phase motor and is essential for the motor's proper operation.