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A sinusoidal function is a mathematical function that describes a smooth periodic oscillation. It can be in the form of a sine or cosine function. In engineering and physics, sinusoidal functions are essential for describing alternating current (AC) waveforms.

The general representation of a sinusoidal function is: \[ y = A \, \text{sin} \, (\theta) \] where:

• \( A \) is the amplitude, which represents the peak value of the oscillation.
• \( \theta \) is the phase angle, which represents the initial angle at time zero.
• 'sin' (sine) determines the shape of the waveform.

In the context of AC circuits, it is common to use sinusoidal functions to denote voltages and currents, which alternate between positive and negative values over time.

For the given exercise, we used sinusoidal functions to represent voltages or currents at various phases.

Rectangular form is a way to represent complex numbers, which can be especially useful in electrical engineering when dealing with sinusoidal functions. This form represents a complex number as: \[ z = x + jy \]

Here:

• \( x \) is the real part.
• \( y \) is the imaginary part.
• \( j \) is the imaginary unit, equivalent to \( \text{√} {-1} \).

Using the rectangular form is particularly advantageous when adding or subtracting complex numbers.

For the given exercise, to add sinusoidal functions, we converted phasors to rectangular form using the formula: \[ A \, \text{∠} \, \theta = A \, (\text{cos} \, \theta + j \, \text{sin} \, \theta) \] This conversion allowed us to combine the real and imaginary parts separately, thereby simplifying the addition process.

Polar form represents a complex number in terms of its magnitude and angle. This is highly useful in AC circuit analysis and when working with phasors. A complex number in polar form is written as: \[ z = R \, \text{∠} \, \theta \]

Here:

• \( R \) is the magnitude and can be calculated as \( R = \text{√} (x^2 + y^2) \).
• \( \theta \) is the phase angle and can be calculated as \( \theta = \text{tan}^{-1} \, (y/x) \).

The magnitude represents the length of the vector in the complex plane, while the phase angle represents the orientation of the vector.

In the exercise, converting the resultant rectangular form phasor to polar form involved calculating these magnitude and angle values to write the final phasor representation.

Phasor representation is a way to simplify the analysis of AC circuits by converting sinusoidal functions to complex numbers. Phasors translate the time domain problem to the frequency domain, making the math easier.

A phasor is generally represented as: \[ A \, \text{∠} \, \theta \] where:

• \( A \) is the amplitude.
• \( \theta \) is the phase angle.

Phasors make use of Euler's formula, \( e^{j\theta} = \text{cos} \, \theta + j \, \text{sin} \, \theta \), to transform sinusoidal functions into complex exponential form.

To solve the given problem, we first converted sinusoidal functions to phasors and then manipulated them using the phasor method. This method simplifies the addition, subtraction, and multiplication of sinusoidal functions, making it a crucial tool in electrical engineering.