Euler’s Formula, expressed as e^(iθ) = cos(θ) + i sin(θ), provides a profound connection between exponential and trigonometric functions in the context of complex numbers. When a complex number is expressed in its trigonometric form, z = r(cos(θ) + i sin(θ)), Euler’s Formula allows us to rewrite it as z = r e^(iθ). This form is particularly useful in simplifying operations such as multiplication, division, and exponentiation of complex numbers, as it allows for easy manipulation of the moduli and arguments, thereby providing a seamless bridge between algebraic and geometric interpretations of complex numbers.

No, a complex number cannot have a negative modulus. The modulus of a complex number, defined as the square root of the sum of the squares of its real and imaginary parts, is always non-negative. Mathematically, if z = a + bi, then |z| = sqrt(a² + b²), which is always greater than or equal to zero. The modulus represents the distance of the point representing the complex number from the origin in the complex plane, and distance cannot be negative. Thus, the modulus of a complex number is always a non-negative real number.

The principal argument is significant in ensuring consistency and uniformity in mathematical communication and computation. When performing operations like multiplication and division of complex numbers in trigonometric form, the arguments of the numbers are added or subtracted. The principal argument ensures that the resulting argument of the operation remains within a standardised interval, typically (-π, π]. This avoids ambiguity and maintains a standard form, making mathematical work more coherent and interpretable across various contexts and applications, thereby avoiding potential misinterpretations or mathematical errors.

The argument of a complex number is restricted to a specific interval to ensure uniqueness. Without such a restriction, the argument could have infinitely many values, differing by multiples of 2π, which could lead to ambiguity. The typical interval chosen is (-π, π] because it symmetrically surrounds the real axis, providing a balanced perspective in both the positive and negative imaginary directions. This choice is somewhat arbitrary and is made for the sake of mathematical consistency and convention, ensuring that computations and communications among mathematicians and scientists are coherent and unified.

In electrical engineering, the trigonometric form of complex numbers is pivotal in analysing AC circuits. The voltage and current in an AC circuit can be represented as phasors, which are complex numbers. The modulus of the complex number represents the amplitude of the wave, while the argument represents the phase shift. When multiplying phasors (complex numbers), engineers can easily determine the resultant wave's amplitude and phase shift by multiplying the amplitudes and adding the phase shifts, respectively. This simplifies the analysis of AC circuits, particularly when dealing with impedance and calculating the voltage and current across circuit elements.