In polar form, how is a complex number represented?

In polar form, a complex number is represented as ( z = r(\cos(\theta) + i\sin(\theta)) ), where ( r ) is the magnitude (or modulus) of the complex number and ( \theta ) is the argument (or angle) in radians. This representation allows for a clear and effective visualization of complex numbers in the polar coordinate plane.

The term ( r ) corresponds to the distance from the origin to the point representing the complex number, and the terms ( \cos(\theta) ) and ( \sin(\theta) ) describe how this point is positioned in relation to the angles formed with the horizontal axis. The use of ( i ) denotes the imaginary unit.

The structure of the polar form effectively combines both the magnitude and the direction of the complex number, facilitating easier multiplication and division of complex numbers compared to their Cartesian (rectangular) form. This is particularly advantageous in various applications, such as electrical engineering and physics, where complex numbers are often used to represent oscillations and phase shifts.

Other representations, such as the second choice using a minus sign, do not align with the standard polar form definition. Similarly, the third and fourth options introduce inaccuracies in the representation of