The argument of the complex number \(\left(\frac{i}{2}-\frac{2}{i}\right)\) is equal to

Concept Used:

The argument of a complex number z = x + iy is the angle θ such that x = r cos θ and y = r sin θ, where r = √(x² + y²) is the modulus of z.

To find the argument, we first simplify the complex number and then use the formula: θ = arctan(y/x).

Calculation:

First, simplify the complex number:

\(\frac{i}{2} - \frac{2}{i} = \frac{i}{2} - \frac{2i}{i^2} = \frac{i}{2} - \frac{2i}{-1} = \frac{i}{2} + 2i = \frac{5i}{2}\)

This complex number is \(0 + \frac{5}{2}i\), so x = 0 and y = \(\frac{5}{2}\).

Now, we find the argument θ:

Since x = 0 and y > 0, the complex number lies on the positive imaginary axis.

Therefore, the argument is \(\frac{\pi}{2}\).

∴ The argument of the complex number is \(\frac{\pi}{2}\).

Hence option 4 is correct