The argument of the complex number`((i)/(2)-(2)/(i))` is equal to

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `(pi)/(3)`
D. `(pi)/(12)`

Answer

To find the argument of the complex number \(\frac{i}{2} - \frac{2}{i}\), we will follow these steps:

### Step 1: Simplify the complex number
We start with the expression:
\[ \frac{i}{2} - \frac{2}{i} \]
To simplify, we can rewrite \(\frac{2}{i}\) by multiplying the numerator and denominator by \(i\):
\[ \frac{2}{i} = \frac{2i}{i^2} = \frac{2i}{-1} = -2i \]
Now, substituting this back into the expression, we have:
\[ \frac{i}{2} - (-2i) = \frac{i}{2} + 2i \]
Next, we need a common denominator to combine the terms:
\[ \frac{i}{2} + \frac{4i}{2} = \frac{i + 4i}{2} = \frac{5i}{2} \]

### Step 2: Identify the real and imaginary parts
Now, we have the complex number in the form:
\[ \frac{5i}{2} \]
Here, the real part \(x = 0\) and the imaginary part \(y = \frac{5}{2}\).

### Step 3: Calculate the argument
The argument of a complex number is given by:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Substituting \(y\) and \(x\):
\[ \theta = \tan^{-1}\left(\frac{\frac{5}{2}}{0}\right) \]
Since the denominator is zero, this indicates that the complex number lies on the positive imaginary axis. Therefore, the argument is:
\[ \theta = \frac{\pi}{2} \text{ (or } 90^\circ\text{)} \]

### Final Answer
The argument of the complex number \(\frac{i}{2} - \frac{2}{i}\) is:
\[ \frac{\pi}{2} \]