The argument of the complex number `(i/2-2/i)` is equal to (a)`pi/2` (b)`pi/4` (c)`pi/12` (d)`(3pi)/4`

Options

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

Answer

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

### Step 1: Simplify the complex number
We start with the expression:
\[ z = \frac{i}{2} - \frac{2}{i} \]
To simplify \(-\frac{2}{i}\), we multiply the numerator and denominator by \(i\):
\[ -\frac{2}{i} = -\frac{2i}{i^2} = -\frac{2i}{-1} = 2i \]
Now we can rewrite \(z\):
\[ z = \frac{i}{2} + 2i \]

### Step 2: Combine the terms
Next, we need to combine the terms:
\[ z = \frac{i}{2} + 2i = \frac{i}{2} + \frac{4i}{2} = \frac{5i}{2} \]

### Step 3: Identify the real and imaginary parts
Now, we identify the real part \(a\) and the imaginary part \(b\) of the complex number:
\[ a = 0, \quad b = \frac{5}{2} \]

### Step 4: Calculate the argument
The argument \(\theta\) of a complex number \(z = a + bi\) is given by:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
In our case, since \(a = 0\), we have:
\[ \theta = \tan^{-1}\left(\frac{\frac{5}{2}}{0}\right) \]
This expression tends to infinity, which corresponds to:
\[ \theta = \frac{\pi}{2} \]

### Conclusion
Thus, the argument of the complex number \(z\) is:
\[ \theta = \frac{\pi}{2} \]
So, the correct answer is (a) \(\frac{\pi}{2}\).

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