The amplitude of `(1)/(i)` is

Options

A. (a) 0
B. (b) `pi`
C. (c) `(pi)/(2)`
D. (d) `-(pi)/(2)`

Answer

To find the amplitude of \( \frac{1}{i} \), we can follow these steps:

### Step 1: Define the complex number
Let \( Z = \frac{1}{i} \).

### Step 2: Multiply by the conjugate
To simplify \( Z \), we can multiply the numerator and the denominator by \( i \):
\[ Z = \frac{1}{i} \cdot \frac{i}{i} = \frac{i}{i^2} \]

### Step 3: Substitute the value of \( i^2 \)
We know that \( i^2 = -1 \). Therefore,
\[ Z = \frac{i}{-1} = -i \]

### Step 4: Identify the real and imaginary parts
Now, we can express \( Z \) in terms of its real and imaginary parts:
\[ Z = 0 - i \]
Here, the real part \( a = 0 \) and the imaginary part \( b = -1 \).

### Step 5: Calculate the amplitude
The amplitude (or argument) \( \theta \) of a complex number \( Z = a + bi \) is given by:
\[ \tan \theta = \frac{b}{a} \]
Substituting the values of \( a \) and \( b \):
\[ \tan \theta = \frac{-1}{0} \]
Since division by zero is undefined, we interpret this as \( \tan \theta \) approaching negative infinity.

### Step 6: Determine the angle
The angle \( \theta \) for which \( \tan \theta \) approaches negative infinity is:
\[ \theta = -\frac{\pi}{2} \]

### Conclusion
Thus, the amplitude of \( \frac{1}{i} \) is:
\[ \text{Amplitude} = -\frac{\pi}{2} \]

### Final Answer
The correct option is \( \text{D} \, -\frac{\pi}{2} \).

---