The amplitude of `1/i` is equal to `0` b. `pi/2` c. `-pi/2` d. `pi`

Options

A. 0
B. `pi/2`
C. `-pi/2`
D. `pi`

Answer

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

### Step 1: Rewrite the expression
We start with the expression \( z = \frac{1}{i} \). To simplify this, we can multiply the numerator and denominator by \( i \) (the complex unit).

\[ z = \frac{1}{i} \cdot \frac{i}{i} = \frac{i}{i^2} \]

### Step 2: Simplify using \( i^2 \)
We know that \( i^2 = -1 \). Therefore, we can substitute this into our expression:

\[ z = \frac{i}{-1} = -i \]

### Step 3: Identify the real and imaginary parts
Now we have \( z = -i \). In standard form, this can be expressed as:

\[ z = 0 + (-1)i \]

From this expression, we can identify:
- Real part \( a = 0 \)
- Imaginary part \( b = -1 \)

### Step 4: Determine the quadrant
Since the real part \( a = 0 \) and the imaginary part \( b = -1 \), the point lies on the negative imaginary axis. This corresponds to the fourth quadrant.

### Step 5: Calculate the amplitude
The amplitude (or argument) of a complex number in the fourth quadrant is calculated as:

\[ \text{Amplitude} = \tan^{-1}\left(\frac{b}{a}\right) \]

However, since \( a = 0 \), we have:

\[ \frac{b}{a} = \frac{-1}{0} \]

This indicates that the angle approaches \( -\frac{\pi}{2} \) (or \( \frac{3\pi}{2} \) in standard position, but we take the negative for the fourth quadrant).

### Conclusion
Thus, the amplitude of \( \frac{1}{i} \) is:

\[ -\frac{\pi}{2} \]

### Final Answer
The correct option is **c. \(-\frac{\pi}{2}\)**.

---