the vaule of arg (x) when `x lt 0` is

Options

A. 0
B. `(pi)/(2)`
C. `pi`
D. None of these

Explanation

To find the value of the argument of \( x \) when \( x < 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Argument of a Complex Number**:
The argument of a complex number \( z = x + iy \) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]
where \( x \) is the real part and \( y \) is the imaginary part.

2. **Setting the Values**:
In our case, since we are dealing with a real number \( x \) and it is given that \( x < 0 \), we can express it as:
\[ z = x + 0i \]
Here, \( y = 0 \) and \( x < 0 \).

3. **Calculating the Argument**:
Plugging the values into the argument formula:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{0}{x}\right) \]
Since \( y = 0 \), this simplifies to:
\[ \text{arg}(z) = \tan^{-1}(0) = 0 \]

4. **Determining the Quadrant**:
However, we must consider the sign of \( x \). Since \( x < 0 \), the point \( (x, 0) \) lies on the negative real axis. The angle corresponding to the negative real axis is:
\[ \pi \text{ (or } 180^\circ\text{)} \]

5. **Final Result**:
Therefore, the value of \( \text{arg}(x) \) when \( x < 0 \) is:
\[ \text{arg}(x) = \pi \]

### Conclusion:
Thus, the value of \( \text{arg}(x) \) when \( x < 0 \) is \( \pi \).