If `z_(1)=-1 and z_(2)=-i`, then find Arg `(z_(1)z_(2))`

Answer

To solve the problem, we need to find the argument of the product of two complex numbers \( z_1 \) and \( z_2 \), where \( z_1 = -1 \) and \( z_2 = -i \).

### Step-by-Step Solution:

1. **Identify the complex numbers**:
- Let \( z_1 = -1 \)
- Let \( z_2 = -i \)

2. **Find the arguments of \( z_1 \) and \( z_2 \)**:
- The argument of \( z_1 = -1 \) is \( \text{Arg}(z_1) = \pi \) (since it lies on the negative real axis).
- The argument of \( z_2 = -i \) is \( \text{Arg}(z_2) = -\frac{\pi}{2} \) (since it lies on the negative imaginary axis).

3. **Use the property of arguments**:
- The argument of the product of two complex numbers is given by:
\[ \text{Arg}(z_1 z_2) = \text{Arg}(z_1) + \text{Arg}(z_2) \]

4. **Calculate the argument of the product**:
- Substitute the values:
\[ \text{Arg}(z_1 z_2) = \pi + \left(-\frac{\pi}{2}\right) \]
- Simplifying this gives:
\[ \text{Arg}(z_1 z_2) = \pi - \frac{\pi}{2} = \frac{\pi}{2} \]

5. **Final answer**:
- Therefore, the argument of \( z_1 z_2 \) is:
\[ \text{Arg}(z_1 z_2) = \frac{\pi}{2} \]