Let z and `omega` be two non-zero complex numbers, such that `|z|=|omega|` and `"arg"(z)+"arg"(omega)=pi`. Then, z equals

Options

A. `omega`
B. `-omega`
C. `baromega`
D. `-baromega`

Answer

To find the value of the complex number \( z \) given the conditions \( |z| = |\omega| \) and \( \text{arg}(z) + \text{arg}(\omega) = \pi \), we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Given Conditions**:
We know that \( |z| = |\omega| \) implies that both complex numbers have the same magnitude. The second condition, \( \text{arg}(z) + \text{arg}(\omega) = \pi \), indicates that the angles (arguments) of \( z \) and \( \omega \) sum to \( \pi \), meaning they are in opposite directions in the complex plane.

2. **Expressing \( z \) and \( \omega \)**:
We can express \( z \) and \( \omega \) in terms of their magnitudes and arguments:
\[ z = |z| \left( \cos(\theta) + i \sin(\theta) \right) = |z| e^{i\theta} \]
\[ \omega = |\omega| \left( \cos(\theta_1) + i \sin(\theta_1) \right) = |\omega| e^{i\theta_1} \]
where \( \theta = \text{arg}(z) \) and \( \theta_1 = \text{arg}(\omega) \).

3. **Using the Argument Condition**:
From the condition \( \text{arg}(z) + \text{arg}(\omega) = \pi \), we can express \( \theta_1 \) in terms of \( \theta \):
\[ \theta + \theta_1 = \pi \implies \theta_1 = \pi - \theta \]

4. **Substituting the Argument**:
Since \( |z| = |\omega| \), we can denote this common magnitude as \( r \). Thus, we can write:
\[ z = r e^{i\theta} \]
\[ \omega = r e^{i(\pi - \theta)} = r \left( \cos(\pi - \theta) + i \sin(\pi - \theta) \right) \]

5. **Calculating \( \cos(\pi - \theta) \) and \( \sin(\pi - \theta) \)**:
Using trigonometric identities:
\[ \cos(\pi - \theta) = -\cos(\theta) \]
\[ \sin(\pi - \theta) = \sin(\theta) \]
Therefore, we can express \( \omega \) as:
\[ \omega = r \left( -\cos(\theta) + i \sin(\theta) \right) = -r \left( \cos(\theta) - i \sin(\theta) \right) \]

6. **Expressing \( z \) in Terms of \( \omega \)**:
Since \( z = r e^{i\theta} \) and \( \omega = r e^{i(\pi - \theta)} \), we can write:
\[ z = -\omega \]

7. **Final Result**:
Thus, we conclude that:
\[ z = -\omega \]

### Conclusion
The final answer is:
\[ z = -\omega \]