Let `z_(1),z_(2)` and `z_(3)` be three points on `|z|=1`. If `theta_(1), theta_(2)` and `theta_(3)` be the arguments of `z_(1),z_(2),z_(3)` respectively, then `cos(theta_(1)-theta_(2))+cos(theta_(2)-theta_(3))+cos(theta_(3)-theta_(1))`

Options

A. `ge -3/2`
B. `le-3/2`
C. `ge 3/2`
D. `le 2`

Answer

To solve the problem, we need to analyze the expression \( \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) \) given that \( z_1, z_2, z_3 \) are points on the unit circle (i.e., \( |z| = 1 \)).

### Step-by-Step Solution:

1. **Understanding the Points on the Unit Circle**:
Since \( z_1, z_2, z_3 \) lie on the unit circle, we can express them in terms of their arguments:
\[ z_1 = e^{i\theta_1}, \quad z_2 = e^{i\theta_2}, \quad z_3 = e^{i\theta_3} \]
where \( \theta_1, \theta_2, \theta_3 \) are the angles corresponding to \( z_1, z_2, z_3 \).

2. **Using the Cosine Addition Formula**:
We know from trigonometry that:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
Thus, we can express the cosines in our expression:
\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) \]

3. **Expressing in Terms of Complex Numbers**:
The sum \( z_1 + z_2 + z_3 \) can be analyzed:
\[ z_1 + z_2 + z_3 = e^{i\theta_1} + e^{i\theta_2} + e^{i\theta_3} \]
The modulus of this sum squared is:
\[ |z_1 + z_2 + z_3|^2 = (z_1 + z_2 + z_3)(\overline{z_1 + z_2 + z_3}) = z_1\overline{z_1} + z_2\overline{z_2} + z_3\overline{z_3} + 2 \text{Re}(z_1\overline{z_2} + z_2\overline{z_3} + z_3\overline{z_1}) \]
Since \( |z_1| = |z_2| = |z_3| = 1 \), we have:
\[ |z_1 + z_2 + z_3|^2 = 3 + 2 \text{Re}(z_1\overline{z_2} + z_2\overline{z_3} + z_3\overline{z_1}) \]

4. **Setting Up the Inequality**:
Since \( |z_1 + z_2 + z_3|^2 \geq 0 \), we have:
\[ 3 + 2 \text{Re}(z_1\overline{z_2} + z_2\overline{z_3} + z_3\overline{z_1}) \geq 0 \]
Rearranging gives:
\[ 2 \text{Re}(z_1\overline{z_2} + z_2\overline{z_3} + z_3\overline{z_1}) \geq -3 \]
Dividing by 2:
\[ \text{Re}(z_1\overline{z_2} + z_2\overline{z_3} + z_3\overline{z_1}) \geq -\frac{3}{2} \]

5. **Relating to Cosines**:
The real part can be expressed in terms of cosines:
\[ \text{Re}(z_1\overline{z_2}) = \cos(\theta_1 - \theta_2), \quad \text{Re}(z_2\overline{z_3}) = \cos(\theta_2 - \theta_3), \quad \text{Re}(z_3\overline{z_1}) = \cos(\theta_3 - \theta_1) \]
Thus, we have:
\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) \geq -\frac{3}{2} \]

### Conclusion:
The final result is:
\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) \geq -\frac{3}{2} \]