Let `t_1,t_2,t_3` be the three distinct points on circle |t|=1. if `theta_1,theta_2` and `theta_3` be the arguments of `t_1,t_2,t_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 evaluate the expression:

\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) \]

where \(\theta_1\), \(\theta_2\), and \(\theta_3\) are the arguments of the points \(t_1\), \(t_2\), and \(t_3\) on the unit circle \(|t| = 1\).

### Step 1: Understanding the Geometry
Since \(t_1\), \(t_2\), and \(t_3\) are distinct points on the unit circle, we can visualize these points as dividing the circle into three sectors. The angles between these points will be important for our calculations.

### Step 2: Setting Up the Angles
Assume that the angles are evenly spaced around the circle. This means that the angle between each pair of points can be expressed as:

\[ \theta_2 - \theta_1 = \frac{2\pi}{3}, \quad \theta_3 - \theta_2 = \frac{2\pi}{3}, \quad \theta_1 - \theta_3 = \frac{2\pi}{3} \]

### Step 3: Evaluating the Cosine Terms
Using the cosine of the angles, we can write:

\[ \cos(\theta_1 - \theta_2) = \cos\left(-\frac{2\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) \]
\[ \cos(\theta_2 - \theta_3) = \cos\left(-\frac{2\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) \]
\[ \cos(\theta_3 - \theta_1) = \cos\left(-\frac{2\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) \]

### Step 4: Summing the Cosine Values
Now, we can sum these values:

\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) = 3 \cos\left(\frac{2\pi}{3}\right) \]

### Step 5: Calculating \(\cos\left(\frac{2\pi}{3}\right)\)
The value of \(\cos\left(\frac{2\pi}{3}\right)\) is:

\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]

Thus, we have:

\[ 3 \cos\left(\frac{2\pi}{3}\right) = 3 \left(-\frac{1}{2}\right) = -\frac{3}{2} \]

### Conclusion
Therefore, we conclude that:

\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) = -\frac{3}{2} \]

This value indicates that the expression is equal to \(-\frac{3}{2}\), which matches with one of the options provided in the question.

### Final Answer
\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) = -\frac{3}{2} \]