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 find the value of the expression:

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

where \( t_1, t_2, t_3 \) are points on the unit circle, and \( \theta_1, \theta_2, \theta_3 \) are their respective arguments.

### Step 1: Understanding the Geometry
Since \( t_1, t_2, t_3 \) are distinct points on the unit circle, we can represent them in terms of their angles \( \theta_1, \theta_2, \theta_3 \). The angles will be spaced around the circle.

### Step 2: Considering Equidistant Points
To simplify our calculations, we can consider the case where the points are equidistant on the circle. For three points on the circle, the angles can be taken as:

\[ \theta_1 = 0, \quad \theta_2 = \frac{2\pi}{3}, \quad \theta_3 = \frac{4\pi}{3} \]

This gives us three points that are equally spaced, each separated by an angle of \( \frac{2\pi}{3} \).

### Step 3: Calculating the Cosine Differences
Now we can compute the cosine differences:

1. \( \cos(\theta_1 - \theta_2) = \cos(0 - \frac{2\pi}{3}) = \cos(-\frac{2\pi}{3}) = -\frac{1}{2} \)
2. \( \cos(\theta_2 - \theta_3) = \cos(\frac{2\pi}{3} - \frac{4\pi}{3}) = \cos(-\frac{2\pi}{3}) = -\frac{1}{2} \)
3. \( \cos(\theta_3 - \theta_1) = \cos(\frac{4\pi}{3} - 0) = \cos(\frac{4\pi}{3}) = -\frac{1}{2} \)

### Step 4: Summing the Results
Now we sum these results:

\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) = -\frac{1}{2} - \frac{1}{2} - \frac{1}{2} = -\frac{3}{2} \]

### Conclusion
Thus, the value of the expression is:

\[ \boxed{-\frac{3}{2}} \]