If `(a cos theta, a sin theta,) I =1,2,3` represent the vertices of an equilateral triange inscribed in a circle, then

Options

A. `cos theta _(1) + cos theta _(2) + cos theta _(3) =0`
B. `sin theta _(1) + sin theta _(2) + sin theta _(3) ne 0`
C. `tan theta _(1) + tan theta _(2) + tan theta _(3) =0`
D. `cot theta _(1) + cot theta _(2) + cot theta _(3) =0`

Answer

To solve the problem, we need to find the value of \( \cos \theta_1 + \cos \theta_2 + \cos \theta_3 \) given that the points \( (a \cos \theta_1, a \sin \theta_1) \), \( (a \cos \theta_2, a \sin \theta_2) \), and \( (a \cos \theta_3, a \sin \theta_3) \) represent the vertices of an equilateral triangle inscribed in a circle.

### Step-by-Step Solution:

1. **Identify the Circle and Triangle Properties**:
The vertices of the equilateral triangle are given in polar coordinates as \( (a \cos \theta_i, a \sin \theta_i) \) for \( i = 1, 2, 3 \). Since these points lie on a circle, we can assume the circle has a center at the origin (0, 0) and a radius \( a \).

2. **Use the Centroid and Circumcenter Relationship**:
For an equilateral triangle, the centroid (which is also the circumcenter) is located at the average of the vertices. Therefore, the centroid \( (G_x, G_y) \) can be calculated as follows:
\[ G_x = \frac{a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3}{3} \]
\[ G_y = \frac{a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3}{3} \]

3. **Set the Centroid to the Origin**:
Since the centroid of the triangle is at the origin (0, 0), we set the equations for \( G_x \) and \( G_y \) to zero:
\[ \frac{a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3}{3} = 0 \]
\[ \frac{a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3}{3} = 0 \]

4. **Simplify the Equations**:
Multiplying both equations by 3 gives:
\[ a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3 = 0 \]
\[ a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3 = 0 \]

5. **Divide by \( a \)** (assuming \( a \neq 0 \)):
\[ \cos \theta_1 + \cos \theta_2 + \cos \theta_3 = 0 \]
\[ \sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 0 \]

6. **Conclusion**:
From the first equation, we find that:
\[ \cos \theta_1 + \cos \theta_2 + \cos \theta_3 = 0 \]

Thus, the value of \( \cos \theta_1 + \cos \theta_2 + \cos \theta_3 \) is **0**.