If `(a cos theta_(1), a sin theta_(1)), ( a cos theta_(2), a sin theta_(2)), (a costheta_(3), a sin theta_(3))` represents the vertices of an equilateral triangle inscribed in `x^(2) + y^(2) = a^(2)`, then

Options

A. `sum costheta_(i)=0`
B. ` sum sin theta_(i) = 0`
C. ` sum tan theta_(i) = 0`
D. `sum cot theta_(i) = 0`

Answer

To solve the problem, we need to find the conditions for the angles \(\theta_1\), \(\theta_2\), and \(\theta_3\) such 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 the circle defined by the equation \(x^2 + y^2 = a^2\).

### Step-by-Step Solution:

1. **Understanding the Circle and Vertices**:
The given equation \(x^2 + y^2 = a^2\) represents a circle centered at the origin with radius \(a\). The points \((a \cos \theta_1, a \sin \theta_1)\), \((a \cos \theta_2, a \sin \theta_2)\), and \((a \cos \theta_3, a \sin \theta_3)\) lie on this circle.

2. **Finding the Centroid**:
The centroid \(G\) of the triangle formed by these three points is given by:
\[ G_x = \frac{a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3}{3} \]
\[ G_y = \frac{a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3}{3} \]

3. **Setting the Centroid to the Origin**:
Since the triangle is inscribed in the circle and is equilateral, the centroid coincides with the circumcenter, which is the origin (0, 0). Therefore, we set both coordinates of the centroid to zero:
\[ G_x = 0 \quad \text{and} \quad G_y = 0 \]

4. **Equating the X-Coordinate to Zero**:
From \(G_x = 0\):
\[ \frac{a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3}{3} = 0 \]
This simplifies to:
\[ \cos \theta_1 + \cos \theta_2 + \cos \theta_3 = 0 \]

5. **Equating the Y-Coordinate to Zero**:
From \(G_y = 0\):
\[ \frac{a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3}{3} = 0 \]
This simplifies to:
\[ \sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 0 \]

6. **Conclusion**:
The conditions derived from the centroid being at the origin lead us to the following results:
- \(\cos \theta_1 + \cos \theta_2 + \cos \theta_3 = 0\)
- \(\sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 0\)

Thus, the correct options are that the summation of the cosines and sines of the angles equals zero.