To find the distance between the centroid and the circumcenter of triangle \( ABC \) with vertices \( A \) and \( B \) given in the problem, we will follow these steps:
### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- \( A = (-3 + 2\sin\theta, 4 + 2\cos\theta) \)
- \( B = (-3 + 2\cos\theta, 4 - 2\cos\theta) \)
We need to find the coordinates of the third vertex \( C \). However, since \( C \) is not provided, we will assume it lies on a circle centered at \( (-3, 4) \) with a radius of 2.
### Step 2: Find the circumcenter of triangle \( ABC \)
The circumcenter of a triangle inscribed in a circle is the center of the circle. Therefore, the circumcenter \( O \) is:
\[
O = (-3, 4)
\]
### Step 3: Find the centroid of triangle \( ABC \)
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
Substituting the coordinates of \( A \) and \( B \):
\[
G = \left( \frac{(-3 + 2\sin\theta) + (-3 + 2\cos\theta) + x_3}{3}, \frac{(4 + 2\cos\theta) + (4 - 2\cos\theta) + y_3}{3} \right)
\]
Simplifying:
\[
G = \left( \frac{-6 + 2\sin\theta + 2\cos\theta + x_3}{3}, \frac{8 + y_3}{3} \right)
\]
### Step 4: Find the distance between the centroid \( G \) and circumcenter \( O \)
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting \( O = (-3, 4) \) and \( G \):
\[
d = \sqrt{\left( \frac{-6 + 2\sin\theta + 2\cos\theta + x_3 + 3}{3} \right)^2 + \left( \frac{8 + y_3 - 12}{3} \right)^2}
\]
This simplifies to:
\[
d = \sqrt{\left( \frac{-3 + 2\sin\theta + 2\cos\theta + x_3}{3} \right)^2 + \left( \frac{y_3 - 4}{3} \right)^2}
\]
### Step 5: Use the property of the circumradius and centroid
Since triangle \( ABC \) is inscribed in a circle of radius 2, we can use the properties of triangles to find the distance between the centroid and circumcenter. The distance between the centroid \( G \) and circumcenter \( O \) can be expressed as:
\[
d = \frac{R}{\sqrt{3}} \quad \text{where } R = 2
\]
Thus:
\[
d = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\]
### Final Answer
The distance between the centroid and the circumcenter of triangle \( ABC \) is:
\[
\frac{2\sqrt{3}}{3}
\]
