Find the angle between the tangents
drawn from `(3, 2)` to the circle
` x^(2) + y^(2) - 6x + 4y - 2 = 0`

Answer

To find the angle between the tangents drawn from the point \( P(3, 2) \) to the circle given by the equation \( x^2 + y^2 - 6x + 4y - 2 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation of the circle in standard form
The given equation of the circle is:
\[ x^2 + y^2 - 6x + 4y - 2 = 0 \]
To rewrite it in standard form, we complete the square for \( x \) and \( y \).

1. For \( x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]

2. For \( y \):
\[ y^2 + 4y = (y + 2)^2 - 4 \]

Substituting these back into the equation:
\[ (x - 3)^2 - 9 + (y + 2)^2 - 4 - 2 = 0 \]
\[ (x - 3)^2 + (y + 2)^2 - 15 = 0 \]
\[ (x - 3)^2 + (y + 2)^2 = 15 \]

This shows that the center of the circle is \( O(3, -2) \) and the radius \( r = \sqrt{15} \).

### Step 2: Calculate the distance \( OP \)
Now we calculate the distance \( OP \) from the point \( P(3, 2) \) to the center \( O(3, -2) \):
\[ OP = \sqrt{(3 - 3)^2 + (2 - (-2))^2} = \sqrt{0 + (2 + 2)^2} = \sqrt{4^2} = 4 \]

### Step 3: Use the tangent length formula
The length of the tangents \( PA \) from the point \( P \) to the circle can be calculated using the formula:
\[ PA = \sqrt{OP^2 - r^2} \]
Substituting the values we have:
\[ PA = \sqrt{4^2 - (\sqrt{15})^2} = \sqrt{16 - 15} = \sqrt{1} = 1 \]

### Step 4: Find the angle between the tangents
The angle \( \theta \) between the two tangents can be found using the formula:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{PA}{r} \]
Substituting the values:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{1}{\sqrt{15}} \]

Now, we can find \( \theta \):
\[ \frac{\theta}{2} = \tan^{-1}\left(\frac{1}{\sqrt{15}}\right) \]
Thus,
\[ \theta = 2 \tan^{-1}\left(\frac{1}{\sqrt{15}}\right) \]

### Final Answer
The angle between the tangents drawn from the point \( (3, 2) \) to the circle is:
\[ \theta = 2 \tan^{-1}\left(\frac{1}{\sqrt{15}}\right) \]