If `sqrt(3)tan2theta=3 and 0^(@) lt theta le 90^(@)`, find the value of `3sqrt(3)costheta+2sintheta-6tan^(2)theta`.

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Solve for \( \theta \)
Given the equation:
\[ \sqrt{3} \tan 2\theta = 3 \]
We can divide both sides by \( \sqrt{3} \):
\[ \tan 2\theta = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 2: Find the angle corresponding to \( \tan 2\theta = \sqrt{3} \)
We know that:
\[ \tan 60^\circ = \sqrt{3} \]
Thus, we can equate:
\[ 2\theta = 60^\circ \]
Now, divide both sides by 2:
\[ \theta = 30^\circ \]

### Step 3: Substitute \( \theta \) into the expression
We need to find the value of:
\[ 3\sqrt{3} \cos \theta + 2 \sin \theta - 6 \tan^2 \theta \]
Substituting \( \theta = 30^\circ \):
\[ 3\sqrt{3} \cos 30^\circ + 2 \sin 30^\circ - 6 \tan^2 30^\circ \]

### Step 4: Calculate the trigonometric values
We know:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus:
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 5: Substitute the values into the expression
Now substituting these values into the expression:
\[ 3\sqrt{3} \left(\frac{\sqrt{3}}{2}\right) + 2 \left(\frac{1}{2}\right) - 6 \left(\frac{1}{3}\right) \]

### Step 6: Simplify the expression
Calculating each term:
1. \( 3\sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{3 \cdot 3}{2} = \frac{9}{2} \)
2. \( 2 \cdot \frac{1}{2} = 1 \)
3. \( 6 \cdot \frac{1}{3} = 2 \)

Putting it all together:
\[ \frac{9}{2} + 1 - 2 \]
Convert \( 1 \) to a fraction:
\[ 1 = \frac{2}{2} \]
So,
\[ \frac{9}{2} + \frac{2}{2} - \frac{4}{2} = \frac{9 + 2 - 4}{2} = \frac{7}{2} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{7}{2} \]