If `tan theta = 4/3`,then the value of `(3 sin theta + 2cos theta)/(3 sin theta - 2 cos theta)` is

Options

A. 0.5
B. `-0.5`
C. 3
D. `-3.0`

Answer

To solve the problem, we start with the given information that \( \tan \theta = \frac{4}{3} \). We need to find the value of

\[ \frac{3 \sin \theta + 2 \cos \theta}{3 \sin \theta - 2 \cos \theta}. \]

### Step 1: Express \(\sin \theta\) and \(\cos \theta\) in terms of \(\tan \theta\)

Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can express \(\sin \theta\) and \(\cos \theta\) using the relationship of sine and cosine with tangent.

Let:
- \(\sin \theta = 4k\)
- \(\cos \theta = 3k\)

where \(k\) is a positive constant.

### Step 2: Calculate \(3 \sin \theta\) and \(2 \cos \theta\)

Now, substituting the values of \(\sin \theta\) and \(\cos \theta\):

\[ 3 \sin \theta = 3(4k) = 12k, \]
\[ 2 \cos \theta = 2(3k) = 6k. \]

### Step 3: Substitute into the expression

Now we can substitute these into the original expression:

\[ \frac{3 \sin \theta + 2 \cos \theta}{3 \sin \theta - 2 \cos \theta} = \frac{12k + 6k}{12k - 6k} = \frac{18k}{6k}. \]

### Step 4: Simplify the expression

Now we simplify the fraction:

\[ \frac{18k}{6k} = 3. \]

### Conclusion

Thus, the value of

\[ \frac{3 \sin \theta + 2 \cos \theta}{3 \sin \theta - 2 \cos \theta} = 3. \]

### Final Answer

The correct answer is option 3, which is \(3\).

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