If, 5cot`theta`=3 ,Find the value of `(6sintheta-3costheta)/(7sintheta+3costheta)`

Options

A. `(20)/(41)`
B. `(44)/(21)`
C. `(11)/(40)`
D. `(21)/(44)`

Answer

To solve the problem, we start with the given equation:

1. **Given**: \( 5 \cot \theta = 3 \)

From this, we can express \( \cot \theta \):
\[ \cot \theta = \frac{3}{5} \]

2. **Using the definition of cotangent**:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Therefore, we have:
\[ \frac{\cos \theta}{\sin \theta} = \frac{3}{5} \]
This implies:
\[ \cos \theta = 3k \quad \text{and} \quad \sin \theta = 5k \]
for some positive constant \( k \).

3. **Finding the value of the expression**:
We need to evaluate:
\[ \frac{6 \sin \theta - 3 \cos \theta}{7 \sin \theta + 3 \cos \theta} \]

Substituting the values of \( \sin \theta \) and \( \cos \theta \):
\[ = \frac{6(5k) - 3(3k)}{7(5k) + 3(3k)} \]

4. **Calculating the numerator**:
\[ = \frac{30k - 9k}{35k + 9k} \]
\[ = \frac{21k}{44k} \]

5. **Simplifying the expression**:
The \( k \) cancels out:
\[ = \frac{21}{44} \]

Thus, the final answer is:
\[ \frac{21}{44} \]