If `tantheta = frac{3}{4}` then `(4sintheta-costheta)/(4sintheta+costheta)` is equal. to:

Options

A. `3/5`
B. `1/4`
C. `1/2`
D. `2/5`

Answer

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

\[ \frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta} \]

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

Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can represent \(\sin \theta\) and \(\cos \theta\) using a right triangle. Let the opposite side be 3 and the adjacent side be 4. The hypotenuse can be calculated using the Pythagorean theorem:

\[ \text{Hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

From this triangle, we can find:

\[ \sin \theta = \frac{3}{5}, \quad \cos \theta = \frac{4}{5} \]

### Step 2: Substitute \(\sin \theta\) and \(\cos \theta\) into the expression

Now we substitute \(\sin \theta\) and \(\cos \theta\) into the expression:

\[ \frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta} = \frac{4 \left(\frac{3}{5}\right) - \frac{4}{5}}{4 \left(\frac{3}{5}\right) + \frac{4}{5}} \]

### Step 3: Simplify the numerator and denominator

Calculating the numerator:

\[ 4 \left(\frac{3}{5}\right) - \frac{4}{5} = \frac{12}{5} - \frac{4}{5} = \frac{12 - 4}{5} = \frac{8}{5} \]

Calculating the denominator:

\[ 4 \left(\frac{3}{5}\right) + \frac{4}{5} = \frac{12}{5} + \frac{4}{5} = \frac{12 + 4}{5} = \frac{16}{5} \]

### Step 4: Form the final expression

Now we have:

\[ \frac{\frac{8}{5}}{\frac{16}{5}} = \frac{8}{5} \cdot \frac{5}{16} = \frac{8}{16} = \frac{1}{2} \]

### Conclusion

Thus, the value of the expression \( \frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta} \) is:

\[ \frac{1}{2} \]