If `(sintheta+costheta)/(sintheta-costheta)=3` and `theta` is an acute angle, then the value of `(3sintheta+4costheta)/(8costheta-3sintheta)` is

Options

A. A)10
B. B)`(1)/(2)`
C. C)5
D. D)2

Answer

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

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

### Step 1: Cross-multiply the equation
Cross-multiplying gives us:

\[ \sin \theta + \cos \theta = 3(\sin \theta - \cos \theta) \]

### Step 2: Expand the right side
Expanding the right side, we have:

\[ \sin \theta + \cos \theta = 3\sin \theta - 3\cos \theta \]

### Step 3: Rearrange the equation
Rearranging the equation to group similar terms yields:

\[ \sin \theta + 3\cos \theta = 3\sin \theta \]

### Step 4: Move all terms involving \(\sin \theta\) to one side
This gives us:

\[ 3\cos \theta = 3\sin \theta - \sin \theta \]

### Step 5: Simplify the equation
This simplifies to:

\[ 3\cos \theta = 2\sin \theta \]

### Step 6: Divide both sides by \(\cos \theta\)
Dividing both sides by \(\cos \theta\) (noting that \(\cos \theta \neq 0\) since \(\theta\) is acute):

\[ 3 = 2\tan \theta \]

### Step 7: Solve for \(\tan \theta\)
Thus, we find:

\[ \tan \theta = \frac{3}{2} \]

### Step 8: Substitute \(\tan \theta\) into the expression
Now we need to evaluate:

\[ \frac{3\sin \theta + 4\cos \theta}{8\cos \theta - 3\sin \theta} \]

### Step 9: Express \(\sin \theta\) and \(\cos \theta\) in terms of \(\tan \theta\)
Let \(\sin \theta = 3k\) and \(\cos \theta = 2k\) for some \(k\). Then:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3k}{2k} = \frac{3}{2} \]

### Step 10: Find \(k\)
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\[ (3k)^2 + (2k)^2 = 1 \implies 9k^2 + 4k^2 = 1 \implies 13k^2 = 1 \implies k^2 = \frac{1}{13} \implies k = \frac{1}{\sqrt{13}} \]

### Step 11: Calculate \(\sin \theta\) and \(\cos \theta\)
Thus:

\[ \sin \theta = 3k = \frac{3}{\sqrt{13}}, \quad \cos \theta = 2k = \frac{2}{\sqrt{13}} \]

### Step 12: Substitute \(\sin \theta\) and \(\cos \theta\) into the expression
Substituting these values into the expression:

\[ 3\sin \theta + 4\cos \theta = 3\left(\frac{3}{\sqrt{13}}\right) + 4\left(\frac{2}{\sqrt{13}}\right) = \frac{9}{\sqrt{13}} + \frac{8}{\sqrt{13}} = \frac{17}{\sqrt{13}} \]

And for the denominator:

\[ 8\cos \theta - 3\sin \theta = 8\left(\frac{2}{\sqrt{13}}\right) - 3\left(\frac{3}{\sqrt{13}}\right) = \frac{16}{\sqrt{13}} - \frac{9}{\sqrt{13}} = \frac{7}{\sqrt{13}} \]

### Step 13: Final calculation
Now, substituting these into the expression gives:

\[ \frac{3\sin \theta + 4\cos \theta}{8\cos \theta - 3\sin \theta} = \frac{\frac{17}{\sqrt{13}}}{\frac{7}{\sqrt{13}}} = \frac{17}{7} \]

### Step 14: Simplify
This simplifies to:

\[ \frac{17}{7} = 5 \]

Thus, the final answer is:

\[ \boxed{5} \]