If `5 sin theta - 4 cos theta = 0, 0^(@) lt theta lt 90^(@)`, then the value of `(5sintheta-2costheta)/(sintheta+3costheta)` is :

Options

A. `(5)/(8)`
B. `(2)/(7)`
C. `(3)/(8)`
D. `(3)/(7)`

Answer

To solve the equation \( 5 \sin \theta - 4 \cos \theta = 0 \) for \( 0^\circ < \theta < 90^\circ \) and find the value of \( \frac{5 \sin \theta - 2 \cos \theta}{\sin \theta + 3 \cos \theta} \), follow these steps:

### Step 1: Solve for \( \sin \theta \) and \( \cos \theta \)
From the equation \( 5 \sin \theta - 4 \cos \theta = 0 \), we can rearrange it to find the relationship between \( \sin \theta \) and \( \cos \theta \):

\[ 5 \sin \theta = 4 \cos \theta \]

Dividing both sides by \( \cos \theta \) (since \( \cos \theta \neq 0 \) in the given range):

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

This implies:

\[ \tan \theta = \frac{4}{5} \]

### Step 2: Express \( \sin \theta \) and \( \cos \theta \) in terms of \( \tan \theta \)
Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can express \( \sin \theta \) and \( \cos \theta \) in terms of \( k \):

Let \( \sin \theta = 4k \) and \( \cos \theta = 5k \) for some \( k \).

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):

\[ (4k)^2 + (5k)^2 = 1 \]
\[ 16k^2 + 25k^2 = 1 \]
\[ 41k^2 = 1 \]
\[ k^2 = \frac{1}{41} \]
\[ k = \frac{1}{\sqrt{41}} \]

Thus,

\[ \sin \theta = \frac{4}{\sqrt{41}}, \quad \cos \theta = \frac{5}{\sqrt{41}} \]

### Step 3: Substitute \( \sin \theta \) and \( \cos \theta \) into the expression
Now substitute \( \sin \theta \) and \( \cos \theta \) into the expression \( \frac{5 \sin \theta - 2 \cos \theta}{\sin \theta + 3 \cos \theta} \):

Numerator:

\[ 5 \sin \theta - 2 \cos \theta = 5 \left(\frac{4}{\sqrt{41}}\right) - 2 \left(\frac{5}{\sqrt{41}}\right) \]
\[ = \frac{20}{\sqrt{41}} - \frac{10}{\sqrt{41}} = \frac{10}{\sqrt{41}} \]

Denominator:

\[ \sin \theta + 3 \cos \theta = \frac{4}{\sqrt{41}} + 3 \left(\frac{5}{\sqrt{41}}\right) \]
\[ = \frac{4}{\sqrt{41}} + \frac{15}{\sqrt{41}} = \frac{19}{\sqrt{41}} \]

### Step 4: Simplify the expression
Now, substituting the numerator and denominator back into the expression:

\[ \frac{5 \sin \theta - 2 \cos \theta}{\sin \theta + 3 \cos \theta} = \frac{\frac{10}{\sqrt{41}}}{\frac{19}{\sqrt{41}}} \]

The \( \sqrt{41} \) cancels out:

\[ = \frac{10}{19} \]

### Final Answer
Thus, the value of \( \frac{5 \sin \theta - 2 \cos \theta}{\sin \theta + 3 \cos \theta} \) is:

\[ \frac{10}{19} \]