If `5tantheta =4," then " (5sintheta-3costheta)/(5sintheta+2costheta)` is equal to

Options

A. 0
B. 1
C. `1/6`
D. 6

Explanation

To solve the problem, we start with the given equation and manipulate it step by step.

**Step 1: Start with the given equation.**
We know that:
\[ 5 \tan \theta = 4 \]
From this, we can find:
\[ \tan \theta = \frac{4}{5} \]

**Step 2: Rewrite the expression we need to evaluate.**
We need to find the value of:
\[ \frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta} \]

**Step 3: Divide both the numerator and the denominator by \(\cos \theta\).**
This gives us:
\[ \frac{\frac{5 \sin \theta}{\cos \theta} - \frac{3 \cos \theta}{\cos \theta}}{\frac{5 \sin \theta}{\cos \theta} + \frac{2 \cos \theta}{\cos \theta}} \]

**Step 4: Simplify the fractions.**
Using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we can rewrite the expression as:
\[ \frac{5 \tan \theta - 3}{5 \tan \theta + 2} \]

**Step 5: Substitute the value of \(\tan \theta\).**
Now, substitute \(\tan \theta = \frac{4}{5}\):
\[ \frac{5 \left(\frac{4}{5}\right) - 3}{5 \left(\frac{4}{5}\right) + 2} \]

**Step 6: Simplify the expression.**
Calculating the numerator:
\[ 5 \left(\frac{4}{5}\right) - 3 = 4 - 3 = 1 \]
Calculating the denominator:
\[ 5 \left(\frac{4}{5}\right) + 2 = 4 + 2 = 6 \]

**Step 7: Write the final result.**
Thus, we have:
\[ \frac{1}{6} \]

Therefore, the value of the expression \(\frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta}\) is \(\frac{1}{6}\).

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