If `cot theta = 4,` then the velue of `(5 sin theta + 3 cos theta)/(5 sin theta - 3 cos theta )`

Options

A. `1/9`
B. `1/3`
C. `3`
D. 9

Answer

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

**Given:** \( \cot \theta = 4 \)

We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
From this, we can express \( \cos \theta \) in terms of \( \sin \theta \):
\[ \cos \theta = 4 \sin \theta \]

Now, we need to find the value of:
\[ \frac{5 \sin \theta + 3 \cos \theta}{5 \sin \theta - 3 \cos \theta} \]

**Step 1:** Substitute \( \cos \theta \) in the expression.

Using \( \cos \theta = 4 \sin \theta \), we can substitute this into the expression:
\[ \frac{5 \sin \theta + 3(4 \sin \theta)}{5 \sin \theta - 3(4 \sin \theta)} \]

**Step 2:** Simplify the numerator and the denominator.

The numerator becomes:
\[ 5 \sin \theta + 12 \sin \theta = 17 \sin \theta \]

The denominator becomes:
\[ 5 \sin \theta - 12 \sin \theta = -7 \sin \theta \]

So, we can rewrite the expression as:
\[ \frac{17 \sin \theta}{-7 \sin \theta} \]

**Step 3:** Cancel \( \sin \theta \) (assuming \( \sin \theta \neq 0 \)).

This simplifies to:
\[ \frac{17}{-7} = -\frac{17}{7} \]

Thus, the final answer is:
\[ -\frac{17}{7} \]

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