Question

For an acute angle \(\theta\), if \(\sin \theta = \frac{1}{9}\), then value of \(\frac{9 \csc \theta + 1}{9 \csc \theta - 1}\) is

• A. \(0\)
• B. \(\frac{80}{81}\)
• C. \(1\)
• D. \(\frac{82}{80}\)

Hint:

Direct substitution is fastest here. Don't waste time trying to find \(\cos \theta\) or other ratios.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
\(\csc \theta\) is the reciprocal of \(\sin \theta\). If \(\sin \theta = \frac{1}{x}\), then \(\csc \theta = x\).
Step 2: Detailed Explanation:
Given: \(\sin \theta = \frac{1}{9}\)
Therefore, \(\csc \theta = \frac{1}{\sin \theta} = 9\).
Now, evaluate the given expression:
\[ \frac{9 \csc \theta + 1}{9 \csc \theta - 1} \]
Substitute \(\csc \theta = 9\):
\[ = \frac{9(9) + 1}{9(9) - 1} \]
\[ = \frac{81 + 1}{81 - 1} \]
\[ = \frac{82}{80} \]
Step 3: Final Answer:
The value is \(\frac{82}{80}\).