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

To solve the given problem, we need to calculate the value of the expression \(\frac{9 \csc \theta + 1}{9 \csc \theta - 1}\) for an acute angle \(\theta\) where \(\sin \theta = \frac{1}{9}\). Let's solve this step-by-step:

1. The cosecant function is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\).
2. Given that \(\sin \theta = \frac{1}{9}\), we find \(\csc \theta\) as follows: \(\csc \theta = \frac{1}{\frac{1}{9}} = 9\).
3. Substitute \(\csc \theta = 9\) into the expression \(\frac{9 \csc \theta + 1}{9 \csc \theta - 1}\): \(\frac{9 \times 9 + 1}{9 \times 9 - 1}\).
4. Calculate the numerator: \(9 \times 9 + 1 = 81 + 1 = 82\).
5. Calculate the denominator: \(9 \times 9 - 1 = 81 - 1 = 80\).
6. Substitute the calculated values back into the expression: \(\frac{82}{80}\).

Thus, the value of the expression \(\frac{9 \csc \theta + 1}{9 \csc \theta - 1}\) is \(\frac{82}{80}\).

This confirms that the correct answer is \(\frac{82}{80}\).