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:

Always convert complex trigonometric expressions into their basic counterparts.
Since $\csc\theta = \frac{1}{\sin\theta}$, substituting this directly yields $\frac{\frac{9}{\sin\theta} + 1}{\frac{9}{\sin\theta} - 1} = \frac{9 + \sin\theta}{9 - \sin\theta}$, which is extremely quick to compute!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the sine of an acute angle, $\sin\theta = \frac{1}{9}$.
We need to calculate the value of the algebraic trigonometric expression:
\[ \frac{9\csc\theta + 1}{9\csc\theta - 1} \]

Step 2: Key Formula or Approach:
The cosecant function ($\csc\theta$) is the reciprocal of the sine function ($\sin\theta$):
\[ \csc\theta = \frac{1}{\sin\theta} \]
We can compute the value of $\csc\theta$ directly and substitute it into the given expression.

Step 3: Detailed Explanation:

• Given:
\[ \sin\theta = \frac{1}{9} \]

• Find the value of $\csc\theta$ using the reciprocal identity:
\[ \csc\theta = \frac{1}{\sin\theta} = \frac{1}{\frac{1}{9}} = 9 \]

• Write down the target expression:
\[ \text{Expression} = \frac{9\csc\theta + 1}{9\csc\theta - 1} \]

• Substitute $\csc\theta = 9$ into the expression:
\[ \text{Expression} = \frac{9(9) + 1}{9(9) - 1} \]
\[ \text{Expression} = \frac{81 + 1}{81 - 1} \]
\[ \text{Expression} = \frac{82}{80} \]


Step 4: Final Answer:
The value of the expression is $\frac{82}{80}$.