The domain of the function `f(x)=sqrt(log((1)/(|sinx|)))`

Options

A. `R-{-pi,pi}`
B. `R-{n pi|n in Z}`
C. `R-{2n pi|n in z}`
D. `(-oo,oo)`

Explanation

To find the domain of the function \( f(x) = \sqrt{\log\left(\frac{1}{|\sin x|}\right)} \), we need to ensure that the expression inside the square root is non-negative. This means we need to satisfy the following conditions:

1. The logarithm must be defined and greater than or equal to zero.
2. The argument of the logarithm must be positive.

### Step 1: Ensure the logarithm is defined
The logarithm function \( \log(a) \) is defined for \( a > 0 \). Therefore, we need:
\[ \frac{1}{|\sin x|} > 0 \]
This condition is satisfied as long as \( |\sin x| \neq 0 \).

### Step 2: Ensure the logarithm is non-negative
Next, we need the logarithm to be non-negative:
\[ \log\left(\frac{1}{|\sin x|}\right) \geq 0 \]
This implies:
\[ \frac{1}{|\sin x|} \geq 1 \]
Taking the reciprocal (and flipping the inequality since both sides are positive), we get:
\[ |\sin x| \leq 1 \]
This condition is always satisfied since the sine function oscillates between -1 and 1.

### Step 3: Find where \( |\sin x| \) is zero
The sine function is zero at:
\[ \sin x = 0 \implies x = n\pi \quad (n \in \mathbb{Z}) \]
Thus, we need to exclude these points from our domain.

### Step 4: State the domain
The domain of \( f(x) \) is all real numbers except where \( \sin x = 0 \):
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \{ n\pi \mid n \in \mathbb{Z} \} \]

### Final Answer
The domain of the function \( f(x) = \sqrt{\log\left(\frac{1}{|\sin x|}\right)} \) is:
\[ \text{Domain} = \mathbb{R} - \{ n\pi \mid n \in \mathbb{Z} \} \]

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