Find the domain of
`y=sqrt(log_(3)(cos(sinx)))`

Answer

To find the domain of the function \( y = \sqrt{\log_{3}(\cos(\sin x))} \), we need to ensure that the expression inside the square root is non-negative and that the logarithm is defined.

### Step-by-Step Solution:

1. **Identify the conditions for the square root**:
The expression inside the square root must be greater than or equal to zero:
\[ \log_{3}(\cos(\sin x)) \geq 0 \]

2. **Convert the logarithmic inequality**:
The inequality \( \log_{3}(\cos(\sin x)) \geq 0 \) implies:
\[ \cos(\sin x) \geq 3^{0} = 1 \]

3. **Analyze the cosine function**:
The cosine function, \( \cos(\theta) \), has a maximum value of 1. Therefore, \( \cos(\sin x) = 1 \) is the only solution that satisfies the inequality \( \cos(\sin x) \geq 1 \).

4. **Set up the equation**:
We need to find when \( \cos(\sin x) = 1 \). The cosine function equals 1 at integer multiples of \( 2\pi \):
\[ \sin x = 2n\pi \quad \text{for } n \in \mathbb{Z} \]

5. **Find the values of \( x \)**:
The sine function equals 0 at integer multiples of \( \pi \):
\[ x = n\pi \quad \text{for } n \in \mathbb{Z} \]

6. **Conclusion**:
Thus, the domain of the function \( y = \sqrt{\log_{3}(\cos(\sin x))} \) is:
\[ x \in \{ n\pi \mid n \in \mathbb{Z} \} \]

### Final Answer:
The domain of \( y = \sqrt{\log_{3}(\cos(\sin x))} \) is \( x = n\pi \) where \( n \) is any integer.