The set of values of x satisfying `["sin"("cos" x)] =-1` is `([.]` denotes the greatest integer function)

Options

A. `((4n + 1) pi, (4n + 3) pi), n in Z`
B. `[(n + 1) (pi)/(2), (4n +3)(pi)/(2)], n in Z`
C. `((4n + 1)(pi)/(2), (4n + 3)(pi)/(2)), n inZ`
D. none of these

Explanation

To solve the equation \(\lfloor \sin(\cos x) \rfloor = -1\), we need to find the values of \(x\) that satisfy this condition.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function, denoted by \(\lfloor y \rfloor\), gives the largest integer less than or equal to \(y\). For \(\lfloor \sin(\cos x) \rfloor = -1\), it implies:
\[ -1 \leq \sin(\cos x) < 0 \]

2. **Analyzing the Range of \(\sin(\cos x)\)**:
The function \(\cos x\) has a range of \([-1, 1]\). Therefore, we need to evaluate \(\sin(y)\) for \(y\) in the interval \([-1, 1]\).

3. **Finding the Values of \(\sin(y)\)**:
The sine function is negative in the interval \((-\frac{\pi}{2}, 0)\). We need to find the values of \(y = \cos x\) such that:
\[ -1 < \cos x < 0 \]
This means \(\cos x\) must be in the range \((-1, 0)\).

4. **Finding the Corresponding Angles**:
The cosine function is negative in the second and third quadrants. Therefore, we can find the intervals for \(x\):
- In the second quadrant, \(x\) is in the interval \((\frac{\pi}{2}, \pi)\).
- In the third quadrant, \(x\) is in the interval \((\pi, \frac{3\pi}{2})\).

5. **Generalizing the Solution**:
The general solutions for \(x\) can be expressed as:
\[ x = 2n\pi + \frac{\pi}{2} \quad \text{and} \quad x = 2n\pi + \frac{3\pi}{2} \quad \text{for any integer } n. \]

6. **Final Answer**:
Thus, the set of values of \(x\) satisfying \(\lfloor \sin(\cos x) \rfloor = -1\) is:
\[ x = 2n\pi + \frac{\pi}{2}, \quad 2n\pi + \frac{3\pi}{2}, \quad n \in \mathbb{Z} \]