If `cos^(-1)|sinx|gesin^(-1)|sinx|`, then the number of integral values of x in the interval `x in [0, 3pi]` are

Options

A. 7
B. 6
C. 4
D. 5

Answer

To solve the inequality \( \cos^{-1} |\sin x| \geq \sin^{-1} |\sin x| \), we will follow these steps:

### Step 1: Understand the Functions
The functions \( \cos^{-1} y \) and \( \sin^{-1} y \) are defined for \( y \) in the range \([-1, 1]\). The function \( |\sin x| \) will also be in the range \([0, 1]\) since it represents the absolute value of the sine function.

### Step 2: Set Up the Inequality
We need to analyze the inequality:
\[ \cos^{-1} |\sin x| \geq \sin^{-1} |\sin x| \]

### Step 3: Use Known Values
From trigonometric identities, we know that:
\[ \cos^{-1} y + \sin^{-1} y = \frac{\pi}{2} \quad \text{for } y \in [0, 1] \]
Thus, we can rewrite the inequality:
\[ \cos^{-1} |\sin x| \geq \sin^{-1} |\sin x| \implies \cos^{-1} |\sin x| \geq \frac{\pi}{2} - \cos^{-1} |\sin x| \]
This simplifies to:
\[ 2\cos^{-1} |\sin x| \geq \frac{\pi}{2} \]
or
\[ \cos^{-1} |\sin x| \geq \frac{\pi}{4} \]

### Step 4: Solve for \( |\sin x| \)
Taking the cosine of both sides, we get:
\[ |\sin x| \leq \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
This means:
\[ |\sin x| \leq \frac{1}{\sqrt{2}} \]

### Step 5: Find the Values of \( x \)
The sine function \( \sin x \) achieves the value \( \frac{1}{\sqrt{2}} \) at:
\[ x = \frac{\pi}{4} + n\pi \quad \text{and} \quad x = \frac{3\pi}{4} + n\pi \quad \text{for integers } n \]
Thus, the intervals where \( |\sin x| \leq \frac{1}{\sqrt{2}} \) are:
\[ x \in \left[0, \frac{\pi}{4}\right] \cup \left[\frac{3\pi}{4}, \frac{5\pi}{4}\right] \cup \left[\frac{7\pi}{4}, 2\pi\right] \]
And similarly for the next cycle up to \( 3\pi \).

### Step 6: Count Integral Values
Now we need to find the integral values of \( x \) in the interval \( [0, 3\pi] \):
- From \( 0 \) to \( \frac{\pi}{4} \): \( 0 \)
- From \( \frac{3\pi}{4} \) to \( \frac{5\pi}{4} \): \( 1 \) (only \( 1 \))
- From \( \frac{7\pi}{4} \) to \( 2\pi \): \( 2, 3, 4, 5, 6 \)
- From \( 2\pi \) to \( 3\pi \): \( 6, 7, 8, 9 \)

The integral values are \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \).

### Conclusion
The total number of integral values of \( x \) in the interval \( [0, 3\pi] \) is **10**.