The number of solutions of the equation `"sin x = |"cos" 3x| "in" [0, pi]`, is

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To find the number of solutions of the equation \( \sin x = |\cos 3x| \) in the interval \( [0, \pi] \), we will analyze the graphs of \( \sin x \) and \( |\cos 3x| \) step by step.

### Step 1: Understand the functions involved
1. **Function \( \sin x \)**:
- The function \( \sin x \) varies from 0 to 1 in the interval \( [0, \pi] \).
- It starts at \( \sin(0) = 0 \), reaches its maximum at \( \sin(\frac{\pi}{2}) = 1 \), and returns to \( \sin(\pi) = 0 \).

2. **Function \( \cos 3x \)**:
- The function \( \cos 3x \) oscillates between -1 and 1 with a period of \( \frac{2\pi}{3} \).
- In the interval \( [0, \pi] \), it will complete one and a half cycles.

### Step 2: Analyze \( |\cos 3x| \)
- Since we are interested in \( |\cos 3x| \), we will take the absolute value of \( \cos 3x \).
- The graph of \( |\cos 3x| \) will reflect any negative values of \( \cos 3x \) above the x-axis, creating a wave-like pattern that oscillates between 0 and 1.

### Step 3: Plot the graphs
- **Graph of \( \sin x \)**:
- Starts at (0,0), peaks at \( (\frac{\pi}{2}, 1) \), and returns to (π,0).

- **Graph of \( |\cos 3x| \)**:
- Starts at (0,1), goes down to (π/6, 0), reaches (π/3, 1), then down to (π/2, 0), up to (2π/3, 1), down to (5π/6, 0), and finally back to (π, 1).

### Step 4: Identify intersection points
- We need to find the points where the two graphs intersect, i.e., where \( \sin x = |\cos 3x| \).
- By observing the graphs, we can count the number of intersection points:
- The graphs intersect at 6 points in total within the interval \( [0, \pi] \).

### Conclusion
Thus, the number of solutions to the equation \( \sin x = |\cos 3x| \) in the interval \( [0, \pi] \) is **6**.