The number of solution of |cos x| =sin x in the interval `0 lt x lt 4pi` is _________

Answer

To find the number of solutions of the equation |cos x| = sin x in the interval \(0 < x < 4\pi\), we can follow these steps:

### Step 1: Understand the Functions
We need to analyze the functions \(y = |cos x|\) and \(y = sin x\). The function \(sin x\) oscillates between -1 and 1, while \(|cos x|\) oscillates between 0 and 1.

### Step 2: Graph the Functions
1. **Graph of \(sin x\)**: The graph of \(sin x\) starts at 0, reaches 1 at \(x = \frac{\pi}{2}\), goes back to 0 at \(x = \pi\), reaches -1 at \(x = \frac{3\pi}{2}\), and returns to 0 at \(x = 2\pi\). This pattern repeats every \(2\pi\).

2. **Graph of \(|cos x|\)**: The graph of \(cos x\) oscillates between -1 and 1. The graph of \(|cos x|\) will reflect the negative part of \(cos x\) above the x-axis. Thus, it will oscillate between 0 and 1.

### Step 3: Identify the Intervals
In the interval \(0 < x < 4\pi\), we will have two complete cycles of \(sin x\) and \(|cos x|\):
- From \(0\) to \(2\pi\)
- From \(2\pi\) to \(4\pi\)

### Step 4: Find Intersections
1. **First Cycle \(0 < x < 2\pi\)**:
- The graphs intersect where \(sin x = |cos x|\).
- The points of intersection occur when:
- \(sin x = cos x\) → \(x = \frac{\pi}{4}\) and \(x = \frac{5\pi}{4}\)
- \(sin x = -cos x\) → \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\)
- Thus, in the interval \(0 < x < 2\pi\), there are 4 intersections.

2. **Second Cycle \(2\pi < x < 4\pi\)**:
- The behavior of the functions is periodic, so the intersections will repeat in the same manner.
- Therefore, in the interval \(2\pi < x < 4\pi\), we again have 4 intersections.

### Step 5: Total Solutions
Adding the intersections from both cycles:
- First cycle: 4 solutions
- Second cycle: 4 solutions

Thus, the total number of solutions in the interval \(0 < x < 4\pi\) is \(4 + 4 = 8\).

### Final Answer
The number of solutions of \(|cos x| = sin x\) in the interval \(0 < x < 4\pi\) is **8**.

---