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 to 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 \(|\cos x|\) and \(\sin x\). The function \(|\cos x|\) will take the positive values of \(\cos x\) and reflect the negative values above the x-axis.

### Step 2: Graph the Functions
1. **Graph of \(\sin x\)**: The sine function oscillates between -1 and 1, completing one full cycle from \(0\) to \(2\pi\). In the interval \(0 < x < 4\pi\), it will complete two full cycles.

2. **Graph of \(|\cos x|\)**: The cosine function also oscillates between -1 and 1. However, since we are looking at \(|\cos x|\), all negative values will be reflected above the x-axis. The graph of \(|\cos x|\) will have peaks at \(0\), \(\pi\), \(2\pi\), \(3\pi\), and \(4\pi\) and will touch the x-axis at \(\frac{\pi}{2}\), \(\frac{3\pi}{2}\), \(\frac{5\pi}{2}\), and \(\frac{7\pi}{2}\).

### Step 3: Find Intersections
To find the number of solutions to \(|\cos x| = \sin x\), we need to find the points where the graphs of \(|\cos x|\) and \(\sin x\) intersect.

1. **First Cycle (from \(0\) to \(2\pi\))**:
- The graphs intersect at four points:
- Between \(0\) and \(\frac{\pi}{2}\)
- At \(\frac{\pi}{2}\)
- Between \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\)
- At \(\frac{3\pi}{2}\)

2. **Second Cycle (from \(2\pi\) to \(4\pi\))**:
- The same reasoning applies, and we will again find four intersections.

### Step 4: Count the Solutions
Adding the intersections from both cycles:
- From \(0\) to \(2\pi\): 4 solutions
- From \(2\pi\) to \(4\pi\): 4 solutions

### Final Count
Thus, the total number of solutions in the interval \(0 < x < 4\pi\) is:
\[ \text{Total Solutions} = 4 + 4 = 8 \]

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