The number of solutions of `2 cosx=|sinx|, 0 le x le 4 pi,` is (a) `0` (b) `2` (c) `4` (d) infinite

Options

A. 0
B. 2
C. 4
D. infinite

Explanation

To solve the equation \( 2 \cos x = |\sin x| \) for \( 0 \leq x \leq 4\pi \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2 \cos x = |\sin x| \]
This means that we need to consider two cases for \( |\sin x| \):
1. \( \sin x \) when \( \sin x \geq 0 \)
2. \( -\sin x \) when \( \sin x < 0 \)

### Step 2: Analyze the first case
For the first case, where \( \sin x \geq 0 \):
\[ 2 \cos x = \sin x \]
This can be rearranged to:
\[ 2 \cos x - \sin x = 0 \]
or
\[ \tan x = 2 \cos x \]
This equation can be solved within the interval \( 0 \leq x \leq 4\pi \).

### Step 3: Analyze the second case
For the second case, where \( \sin x < 0 \):
\[ 2 \cos x = -\sin x \]
This can be rearranged to:
\[ 2 \cos x + \sin x = 0 \]
or
\[ \tan x = -2 \cos x \]
This equation can also be solved within the interval \( 0 \leq x \leq 4\pi \).

### Step 4: Graphical Interpretation
To find the number of solutions, we can graph both sides of the equation:
- The left side \( y = 2 \cos x \) oscillates between -2 and 2.
- The right side \( y = |\sin x| \) oscillates between 0 and 1.

### Step 5: Finding intersections
1. **For \( 2 \cos x = \sin x \)**:
- The maximum value of \( 2 \cos x \) is 2, which intersects \( |\sin x| \) at points where \( \sin x \) reaches its maximum.
- The intersections occur at \( x = 0, \pi, 2\pi, 3\pi, 4\pi \) within the interval \( 0 \leq x \leq 4\pi \).

2. **For \( 2 \cos x = -\sin x \)**:
- The intersections occur at points where \( \sin x \) is negative, which happens between \( \pi \) and \( 2\pi \), and between \( 3\pi \) and \( 4\pi \).

### Step 6: Count the solutions
By analyzing the graphs and the equations, we find that:
- From \( 0 \) to \( 4\pi \), there are 4 points where \( 2 \cos x = |\sin x| \) intersects.

Thus, the total number of solutions is:
\[ \boxed{4} \]