To find the number of solutions for the equation \( | \cos x | = \cos x - 2 \sin x \) in the interval \([0, 6\pi]\), we will analyze the equation by considering two cases based on the definition of the absolute value.
### Step 1: Case 1 - \( \cos x \geq 0 \)
In this case, the absolute value function can be removed without changing the sign:
\[
| \cos x | = \cos x
\]
Thus, the equation becomes:
\[
\cos x = \cos x - 2 \sin x
\]
Subtracting \( \cos x \) from both sides gives:
\[
0 = -2 \sin x
\]
This simplifies to:
\[
\sin x = 0
\]
The general solutions for \( \sin x = 0 \) are:
\[
x = n\pi \quad \text{where } n \text{ is an integer}
\]
In the interval \([0, 6\pi]\), the values of \( n \) can be \( 0, 1, 2, 3, 4, 5, 6 \), which gives us the solutions:
\[
x = 0, \pi, 2\pi, 3\pi, 4\pi, 5\pi, 6\pi
\]
This results in **7 solutions**.
### Step 2: Case 2 - \( \cos x < 0 \)
In this case, the absolute value function changes the sign:
\[
| \cos x | = -\cos x
\]
Thus, the equation becomes:
\[
-\cos x = \cos x - 2 \sin x
\]
Rearranging gives:
\[
-\cos x - \cos x = -2 \sin x
\]
This simplifies to:
\[
-2\cos x = -2 \sin x
\]
Dividing both sides by -2 (assuming \( \sin x \neq 0 \)) results in:
\[
\cos x = \sin x
\]
This implies:
\[
\tan x = 1
\]
The general solutions for \( \tan x = 1 \) are:
\[
x = \frac{\pi}{4} + n\pi \quad \text{where } n \text{ is an integer}
\]
Now, we need to find the solutions in the interval \([0, 6\pi]\):
1. For \( n = 0 \): \( x = \frac{\pi}{4} \)
2. For \( n = 1 \): \( x = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \)
3. For \( n = 2 \): \( x = \frac{\pi}{4} + 2\pi = \frac{9\pi}{4} \)
4. For \( n = 3 \): \( x = \frac{\pi}{4} + 3\pi = \frac{13\pi}{4} \)
5. For \( n = 4 \): \( x = \frac{\pi}{4} + 4\pi = \frac{17\pi}{4} \)
Now we check which of these values fall into the interval \([0, 6\pi]\):
- \( \frac{\pi}{4} \) is valid.
- \( \frac{5\pi}{4} \) is valid.
- \( \frac{9\pi}{4} \) is valid.
- \( \frac{13\pi}{4} \) is valid.
- \( \frac{17\pi}{4} \) is valid.
Thus, we have **5 solutions** from this case.
### Final Count of Solutions
Adding the solutions from both cases:
- From Case 1: 7 solutions
- From Case 2: 5 solutions
Total solutions = \( 7 + 5 = 12 \)
### Conclusion
The number of solutions of the equation \( | \cos x | = \cos x - 2 \sin x \) in the interval \([0, 6\pi]\) is **12**.
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