The number of values of `theta in [0,4pi]` satisfying the equation `abs(sqrt3 cos x - sin x) ge 2` is

Options

A. 0
B. 2
C. 4
D. 8

Answer

To solve the equation \( |\sqrt{3} \cos x - \sin x| \geq 2 \) for \( x \) in the interval \( [0, 4\pi] \), we will break it down step by step.

### Step 1: Remove the absolute value
The absolute value inequality \( |\sqrt{3} \cos x - \sin x| \geq 2 \) can be rewritten as two separate inequalities:
\[ \sqrt{3} \cos x - \sin x \geq 2 \quad \text{or} \quad \sqrt{3} \cos x - \sin x \leq -2 \]

### Step 2: Solve the first inequality
Starting with the first inequality:
\[ \sqrt{3} \cos x - \sin x \geq 2 \]
Rearranging gives:
\[ \sqrt{3} \cos x - 2 \geq \sin x \]
This can be rewritten as:
\[ \sqrt{3} \cos x - \sin x \geq 2 \]
Dividing through by 2:
\[ \frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x \geq 1 \]

### Step 3: Use the cosine addition formula
Recall that \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). We can express the left-hand side as:
\[ \cos\left(x + \frac{\pi}{6}\right) \geq 1 \]
This implies:
\[ x + \frac{\pi}{6} = 2n\pi \quad \text{for integers } n \]
Thus:
\[ x = 2n\pi - \frac{\pi}{6} \]

### Step 4: Find values for \( n \)
For \( n = 0 \):
\[ x = -\frac{\pi}{6} \quad (\text{not in } [0, 4\pi]) \]
For \( n = 1 \):
\[ x = 2\pi - \frac{\pi}{6} = \frac{12\pi}{6} - \frac{\pi}{6} = \frac{11\pi}{6} \]
For \( n = 2 \):
\[ x = 4\pi - \frac{\pi}{6} = \frac{24\pi}{6} - \frac{\pi}{6} = \frac{23\pi}{6} \]
For \( n = 3 \):
\[ x = 6\pi - \frac{\pi}{6} = \frac{36\pi}{6} - \frac{\pi}{6} = \frac{35\pi}{6} \quad (\text{not in } [0, 4\pi]) \]

### Step 5: Solve the second inequality
Now, consider the second inequality:
\[ \sqrt{3} \cos x - \sin x \leq -2 \]
Rearranging gives:
\[ \sqrt{3} \cos x + 2 \leq \sin x \]
This can be rewritten as:
\[ \frac{\sqrt{3}}{2} \cos x + 1 \leq \frac{1}{2} \sin x \]
Using the cosine addition formula:
\[ \cos\left(x + \frac{\pi}{6}\right) \leq -1 \]
This implies:
\[ x + \frac{\pi}{6} = (2n + 1)\pi \quad \text{for integers } n \]
Thus:
\[ x = (2n + 1)\pi - \frac{\pi}{6} \]

### Step 6: Find values for \( n \)
For \( n = 0 \):
\[ x = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]
For \( n = 1 \):
\[ x = 3\pi - \frac{\pi}{6} = \frac{18\pi}{6} - \frac{\pi}{6} = \frac{17\pi}{6} \]
For \( n = 2 \):
\[ x = 5\pi - \frac{\pi}{6} = \frac{30\pi}{6} - \frac{\pi}{6} = \frac{29\pi}{6} \quad (\text{not in } [0, 4\pi]) \]

### Step 7: Collect all solutions
The valid solutions in the interval \( [0, 4\pi] \) are:
1. \( \frac{11\pi}{6} \)
2. \( \frac{23\pi}{6} \)
3. \( \frac{5\pi}{6} \)
4. \( \frac{17\pi}{6} \)

### Conclusion
Thus, the total number of values of \( x \) satisfying the equation in the interval \( [0, 4\pi] \) is **4**.

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