` theta in [0,2pi]` and `z_(1)`, `z_(2)`, `z_(3)` are three complex numbers such that they are collinear and `(1+|sin theta|)z_(1)+(|cos theta|-1)z_(2)-sqrt(2)z_(3)=0`. If at least one of the complex numbers `z_(1)`, `z_(2)`, `z_(3)` is nonzero, then number of possible values of `theta` is

Options

A. Infinite
B. `4`
C. `2`
D. `8`

Explanation

To solve the given problem, we need to analyze the equation provided and determine the possible values of \( \theta \) based on the conditions given.

### Step-by-Step Solution:

1. **Understanding the Condition of Collinearity**:
The complex numbers \( z_1, z_2, z_3 \) are collinear if there exist scalars \( a, b, c \) such that:
\[ a z_1 + b z_2 + c z_3 = 0 \]
In our case, we have:
\[ (1 + |\sin \theta|) z_1 + (|\cos \theta| - 1) z_2 - \sqrt{2} z_3 = 0 \]
This implies that the coefficients must satisfy:
\[ 1 + |\sin \theta| + (|\cos \theta| - 1) - \sqrt{2} = 0 \]

2. **Simplifying the Equation**:
Rearranging the equation gives us:
\[ |\sin \theta| + |\cos \theta| - \sqrt{2} = 0 \]
This can be rewritten as:
\[ |\sin \theta| + |\cos \theta| = \sqrt{2} \]

3. **Analyzing the Equation**:
The maximum value of \( |\sin \theta| + |\cos \theta| \) occurs when both sine and cosine are equal, which happens at \( \theta = \frac{\pi}{4} + n\frac{\pi}{2} \) for \( n \in \mathbb{Z} \). The maximum value is:
\[ |\sin \theta| + |\cos \theta| = \sqrt{2} \]

4. **Finding the Specific Angles**:
The angles where \( |\sin \theta| + |\cos \theta| = \sqrt{2} \) are:
- \( \theta = \frac{\pi}{4} \)
- \( \theta = \frac{3\pi}{4} \)
- \( \theta = \frac{5\pi}{4} \)
- \( \theta = \frac{7\pi}{4} \)

5. **Conclusion**:
Therefore, the possible values of \( \theta \) in the interval \( [0, 2\pi] \) are:
\[ \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \]
This gives us a total of 4 possible values for \( \theta \).

### Final Answer:
The number of possible values of \( \theta \) is **4**.