The range of the expression \([ \sin(x) ] + [ \cos(x) ]\), where the square brackets represent the greatest integer function (also known as the floor function), can be derived by analyzing the behavior of both \(\sin(x)\) and \(\cos(x)\) and how the greatest integer function interacts with them.

### Step-by-Step Breakdown:

1. **Range of \(\sin(x)\)**: The sine function, \(\sin(x)\), oscillates between -1 and 1. Therefore, \(\sin(x) \in [-1, 1]\).

2. **Range of \(\cos(x)\)**: Similarly, the cosine function, \(\cos(x)\), also oscillates between -1 and 1. So, \(\cos(x) \in [-1, 1]\).

3. **Effect of the Greatest Integer Function**:
   - For \(\sin(x)\), since it ranges between -1 and 1, the greatest integer (or floor) function applied to \(\sin(x)\) will yield:
     \[
     [\sin(x)] = 
     \begin{cases} 
     -1 & \text{if } \sin(x) \in (-1, 0) \\
     0 & \text{if } \sin(x) \in [0, 1) \\
     \end{cases}
     \]
   - Similarly, for \(\cos(x)\), applying the greatest integer function gives:
     \[
     [\cos(x)] = 
     \begin{cases} 
     -1 & \text{if } \cos(x) \in (-1, 0) \\
     0 & \text{if } \cos(x) \in [0, 1) \\
     \end{cases}
     \]

4. **Range of \([\sin(x)] + [\cos(x)]\)**:
   - \([\sin(x)]\) can be either -1 or 0.
   - \([\cos(x)]\) can also be either -1 or 0.
   
   Therefore, the sum \([\sin(x)] + [\cos(x)]\) can take the following values:
   \[
   [\sin(x)] + [\cos(x)] \in \{-2, -1, 0\}.
   \]

### Conclusion:
The range of \([\sin(x)] + [\cos(x)]\) is \(\{-2, -1, 0\}\).

Would you like more details or have any questions about this? 

Here are five related questions:
1. What is the range of \([ \tan(x) ]\)?
2. How does the greatest integer function affect trigonometric functions?
3. Can you explain the behavior of the sine and cosine functions in terms of periodicity?
4. How do you calculate the greatest integer function for other types of functions?
5. What is the difference between the greatest integer function and the ceiling function?

**Tip:** The greatest integer function always rounds down to the nearest integer, no matter how close the input is to the next integer.

Explore the range of [sin(x)] + [cos(x)], where the greatest integer function is applied to both sine and cosine. This trigonometric problem uses the floor function, with a detailed step-by-step solution. Suitable for students in Grades 10-12.

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