To determine the values that the expression \(\sin(2x) - 1\) can take, we need to analyze the range of the sine function, and how subtracting 1 from it affects this range.

### Step-by-Step Analysis:

1. **Range of \(\sin(2x)\)**:
   The sine function \(\sin(2x)\) oscillates between \(-1\) and \(1\). So:
   \[
   -1 \leq \sin(2x) \leq 1
   \]

2. **Subtract 1 from \(\sin(2x)\)**:
   We subtract 1 from each part of the inequality to find the range of \(\sin(2x) - 1\):
   \[
   -1 - 1 \leq \sin(2x) - 1 \leq 1 - 1
   \]

   Simplifying this:
   \[
   -2 \leq \sin(2x) - 1 \leq 0
   \]

### Conclusion:

The expression \(\sin(2x) - 1\) can take any value within the range \([-2, 0]\).

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Do you need further clarification or have any additional questions? Here are some related questions:

1. What is the range of \(\cos(2x) - 1\)?
2. How does the range of \(\sin(kx) - 1\) change with different values of \(k\)?
3. What are the minimum and maximum values of the function \(f(x) = \sin(2x) - 1\)?
4. How do you graph the function \(y = \sin(2x) - 1\)?
5. How do you determine the amplitude and period of the function \(y = \sin(2x)\)?

**Tip:** When manipulating trigonometric functions, always consider their basic range and how arithmetic operations (like addition, subtraction, multiplication) shift or stretch this range.

Discover the range of values that the expression sin(2x) - 1 can take through a detailed mathematical analysis. Understand how the sine function's range and arithmetic operations influence the resulting range of sin(2x) - 1. Suitable for advanced high school students.

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