Let's analyze the given expression and find its range.

Given expression: 
\[ 2\cos(2\theta) - 6\sin(\theta) + 1 \]

### Step 1: Determine the range of \( \cos(2\theta) \) and \( \sin(\theta) \)
- The range of \( \cos(2\theta) \) is \([-1, 1]\).
- The range of \( \sin(\theta) \) is \([-1, 1]\).

### Step 2: Substitute and find the range of each term
1. **For \( 2\cos(2\theta) \):**
   \[ 2\cos(2\theta) \text{ will range from } 2(-1) \text{ to } 2(1) \text{ which is } [-2, 2] \]

2. **For \( -6\sin(\theta) \):**
   \[ -6\sin(\theta) \text{ will range from } -6(1) \text{ to } -6(-1) \text{ which is } [-6, 6] \]

3. **For the constant term \( +1 \):**
   - This is just a constant and does not change.

### Step 3: Combine the ranges
Let's analyze the extreme values by adding the extremes of each term:

1. **Minimum value:**
   \[
   \text{Min } = -2 + (-6) + 1 = -7
   \]

2. **Maximum value:**
   \[
   \text{Max } = 2 + 6 + 1 = 9
   \]

### Conclusion
The range of the expression \( 2\cos(2\theta) - 6\sin(\theta) + 1 \) is \(\boxed{[-7, 9]}\).

Would you like further details or have any questions?

Here are 8 related questions you might find interesting:
1. What is the range of the function \( 3\cos(2\theta) - 5\sin(\theta) \)?
2. How do you find the range of a function involving both sine and cosine?
3. What is the significance of the amplitude in the range of trigonometric functions?
4. How does the frequency of \( \theta \) affect the range of a trigonometric function?
5. Can the range of a trigonometric function be a single value?
6. What are the maximum and minimum values of \( \cos(\theta) \) and \( \sin(\theta) \)?
7. How do phase shifts affect the range of trigonometric functions?
8. How would you find the range of a function like \( 4\cos^2(\theta) + 2\sin(\theta) \)?

**Tip:** When analyzing the range of trigonometric functions, always consider the extreme values of each trigonometric component separately before combining them.

Learn how to find the range of the expression 2cos(2θ) - 6sin(θ) + 1 with detailed steps and analysis. This problem involves trigonometric functions, including cosine and sine, and covers the range determination using trigonometric identities. Suitable for students in Grades 10-12.

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