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The problem asks for values of \( x_1 \) where \( \sin(x_1) = -\frac{1}{2} \) within the interval \(\left[ \frac{11\pi}{6}, \frac{13\pi}{4} \right]\).

### Step 1: Solve \( \sin(x_1) = -\frac{1}{2} \)
The general solution for \( \sin(x_1) = -\frac{1}{2} \) is:
\[
x_1 = 2k\pi + \frac{7\pi}{6} \quad \text{or} \quad x_1 = 2k\pi + \frac{11\pi}{6}
\]
for any integer \( k \), since these angles correspond to where \( \sin(x) = -\frac{1}{2} \).

### Step 2: Find values of \( x_1 \) in the interval \(\left[ \frac{11\pi}{6}, \frac{13\pi}{4} \right]\)
We need to convert the upper bound \( \frac{13\pi}{4} \) to a more comparable form:
\[
\frac{13\pi}{4} = 3\pi + \frac{\pi}{4} = \frac{12\pi}{4} + \frac{\pi}{4} = \frac{13\pi}{4}
\]
Thus, the interval is approximately \( [5.76, 10.2] \) (in radians).

### Step 3: Determine specific values
By adding or subtracting multiples of \( 2\pi \), we will determine which solutions fall into the given interval.

Let me calculate this for you.The values of \( x_1 \) that satisfy \( \sin(x_1) = -\frac{1}{2} \) within the interval \(\left[ \frac{11\pi}{6}, \frac{13\pi}{4} \right]\) are approximately:

\[
x_1 \approx 5.76 \quad \text{and} \quad x_1 \approx 9.95
\]

These correspond to angles where the sine function equals \(-\frac{1}{2}\) in the given interval.

Would you like further details or have any questions?

Here are 5 related questions to expand on the topic:
1. What is the general solution for the equation \( \sin(x) = -1 \)?
2. How do we determine the period of trigonometric functions like sine and cosine?
3. What is the reference angle for the sine function in the unit circle?
4. How do we solve \( \cos(x) = \frac{1}{2} \) within a specified interval?
5. Can you explain how to convert radians to degrees and vice versa?

**Tip:** When solving trigonometric equations over intervals, remember to adjust for the periodic nature of the function by considering all solutions that fall within the interval.

Using the graph of y = sin(x), list all values on the interval [11π/6, 13π/4] that satisfy the ordered pair (x₁, -1/2).

Learn how to solve sin(x) = -1/2 in the given interval [11π/6, 13π/4]. This problem involves understanding the periodic nature of the sine function and finding all solutions that satisfy the equation within the specified interval.

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