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Given the following information, use a double angle identity to find the exact value of the expression.

To solve this problem, we will use the double angle identity for sine, which is given by:

\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]

### Step 1: Identify \(\sin \theta\)

From the problem, \(\sin \theta = -\frac{\sqrt{203}}{18}\).

### Step 2: Find \(\cos \theta\)

Since \(\theta\) is in the third quadrant (\(180^\circ < \theta < 270^\circ\)), \(\cos \theta\) will be negative. Using the Pythagorean identity:

\[
\sin^2 \theta + \cos^2 \theta = 1
\]

Substitute \(\sin \theta = -\frac{\sqrt{203}}{18}\):

\[
\left(-\frac{\sqrt{203}}{18}\right)^2 + \cos^2 \theta = 1
\]

\[
\frac{203}{324} + \cos^2 \theta = 1
\]

\[
\cos^2 \theta = 1 - \frac{203}{324} = \frac{324 - 203}{324} = \frac{121}{324}
\]

\[
\cos \theta = -\sqrt{\frac{121}{324}} = -\frac{11}{18}
\]

### Step 3: Use the double angle identity

Now, we use the double angle identity:

\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]

Substitute the values of \(\sin \theta\) and \(\cos \theta\):

\[
\sin(2\theta) = 2 \times \left(-\frac{\sqrt{203}}{18}\right) \times \left(-\frac{11}{18}\right)
\]

\[
\sin(2\theta) = 2 \times \frac{\sqrt{203} \times 11}{324} = \frac{22\sqrt{203}}{324} = \frac{11\sqrt{203}}{162}
\]

### Step 4: Match with the options

The value of \(\sin(2\theta)\) is \(\frac{11\sqrt{203}}{162}\), which corresponds to option A.

**Answer:** The correct option is **A**.

---

Would you like more details on any part of this solution or have any further questions?

Here are some related questions you might find useful:
1. How do you determine the sign of cosine in different quadrants?
2. What are the other trigonometric double angle identities?
3. Can you derive the sine double angle identity from sum formulas?
4. How do you apply the double angle identity in calculus?
5. What is the relationship between sine and cosine when finding angles?

**Tip:** Always double-check the quadrant in which the angle lies to determine the correct signs for trigonometric functions.

Learn how to solve a trigonometry problem using the double angle identity for sine. Step-by-step solution provided for finding the exact value of the expression involving sine and cosine. Suitable for high school students.

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