Evaluate
`sin(2cos^(-1)(-(5)/(15)))`

Answer

To evaluate the expression \( \sin(2 \cos^{-1}(-\frac{5}{15})) \), we will follow these steps:

### Step 1: Set up the problem
Let \( \theta = \cos^{-1}(-\frac{5}{15}) \). This means that \( \cos \theta = -\frac{5}{15} = -\frac{1}{3} \).

### Step 2: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta \):
\[ \sin^2 \theta + \left(-\frac{1}{3}\right)^2 = 1 \]
This simplifies to:
\[ \sin^2 \theta + \frac{1}{9} = 1 \]

### Step 3: Solve for \( \sin^2 \theta \)
Rearranging gives:
\[ \sin^2 \theta = 1 - \frac{1}{9} \]
Finding a common denominator:
\[ \sin^2 \theta = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

### Step 4: Find \( \sin \theta \)
Taking the square root:
\[ \sin \theta = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \]
Since \( \theta \) is in the range of \( \cos^{-1} \), \( \sin \theta \) is positive.

### Step 5: Calculate \( \sin(2\theta) \)
Using the double angle formula for sine:
\[ \sin(2\theta) = 2 \sin \theta \cos \theta \]
Substituting the values we have:
\[ \sin(2\theta) = 2 \left(\frac{2\sqrt{2}}{3}\right) \left(-\frac{1}{3}\right) \]
Calculating this gives:
\[ \sin(2\theta) = 2 \cdot \frac{2\sqrt{2}}{3} \cdot -\frac{1}{3} = -\frac{4\sqrt{2}}{9} \]

### Final Answer
Thus, the value of \( \sin(2 \cos^{-1}(-\frac{5}{15})) \) is:
\[ \boxed{-\frac{4\sqrt{2}}{9}} \]