`sin{2 cos^(-1) ((-3)/5)}` is equal to :

Options

A. `6/26`
B. `24/25`
C. `4/5`
D. ` - 24/25`

Answer

To solve the expression \( \sin(2 \cos^{-1}(-\frac{3}{5})) \), we can use the double angle formula for sine, which states that:

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

Here, \( \theta = \cos^{-1}(-\frac{3}{5}) \).

### Step 1: Find \( \sin(\theta) \) and \( \cos(\theta) \)

From the definition of \( \theta \):

\[ \cos(\theta) = -\frac{3}{5} \]

To find \( \sin(\theta) \), we can use the Pythagorean identity:

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

Substituting the value of \( \cos(\theta) \):

\[ \sin^2(\theta) + \left(-\frac{3}{5}\right)^2 = 1 \]

Calculating \( \left(-\frac{3}{5}\right)^2 \):

\[ \sin^2(\theta) + \frac{9}{25} = 1 \]

### Step 2: Solve for \( \sin^2(\theta) \)

Rearranging the equation gives:

\[ \sin^2(\theta) = 1 - \frac{9}{25} \]

Finding a common denominator:

\[ \sin^2(\theta) = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

Taking the square root:

\[ \sin(\theta) = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

Since \( \theta = \cos^{-1}(-\frac{3}{5}) \) is in the second quadrant, \( \sin(\theta) \) is positive.

### Step 3: Calculate \( \sin(2\theta) \)

Now we can substitute \( \sin(\theta) \) and \( \cos(\theta) \) into the double angle formula:

\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \left(\frac{4}{5}\right) \left(-\frac{3}{5}\right) \]

Calculating this gives:

\[ \sin(2\theta) = 2 \cdot \frac{4}{5} \cdot -\frac{3}{5} = -\frac{24}{25} \]

### Final Answer

Thus, the value of \( \sin(2 \cos^{-1}(-\frac{3}{5})) \) is:

\[ \boxed{-\frac{24}{25}} \]

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