Find the exact value of $\sin \left(\tan ^{-1}(8 / 15)\right)$

Step 1 ：Given the problem to find the exact value of \( \sin \left(\tan ^{-1}(8 / 15)\right) \)

Step 2 ：We can use the identity \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{1+x^2}} \). This identity is derived from the Pythagorean theorem and the definitions of sine and tangent in a right triangle.

Step 3 ：Substitute \( x = 8/15 \) into the identity to find the exact value.

Step 4 ：Calculate the value of \( x = 8/15 \) to get \( x = 0.5333333333333333 \)

Step 5 ：Substitute \( x = 0.5333333333333333 \) into the identity \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{1+x^2}} \) to get \( \sin_value = 0.47058823529411764 \)

Step 6 ：Final Answer: The exact value of \( \sin \left(\tan ^{-1}(8 / 15)\right) \) is \( \boxed{0.47058823529411764} \)