Find the exact value of $\sin (\alpha-\beta)$ if $\sin \alpha=\frac{15}{17}$ and $\sin \beta=\frac{3}{5}$ and $\alpha$ and $\beta$ are between 0 and $\pi / 2$.

Step 1 ：We are given that \(\sin \alpha = \frac{15}{17}\) and \(\sin \beta = \frac{3}{5}\). We are also told that \(\alpha\) and \(\beta\) are between 0 and \(\frac{\pi}{2}\).

Step 2 ：We need to find the exact value of \(\sin (\alpha - \beta)\). We know that the formula for \(\sin (\alpha - \beta)\) is \(\sin \alpha \cos \beta - \cos \alpha \sin \beta\).

Step 3 ：We are given the values of \(\sin \alpha\) and \(\sin \beta\), but we need to find the values of \(\cos \alpha\) and \(\cos \beta\).

Step 4 ：Since \(\alpha\) and \(\beta\) are between 0 and \(\frac{\pi}{2}\), we know that \(\cos \alpha\) and \(\cos \beta\) are positive. We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find the values of \(\cos \alpha\) and \(\cos \beta\).

Step 5 ：Using the Pythagorean identity, we find that \(\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - (\frac{15}{17})^2} = 0.4705882352941177\) and \(\cos \beta = \sqrt{1 - \sin^2 \beta} = \sqrt{1 - (\frac{3}{5})^2} = 0.8\).

Step 6 ：Substituting these values into the formula for \(\sin (\alpha - \beta)\), we get \(\sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta = \frac{15}{17} \cdot 0.8 - 0.4705882352941177 \cdot \frac{3}{5} = 0.42352941176470593\).

Step 7 ：Final Answer: The exact value of \(\sin (\alpha - \beta)\) is \(\boxed{0.42352941176470593}\).