Assume that $\sin (x)=\frac{3}{5}$ and $\sin (y)=-\frac{4}{7}$, where $\frac{\pi}{2}< x< \pi$ and $\frac{3 \pi}{2}< y< 2 \pi$. Calculate the indicated function value.
\[
\sin (x-y)
\]
(Give an exact answer. Use symbolic notation and fractions where needed.)
\[
\sin (x-y)=
\]

Step 1 ：Given that \(\sin (x)=\frac{3}{5}\) and \(\sin (y)=-\frac{4}{7}\), where \(\frac{\pi}{2}

Step 2 ：We know that the formula for \(\sin (x-y)\) is \(\sin x \cos y - \cos x \sin y\). We are given the values of \(\sin x\) and \(\sin y\), but we don't have the values of \(\cos x\) and \(\cos y\).

Step 3 ：We can calculate the values of \(\cos x\) and \(\cos y\) using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) (and similarly for y).

Step 4 ：Since we know the quadrants in which x and y lie, we can determine the signs of \(\cos x\) and \(\cos y\).

Step 5 ：Substituting these values into the formula for \(\sin (x-y)\), we get \(\sin (x-y) = \sin x \cos y - \cos x \sin y = \frac{3}{5} \cdot -\sqrt{1 - \left(-\frac{4}{7}\right)^2} - \sqrt{1 - \left(\frac{3}{5}\right)^2} \cdot -\frac{4}{7}\).

Step 6 ：Simplifying the above expression, we get \(\sin (x-y) = -\frac{3}{5} \sqrt{1 - \frac{16}{49}} + \frac{4}{7} \sqrt{1 - \frac{9}{25}}\).

Step 7 ：\(\boxed{\sin (x-y) = -\frac{3}{5} \sqrt{1 - \frac{16}{49}} + \frac{4}{7} \sqrt{1 - \frac{9}{25}}}\)