Find the exact value of $\sin 165^{\circ}$

Step 1 ：We know that \(\sin 165^\circ\) can be expressed as \(\sin (120^\circ + 45^\circ)\).

Step 2 ：Using the sum of angles formula for sine, which is \(\sin (a + b) = \sin a \cos b + \cos a \sin b\), we can substitute \(a = 120^\circ\) and \(b = 45^\circ\) into the formula to find the exact value of \(\sin 165^\circ\).

Step 3 ：We know that the exact values of \(\sin 120^\circ\) and \(\sin 45^\circ\) are \(\frac{\sqrt{3}}{2}\) and \(\frac{1}{\sqrt{2}}\) respectively, and the exact values of \(\cos 120^\circ\) and \(\cos 45^\circ\) are \(-\frac{1}{2}\) and \(\frac{1}{\sqrt{2}}\) respectively.

Step 4 ：Substituting these values into the sum of angles formula will give us the exact value of \(\sin 165^\circ\).

Step 5 ：The exact value of \(\sin 165^\circ\) is \(-\frac{1}{4}\sqrt{2} + \frac{1}{4}\sqrt{6}\).

Step 6 ：Final Answer: The exact value of \(\sin 165^\circ\) is \(\boxed{-\frac{1}{4}\sqrt{2} + \frac{1}{4}\sqrt{6}}\).