Select the exact value of $\sin \frac{7 \pi}{12} \quad$ using the fact that $\frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$

Step 1 ：We are given that \(\frac{7 \pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}\).

Step 2 ：We can use the sum of angles formula for sine, which is \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). Here, \(a = \frac{\pi}{3}\) and \(b = \frac{\pi}{4}\).

Step 3 ：We know the values of \(\sin\) and \(\cos\) for these angles, so we can substitute them into the formula to find the value of \(\sin \frac{7 \pi}{12}\).

Step 4 ：Substituting the values, we get \(\sin \frac{7 \pi}{12} = \sin \frac{\pi}{3} \cos \frac{\pi}{4} + \cos \frac{\pi}{3} \sin \frac{\pi}{4} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\).

Step 5 ：Final Answer: The exact value of \(\sin \frac{7 \pi}{12}\) is \(\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}\).