Calculate an exact answer using a formula learned in class.
\[
\sin \frac{19 \pi}{12}
\]
No decimals

Step 1 ：Given the angle \(\frac{19 \pi}{12}\), which is not a standard angle, we need to express it as a sum or difference of standard angles.

Step 2 ：We can express \(\frac{19 \pi}{12}\) as \(\frac{16 \pi}{12} + \frac{3 \pi}{12}\), which simplifies to \(\frac{4 \pi}{3} + \frac{\pi}{4}\). These are standard angles.

Step 3 ：We can use the sum-to-product identities to find the sine value. The sum-to-product identities are: \[\sin(a + b) = \sin a \cos b + \cos a \sin b\] and \[\sin(a - b) = \sin a \cos b - \cos a \sin b\]

Step 4 ：Let's assign \(a = \frac{4 \pi}{3}\) and \(b = \frac{\pi}{4}\).

Step 5 ：We know that \(\sin a = -\frac{\sqrt{3}}{2}\), \(\cos a = -\frac{1}{2}\), \(\sin b = \frac{\sqrt{2}}{2}\), and \(\cos b = \frac{\sqrt{2}}{2}\).

Step 6 ：Substitute these values into the sum-to-product identity for sine, we get \(\sin(a + b) = \sin a \cos b + \cos a \sin b = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\).

Step 7 ：Thus, the exact value of \(\sin \frac{19 \pi}{12}\) is \(-\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\).

Step 8 ：Final Answer: \(\boxed{-\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}\)