Step 1 :\(\sin(x + 2\pi) = \sin(x)\) for any real number \(x\)
Step 2 :\(\frac{37\pi}{6}\) is greater than \(2\pi\), so we subtract \(2\pi\) from it
Step 3 :\(2\pi = \frac{12\pi}{6}\), so we subtract multiples of \(\frac{12\pi}{6}\) from \(\frac{37\pi}{6}\) until we get a fraction that is less than \(\frac{12\pi}{6}\)
Step 4 :\(\frac{37\pi}{6} - 3*\frac{12\pi}{6} = \frac{37\pi}{6} - \frac{36\pi}{6} = \frac{\pi}{6}\)
Step 5 :\(\sin\frac{37\pi}{6} = \sin\frac{\pi}{6}\)
Step 6 :\(\sin\frac{\pi}{6} = \frac{1}{2}\)
Step 7 :\(\sin\frac{37\pi}{6} = \frac{1}{2}\)
Step 8 :\(\boxed{\sin\frac{37\pi}{6} = \frac{1}{2}}\)