Use the fact that the trigonometric functions are periodic to find the exact value of the given expression. Do not use a calculator.
\[
\sin \frac{37 \pi}{6}
\]
\[
\sin \frac{37 \pi}{6}=\square
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Answer
\(\boxed{\sin\frac{37\pi}{6} = \frac{1}{2}}\)
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 ：\(\boxed{\sin\frac{37\pi}{6} = \frac{1}{2}}\)
