Find the exact value of the real number $y$ if it exists. Do not use a calculator.
\[
y=\sin ^{-1}\left(\frac{1}{2}\right)
\]
Select the correct answer below and, if necessary, fill in the answer box to complete your choice.
A.
\[
y=\sin ^{-1}\left(\frac{1}{2}\right)=
\]
(Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)
B. $\sin ^{-1}\left(\frac{1}{2}\right)$ does not exist.
Answer
Final Answer: The exact value of the real number y is \(\boxed{\frac{\pi}{6}}\).
Step 1 ：The inverse sine function, also known as arcsin, is the inverse function of the sine function. It returns the angle whose sine is a given number. In this case, we are asked to find the angle whose sine is 1/2.
Step 2 ：From the unit circle, we know that \(\sin(\pi/6) = 1/2\) and \(\sin(5\pi/6) = 1/2\). However, the range of the inverse sine function is \([-\pi/2, \pi/2]\), so the only possible value for y is \(\pi/6\).
Step 3 ：Final Answer: The exact value of the real number y is \(\boxed{\frac{\pi}{6}}\).
