Find the exact value of the real number $y$ if it exists. Do not use a calculator.
\[
y=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)
\]
Select the correct answer below and, if necessary, fill in the answer box to complete your choice.
A.
\[
y=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)=
\]
(Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)
$B$.
\[
\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right) \text { does not exist }
\]
Answer
Final Answer: The exact value of the real number \(y\) is \(\boxed{\frac{\pi}{6}}\).
Step 1 ：The cosine function has its maximum value of 1 and minimum value of -1. Since \(\frac{\sqrt{3}}{2}\) is within this range, the inverse cosine of \(\frac{\sqrt{3}}{2}\) does exist.
Step 2 ：The cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. We know that \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\). Therefore, \(\cos^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{6}\).
Step 3 ：Final Answer: The exact value of the real number \(y\) is \(\boxed{\frac{\pi}{6}}\).
