Question 8 (2 points)
Determine an exact value of $\sin \emptyset$ if $\cos \emptyset=\frac{1}{2}$ and $\emptyset$ is an angle in quadrant IV
Answer
So, the exact value of \(\sin \emptyset\) if \(\cos \emptyset=\frac{1}{2}\) and \(\emptyset\) is an angle in quadrant IV is \(\boxed{-0.8660254037844386}\).
Step 1 ：Given that the cosine of an angle in the fourth quadrant is \(\frac{1}{2}\), we can use the Pythagorean identity \(\sin^2 \emptyset + \cos^2 \emptyset = 1\) to find the value of \(\sin \emptyset\).
Step 2 ：Substitute \(\cos \emptyset = \frac{1}{2}\) into the identity to find \(\sin \emptyset\).
Step 3 ：After calculation, we find that \(\sin \emptyset = -0.8660254037844386\).
Step 4 ：So, the exact value of \(\sin \emptyset\) if \(\cos \emptyset=\frac{1}{2}\) and \(\emptyset\) is an angle in quadrant IV is \(\boxed{-0.8660254037844386}\).
