If $\sin ^{2} \theta=1 / 2$ and $\pi< \theta< 3 \pi / 2$ (quadrant III), then $\theta=$
Answer
\(\boxed{\theta = 5\pi/4}\) or \(\boxed{\theta = 2.356194490192345}\) in radians is the final answer.
Step 1 ：The given equation is \(\sin ^{2} \theta=1 / 2\).
Step 2 ：We can take the square root of both sides to get \(\sin \theta = \pm \sqrt{1/2}\).
Step 3 ：However, since \(\theta\) is in the third quadrant where sine is negative, we have \(\sin \theta = -\sqrt{1/2}\).
Step 4 ：We know that \(\sin \theta = -\sqrt{1/2}\) when \(\theta = 7\pi/4\) or \(\theta = 5\pi/4\).
Step 5 ：But since \(\pi<\theta<3 \pi / 2\), the only possible value for \(\theta\) is \(\theta = 5\pi/4\).
Step 6 ：\(\boxed{\theta = 5\pi/4}\) or \(\boxed{\theta = 2.356194490192345}\) in radians is the final answer.
