Given $x=\sin \frac{\pi }{2}$ and $y=\cos \frac{\pi }{2}$, which of the following is correct for $x$ and $y$ where $0\le \theta \le \pi$?

• A. $x=0$ and $y=1$
• B. $x=0$ and $y=-1$
• C. $x=-1$ and $y=0$
• D. $x=1$ and $y=0$

Answer: (D)

Verified step by step guidance

1. Identify the given values: \(x = \sin\left(\frac{\pi}{2}\right)\) and \(y = \cos\left(\frac{\pi}{2}\right)\).
2. Recall the unit circle values for sine and cosine at \(\theta = \frac{\pi}{2}\) radians (90 degrees).
3. Use the fundamental trigonometric values: \(\sin\left(\frac{\pi}{2}\right) = 1\) and \(\cos\left(\frac{\pi}{2}\right) = 0\).
4. Substitute these values back into the expressions for \(x\) and \(y\) to find \(x = 1\) and \(y = 0\).
5. Compare the results with the given options to determine which pair matches the calculated values.