To determine the corresponding point on the unit circle for the angle [tex]\(\theta = \frac{5\pi}{2}\)[/tex], follow these steps:
1. Normalize the Angle:
The given angle is [tex]\(\theta = \frac{5\pi}{2}\)[/tex]. This angle is beyond one full revolution ([tex]\(2\pi\)[/tex]). To find the equivalent angle within a single revolution, we need to reduce the angle:
[tex]\[
\theta \mod 2\pi = \frac{5\pi}{2} \mod 2\pi
\][/tex]
Simplifying,
[tex]\[
\frac{5\pi}{2} = 2\pi + \frac{\pi}{2}
\][/tex]
So,
[tex]\[
\frac{5\pi}{2} \mod 2\pi = \frac{\pi}{2}
\][/tex]
Therefore, [tex]\(\theta = \frac{\pi}{2}\)[/tex] is the equivalent angle within one revolution.
2. Determine the Point on the Unit Circle:
The angle [tex]\(\frac{\pi}{2}\)[/tex] corresponds to [tex]\(\frac{\pi}{2}\)[/tex] radians, which is the upward positive y-direction on the unit circle (90 degrees).
The cosine of [tex]\(\frac{\pi}{2}\)[/tex] is:
[tex]\[
\cos\left(\frac{\pi}{2}\right) = 0
\][/tex]
The sine of [tex]\(\frac{\pi}{2}\)[/tex] is:
[tex]\[
\sin\left(\frac{\pi}{2}\right) = 1
\][/tex]
Therefore, the coordinates corresponding to [tex]\(\theta = \frac{\pi}{2}\)[/tex] on the unit circle are [tex]\((0, 1)\)[/tex].
3. Verify the Point:
Checking against the answer choices:
- A. [tex]\(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex]
- B. [tex]\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)[/tex]
- C. [tex]\(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)[/tex]
- D. [tex]\(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)[/tex]
None of these points match the calculated point [tex]\((0, 1)\)[/tex]. Hence, none of the provided options (A, B, C, or D) are correct.
4. Conclusion:
The corresponding point on the unit circle for [tex]\(\theta = \frac{5\pi}{2}\)[/tex] is [tex]\((0, 1)\)[/tex], which corresponds to none of the answer choices. Thus, the correct point is:
[tex]\[
(0, 1)
\][/tex]
Hence, the provided options do not match the result obtained from normalizing the angle and calculating the trigonometric values.
