To determine the angle [tex]\(\theta\)[/tex] that corresponds to the given point [tex]\(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex] on the unit circle, we will use the properties of the unit circle and the trigonometric functions sine and cosine.
1. Understanding coordinates on the unit circle:
- The point [tex]\(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex] gives us coordinates [tex]\((x,y)\)[/tex] where [tex]\(x = \cos(\theta)\)[/tex] and [tex]\(y = \sin(\theta)\)[/tex].
2. Identifying standard angle values:
- The cosine value ([tex]\(x\)[/tex]-coordinate) is [tex]\(\frac{1}{2}\)[/tex].
- The sine value ([tex]\(y\)[/tex]-coordinate) is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
3. Matching with known unit circle values:
- For [tex]\(\cos(\theta) = \frac{1}{2}\)[/tex], we review the special angles and their cosine values. One such angle is [tex]\(\frac{\pi}{3}\)[/tex] because [tex]\(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)[/tex].
- For [tex]\(\sin(\theta) = \frac{\sqrt{3}}{2}\)[/tex], we also consider special angles. Again, [tex]\(\frac{\pi}{3}\)[/tex] is special because [tex]\(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)[/tex].
4. Confirming the angle:
- Therefore, the angle [tex]\(\theta\)[/tex] that simultaneously satisfies [tex]\(\cos(\theta) = \frac{1}{2}\)[/tex] and [tex]\(\sin(\theta) = \frac{\sqrt{3}}{2}\)[/tex] is [tex]\(\theta = \frac{\pi}{3}\)[/tex].
So, the correct angle [tex]\(\theta\)[/tex] is [tex]\(\frac{\pi}{3}\)[/tex] radians, which corresponds to option:
A. [tex]\(\frac{\pi}{3}\)[/tex] radians
