To determine the angle [tex]\( s \)[/tex] associated with the point [tex]\(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex] on the unit circle, we follow these steps:
1. Identify the specific point:
The given point [tex]\(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex] lies on the unit circle, where [tex]\( x = -\frac{1}{2} \)[/tex] and [tex]\( y = \frac{\sqrt{3}}{2} \)[/tex].
2. Determine the corresponding angle in the unit circle:
On the unit circle, points are represented in the form [tex]\((\cos s, \sin s)\)[/tex].
Given [tex]\( \cos s = -\frac{1}{2} \)[/tex] and [tex]\( \sin s = \frac{\sqrt{3}}{2} \)[/tex], this point corresponds to the standard angle of [tex]\( \frac{2\pi}{3} \)[/tex] radians in the second quadrant.
3. Express the angle in the interval [tex]\([0, 2\pi)\)[/tex]:
The angle [tex]\( s = \frac{2\pi}{3} \)[/tex] is already in the interval [tex]\([0, 2\pi)\)[/tex].
4. Identify other rotationally equivalent angles:
Angles on the unit circle can be represented periodically by adding any integer multiple of [tex]\( 2\pi \)[/tex]. Thus, all angles that correspond to the given point can be written in the form:
[tex]\[
s = \frac{2\pi}{3} + 2k\pi \quad \text{where } k \text{ is any integer}.
\][/tex]
So, the exact radian answer in the interval [tex]\([0, 2\pi)\)[/tex] is:
[tex]\[
s = \frac{2\pi}{3}
\][/tex]
And the general form for all equivalent angles is:
[tex]\[
s = \frac{2\pi}{3} + 2k\pi \quad \text{where } k \text{ is any integer}.
\][/tex]
This encapsulates the solution for all real numbers [tex]\( s \)[/tex] associated with the point [tex]\(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex] on the unit circle.
