To find the value of [tex]\(\sin \frac{2 \pi}{3}\)[/tex], we can use some properties of trigonometric functions and the unit circle.
1. Identifying the angle on the unit circle:
[tex]\(\frac{2\pi}{3}\)[/tex] radians is located in the second quadrant of the unit circle. This is because [tex]\(\frac{2\pi}{3}\)[/tex] radians is more than [tex]\(\frac{\pi}{2}\)[/tex] radians (which is 90 degrees) but less than [tex]\(\pi\)[/tex] radians (which is 180 degrees).
2. Related angle in the unit circle:
The reference angle for [tex]\(\frac{2\pi}{3}\)[/tex] can be found by subtracting it from [tex]\(\pi\)[/tex]:
[tex]\[
\pi - \frac{2\pi}{3} = \frac{\pi}{3}
\][/tex]
So, the reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
3. Using the sine function's property:
The sine of an angle in the second quadrant can be found by recognizing that sine has a positive value there and is equal to the sine of its reference angle. Hence:
[tex]\[
\sin\left( \frac{2\pi}{3} \right) = \sin\left( \pi - \frac{\pi}{3} \right) = \sin\left( \frac{\pi}{3} \right)
\][/tex]
4. Known value of sine for common angles:
The value of [tex]\(\sin \frac{\pi}{3}\)[/tex] is a well-known trigonometric value:
[tex]\[
\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}
\][/tex]
Therefore, the value of [tex]\(\sin \frac{2\pi}{3}\)[/tex] simplifies to:
[tex]\[
\sin \frac{2 \pi}{3} = \frac{\sqrt{3}}{2}
\][/tex]
So, the correct choice is:
A. [tex]\(\sin \frac{2 \pi}{3}= \frac{\sqrt{3}}{2}\)[/tex]
