To find [tex]\(\sin \frac{4 \pi}{3}\)[/tex], follow the steps below:
1. Determine the Angle in Radians:
The angle given is [tex]\(\frac{4 \pi}{3}\)[/tex]. This angle is in the third quadrant of the unit circle.
2. Understand the Reference Angle:
The reference angle for [tex]\(\frac{4 \pi}{3}\)[/tex] in the third quadrant can be found as follows:
[tex]\[
\pi + \frac{\pi}{3} = \frac{4\pi}{3}
\][/tex]
3. Calculate the Sine of the Reference Angle:
The reference angle is [tex]\(\frac{\pi}{3}\)[/tex]. We know that:
[tex]\[
\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}
\][/tex]
4. Consider the Sign in the Third Quadrant:
In the third quadrant, the sine function is negative. Therefore:
[tex]\[
\sin \frac{4 \pi}{3} = -\sin \frac{\pi}{3}
\][/tex]
5. Substitute the Sine Value:
Since [tex]\(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\)[/tex]:
[tex]\[
\sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
6. Verification with Decimal Approximation:
The numerical approximation for [tex]\(-\frac{\sqrt{3}}{2}\)[/tex] is:
[tex]\[
\sin \frac{4 \pi}{3} \approx -0.8660254037844386
\][/tex]
Therefore, the answer is:
[tex]\[
\sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
Breaking it down:
[tex]\[
\sin \frac{4 \pi}{3}=\sin \left(\pi + \frac{\pi}{3} \right)= -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
So the final result is:
[tex]\[
\sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
