Let's find [tex]\(\sin \frac{4\pi}{3}\)[/tex] step by step.
1. Determine the Reference Angle:
The given angle is [tex]\(\frac{4\pi}{3}\)[/tex]. First, let's rewrite this angle in a more familiar form by converting it to degrees if necessary. However, since it's straightforward in radians, we can directly find the reference angle.
[tex]\[
\frac{4\pi}{3} \text{ radians} = 240^\circ
\][/tex]
Since [tex]\(240^\circ\)[/tex] is an angle located in the third quadrant of the unit circle, we first determine the reference angle (the acute angle that the terminal side makes with the x-axis).
The reference angle for [tex]\(240^\circ\)[/tex] is:
[tex]\[
240^\circ - 180^\circ = 60^\circ \quad \text{or} \quad \frac{\pi}{3} \text{ radians}
\][/tex]
2. Sign Determination in the Third Quadrant:
In the third quadrant, the sine function is negative. Therefore, we must apply this knowledge to our reference angle.
[tex]\[
\sin\left(\frac{4\pi}{3}\right) = - \sin\left(\frac{\pi}{3}\right)
\][/tex]
3. Evaluate [tex]\(\sin\left(\frac{\pi}{3}\right)\)[/tex]:
Now we need the sine of the reference angle [tex]\(\frac{\pi}{3}\)[/tex]:
[tex]\[
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
4. Apply the Sign from Step 2:
Since we are in the third quadrant where sine is negative:
[tex]\[
\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}
\][/tex]
Hence, the value of [tex]\(\sin \frac{4\pi}{3}\)[/tex] is:
[tex]\[
\sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
Numerically, this simplifies to approximately:
[tex]\[
\sin \frac{4\pi}{3} \approx -0.8660254037844385
\][/tex]
