To determine the value of [tex]\(\sin \frac{31 \pi}{3}\)[/tex], follow these steps:
1. Convert the Angle to Radians:
The given angle is already given in radians: [tex]\(\frac{31 \pi}{3}\)[/tex].
2. Simplify the Angle:
[tex]\(\frac{31 \pi}{3}\)[/tex] can be simplified by noting that [tex]\(2\pi\)[/tex] radians correspond to one full revolution around the unit circle. Therefore, we can reduce the angle modulo [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{31 \pi}{3} = \frac{30 \pi}{3} + \frac{\pi}{3} = 10\pi + \frac{\pi}{3}
\][/tex]
Since [tex]\(10\pi\)[/tex] is an integer multiple of [tex]\(2\pi\)[/tex] (specifically, [tex]\(5\)[/tex] full revolutions), it does not affect the sine value, and we are essentially left with:
[tex]\[
\frac{\pi}{3}
\][/tex]
3. Calculate the Sine Value:
Now, we need to find [tex]\(\sin \frac{\pi}{3}\)[/tex]. From trigonometric tables or knowledge of special angles:
[tex]\[
\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}
\][/tex]
4. Express the Sine Value Numerically:
The numerical value of [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is approximately:
[tex]\[
0.8660254037844374
\][/tex]
5. Conclusion:
So, the value of [tex]\(\sin \frac{31 \pi}{3}\)[/tex] is:
[tex]\[
0.8660254037844374
\][/tex]
Thus, [tex]\(\sin \frac{31 \pi}{3} = 0.8660254037844374\)[/tex].
