Find the value of sin `(31pi)/3`.

Options

A. `sqrt3/2`
B. `-sqrt3/2`
C. `2/sqrt3`
D. `-2/sqrt3`

Explanation

To find the value of \( \sin\left(\frac{31\pi}{3}\right) \), we can use the periodic property of the sine function. The sine function has a period of \( 2\pi \), which means that \( \sin(\theta + 2n\pi) = \sin(\theta) \) for any integer \( n \).

### Step-by-Step Solution:

1. **Identify the angle**: We start with the angle \( \frac{31\pi}{3} \).

2. **Reduce the angle**: We can express \( \frac{31\pi}{3} \) in terms of \( 2\pi \) to find an equivalent angle within the range of \( 0 \) to \( 2\pi \).
\[ 2\pi = \frac{6\pi}{3} \]
To find how many full rotations of \( 2\pi \) fit into \( \frac{31\pi}{3} \), we can divide \( \frac{31\pi}{3} \) by \( 2\pi \):
\[ \frac{31\pi/3}{2\pi} = \frac{31}{6} \approx 5.1667 \]
This means there are 5 full rotations of \( 2\pi \) and a remainder.

3. **Calculate the remainder**: To find the equivalent angle, we can subtract \( 5 \times 2\pi \) from \( \frac{31\pi}{3} \):
\[ 5 \times 2\pi = 10\pi = \frac{30\pi}{3} \]
Now, we subtract:
\[ \frac{31\pi}{3} - \frac{30\pi}{3} = \frac{1\pi}{3} = \frac{\pi}{3} \]

4. **Use the sine function**: Now we can find:
\[ \sin\left(\frac{31\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) \]

5. **Evaluate \( \sin\left(\frac{\pi}{3}\right) \)**: We know from trigonometric values that:
\[ \sin\left(\frac{\pi}{3}\right) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

### Final Answer:
Thus, the value of \( \sin\left(\frac{31\pi}{3}\right) \) is:
\[ \frac{\sqrt{3}}{2} \]