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 follow these steps:

### Step 1: Simplify the angle
First, we notice that \( \frac{31\pi}{3} \) is greater than \( 2\pi \). To find an equivalent angle within the range of \( [0, 2\pi] \), we can subtract multiples of \( 2\pi \).

\[ 2\pi = \frac{6\pi}{3} \]

Now, we can find how many times \( 2\pi \) fits into \( \frac{31\pi}{3} \):

\[ \frac{31\pi}{3} - 10\pi = \frac{31\pi}{3} - \frac{30\pi}{3} = \frac{\pi}{3} \]

### Step 2: Use the sine function property
Using the property of the sine function, we know that:

\[ \sin(2n\pi + \theta) = \sin(\theta) \]

where \( n \) is an integer. Here, we can let \( n = 5 \) (since \( 10\pi = 2 \times 5\pi \)) and \( \theta = \frac{\pi}{3} \).

Thus, we have:

\[ \sin\left(\frac{31\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) \]

### Step 3: Calculate \( \sin\left(\frac{\pi}{3}\right) \)
Now we need to find \( \sin\left(\frac{\pi}{3}\right) \). The value of \( \sin\left(\frac{\pi}{3}\right) \) is known:

\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Final Answer
Therefore, the value of \( \sin\left(\frac{31\pi}{3}\right) \) is:

\[ \sin\left(\frac{31\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

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