Find the value of
(i) `sin. (5pi)/(3)`

Answer

To find the value of \( \sin\left(\frac{5\pi}{3}\right) \), we can follow these steps:

### Step 1: Rewrite the angle
We can express \( \frac{5\pi}{3} \) in a more manageable form. Notice that:
\[ \frac{5\pi}{3} = 2\pi - \frac{\pi}{3} \]

### Step 2: Use the sine subtraction identity
Using the identity \( \sin(2\pi - \theta) = -\sin(\theta) \), we can rewrite our expression:
\[ \sin\left(\frac{5\pi}{3}\right) = \sin\left(2\pi - \frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) \]

### Step 3: Find the value of \( \sin\left(\frac{\pi}{3}\right) \)
From trigonometric values, we know:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 4: Substitute back into the equation
Now substituting this value back into our expression, we have:
\[ \sin\left(\frac{5\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the value of \( \sin\left(\frac{5\pi}{3}\right) \) is:
\[ \boxed{-\frac{\sqrt{3}}{2}} \]

---