Evaluate : `"sin" (8 pi)/(3) "cos" (23 pi)/(6) + "cos" (13 pi)/(3) "sin" (35 pi)/(6)`

Answer

To evaluate the expression \( \sin\left(\frac{8\pi}{3}\right) \cos\left(\frac{23\pi}{6}\right) + \cos\left(\frac{13\pi}{3}\right) \sin\left(\frac{35\pi}{6}\right) \), we can follow these steps:

### Step 1: Simplify the angles
We first need to simplify the angles to find their equivalent angles within the range of \(0\) to \(2\pi\).

1. **For \( \frac{8\pi}{3} \)**:
\[ \frac{8\pi}{3} = 2\pi + \frac{2\pi}{3} \quad \text{(which is equivalent to } \frac{2\pi}{3} \text{)} \]

2. **For \( \frac{23\pi}{6} \)**:
\[ \frac{23\pi}{6} = 3\pi + \frac{5\pi}{6} \quad \text{(which is equivalent to } \frac{5\pi}{6} \text{)} \]

3. **For \( \frac{13\pi}{3} \)**:
\[ \frac{13\pi}{3} = 4\pi + \frac{\pi}{3} \quad \text{(which is equivalent to } \frac{\pi}{3} \text{)} \]

4. **For \( \frac{35\pi}{6} \)**:
\[ \frac{35\pi}{6} = 5\pi + \frac{5\pi}{6} \quad \text{(which is equivalent to } \frac{5\pi}{6} \text{)} \]

### Step 2: Substitute the simplified angles into the expression
Now we substitute the simplified angles back into the expression:
\[ \sin\left(\frac{2\pi}{3}\right) \cos\left(\frac{5\pi}{6}\right) + \cos\left(\frac{\pi}{3}\right) \sin\left(\frac{5\pi}{6}\right) \]

### Step 3: Evaluate the trigonometric functions
Next, we evaluate the trigonometric functions:

1. **For \( \sin\left(\frac{2\pi}{3}\right) \)**:
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

2. **For \( \cos\left(\frac{5\pi}{6}\right) \)**:
\[ \cos\left(\frac{5\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \]

3. **For \( \cos\left(\frac{\pi}{3}\right) \)**:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

4. **For \( \sin\left(\frac{5\pi}{6}\right) \)**:
\[ \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Step 4: Substitute and simplify
Now, we substitute these values back into the expression:
\[ \frac{\sqrt{3}}{2} \left(-\frac{\sqrt{3}}{2}\right) + \frac{1}{2} \left(\frac{1}{2}\right) \]
This simplifies to:
\[ -\frac{3}{4} + \frac{1}{4} = -\frac{3}{4} + \frac{1}{4} = -\frac{2}{4} = -\frac{1}{2} \]

### Final Answer
Thus, the final answer is:
\[ -\frac{1}{2} \]