`sinpi/10+sin(13pi)/(10)=?`

Options

A. 1
B. `1/2`
C. `-1/2`
D. None of these

Answer

To solve the expression \( \sin\left(\frac{\pi}{10}\right) + \sin\left(\frac{13\pi}{10}\right) \), we can use the properties of the sine function and the sine addition formula.

### Step-by-Step Solution:

1. **Identify the angles**:
We have two angles:
- \( \frac{\pi}{10} \)
- \( \frac{13\pi}{10} \)

2. **Use the sine function property**:
We know that:
\[ \sin(\theta + \pi) = -\sin(\theta) \]
Therefore, we can rewrite \( \sin\left(\frac{13\pi}{10}\right) \):
\[ \sin\left(\frac{13\pi}{10}\right) = \sin\left(\pi + \frac{3\pi}{10}\right) = -\sin\left(\frac{3\pi}{10}\right) \]

3. **Substituting back into the expression**:
Now substitute this back into the original expression:
\[ \sin\left(\frac{\pi}{10}\right) + \sin\left(\frac{13\pi}{10}\right) = \sin\left(\frac{\pi}{10}\right) - \sin\left(\frac{3\pi}{10}\right) \]

4. **Using the sine subtraction formula**:
We can use the sine subtraction formula:
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
Here, let \( A = \frac{\pi}{10} \) and \( B = \frac{3\pi}{10} \).

5. **Calculate \( A + B \) and \( A - B \)**:
- \( A + B = \frac{\pi}{10} + \frac{3\pi}{10} = \frac{4\pi}{10} = \frac{2\pi}{5} \)
- \( A - B = \frac{\pi}{10} - \frac{3\pi}{10} = -\frac{2\pi}{10} = -\frac{\pi}{5} \)

6. **Substituting into the sine subtraction formula**:
Now substitute these values into the formula:
\[ \sin\left(\frac{\pi}{10}\right) - \sin\left(\frac{3\pi}{10}\right) = 2 \cos\left(\frac{2\pi}{5} \cdot \frac{1}{2}\right) \sin\left(-\frac{\pi}{5} \cdot \frac{1}{2}\right) \]
Simplifying gives:
\[ = 2 \cos\left(\frac{\pi}{5}\right) \sin\left(-\frac{\pi}{10}\right) \]
Since \( \sin(-x) = -\sin(x) \):
\[ = -2 \cos\left(\frac{\pi}{5}\right) \sin\left(\frac{\pi}{10}\right) \]

7. **Final Result**:
Thus, the final result is:
\[ \sin\left(\frac{\pi}{10}\right) + \sin\left(\frac{13\pi}{10}\right) = -2 \cos\left(\frac{\pi}{5}\right) \sin\left(\frac{\pi}{10}\right) \]