`sin((pi)/(10))sin((3pi)/(10))=?`

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. `(1)/(4)`
D. 1

Answer

To solve the problem \( \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) \), we can follow these steps:

### Step-by-Step Solution:

1. **Convert Radians to Degrees:**
\[ \sin\left(\frac{\pi}{10}\right) = \sin(18^\circ) \quad \text{and} \quad \sin\left(\frac{3\pi}{10}\right) = \sin(54^\circ) \]

2. **Use the Identity:**
We know that:
\[ \sin(54^\circ) = \cos(36^\circ) \quad \text{(since } 54^\circ + 36^\circ = 90^\circ\text{)} \]
Therefore, we can rewrite the expression:
\[ \sin(18^\circ) \sin(54^\circ) = \sin(18^\circ) \cos(36^\circ) \]

3. **Substitute Known Values:**
We have the known values:
\[ \sin(18^\circ) = \frac{\sqrt{5} - 1}{4} \quad \text{and} \quad \cos(36^\circ) = \frac{\sqrt{5} + 1}{4} \]

4. **Multiply the Values:**
Now we can multiply these two values:
\[ \sin(18^\circ) \cos(36^\circ) = \left(\frac{\sqrt{5} - 1}{4}\right) \left(\frac{\sqrt{5} + 1}{4}\right) \]

5. **Use the Difference of Squares:**
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):
\[ = \frac{(\sqrt{5})^2 - (1)^2}{16} = \frac{5 - 1}{16} = \frac{4}{16} = \frac{1}{4} \]

6. **Conclusion:**
Therefore, the value of \( \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) \) is:
\[ \frac{1}{4} \]

### Final Answer:
\[ \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) = \frac{1}{4} \]