Find the value of `sin(pi/10) sin((13pi)/10)`

Options

A.

`-1/4 `

B.

`1/4 `

C.

`-1/2 `

D. None of these

Explanation

To find the value of \( \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{13\pi}{10}\right) \), we can follow these steps:

### Step 1: Simplify the Angles
We know that \( \frac{13\pi}{10} \) can be rewritten using the identity \( \sin(\theta + \pi) = -\sin(\theta) \). Thus, we can express:
\[ \sin\left(\frac{13\pi}{10}\right) = \sin\left(\pi + \frac{3\pi}{10}\right) = -\sin\left(\frac{3\pi}{10}\right) \]
So, we have:
\[ \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{13\pi}{10}\right) = \sin\left(\frac{\pi}{10}\right) \left(-\sin\left(\frac{3\pi}{10}\right)\right) = -\sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) \]

### Step 2: Use the Product-to-Sum Formula
We can use the product-to-sum identities for sine:
\[ \sin A \sin B = \frac{1}{2} \left[ \cos(A - B) - \cos(A + B) \right] \]
Let \( A = \frac{\pi}{10} \) and \( B = \frac{3\pi}{10} \):
\[ -\sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) = -\frac{1}{2} \left[ \cos\left(\frac{\pi}{10} - \frac{3\pi}{10}\right) - \cos\left(\frac{\pi}{10} + \frac{3\pi}{10}\right) \right] \]
Calculating the angles:
\[ \frac{\pi}{10} - \frac{3\pi}{10} = -\frac{2\pi}{10} = -\frac{\pi}{5} \]
\[ \frac{\pi}{10} + \frac{3\pi}{10} = \frac{4\pi}{10} = \frac{2\pi}{5} \]
Thus, we have:
\[ -\sin\left(\frac{\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right) = -\frac{1}{2} \left[ \cos\left(-\frac{\pi}{5}\right) - \cos\left(\frac{2\pi}{5}\right) \right] \]
Using the property \( \cos(-x) = \cos(x) \):
\[ = -\frac{1}{2} \left[ \cos\left(\frac{\pi}{5}\right) - \cos\left(\frac{2\pi}{5}\right) \right] \]

### Step 3: Use Known Values of Cosine
The values of \( \cos\left(\frac{\pi}{5}\right) \) and \( \cos\left(\frac{2\pi}{5}\right) \) can be derived from the pentagon properties:
\[ \cos\left(\frac{\pi}{5}\right) = \frac{\sqrt{5}+1}{4}, \quad \cos\left(\frac{2\pi}{5}\right) = \frac{\sqrt{5}-1}{4} \]
Substituting these values:
\[ -\frac{1}{2} \left[ \frac{\sqrt{5}+1}{4} - \frac{\sqrt{5}-1}{4} \right] = -\frac{1}{2} \left[ \frac{(\sqrt{5}+1) - (\sqrt{5}-1)}{4} \right] \]
This simplifies to:
\[ -\frac{1}{2} \left[ \frac{2}{4} \right] = -\frac{1}{2} \cdot \frac{1}{2} = -\frac{1}{4} \]

### Final Answer
Thus, we conclude that:
\[ \sin\left(\frac{\pi}{10}\right) \sin\left(\frac{13\pi}{10}\right) = -\frac{1}{4} \]