The value of `sin(pi/14)sin((3pi)/14)sin((5pi)/14)` is

Options

A. `1//16`
B. `1//8`
C. `1//2`
D. `1//4`

Answer

To find the value of \( \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \), we can follow these steps:

### Step 1: Introduce a substitution
Let \( \theta = \frac{\pi}{14} \). Then we can rewrite the expression as:
\[ \sin(\theta) \sin(3\theta) \sin(5\theta) \]

### Step 2: Multiply and divide by \( 2 \cos(\theta) \)
We can multiply and divide the expression by \( 2 \cos(\theta) \):
\[ \sin(\theta) \sin(3\theta) \sin(5\theta) = \frac{2 \cos(\theta) \sin(\theta) \sin(3\theta) \sin(5\theta)}{2 \cos(\theta)} \]

### Step 3: Use the sine double angle identity
Using the identity \( 2 \sin(a) \cos(a) = \sin(2a) \), we can write:
\[ 2 \cos(\theta) \sin(\theta) = \sin(2\theta) \]
Thus, we have:
\[ \frac{\sin(2\theta) \sin(3\theta) \sin(5\theta)}{2 \cos(\theta)} \]

### Step 4: Multiply and divide again by \( 2 \)
Now we multiply and divide by \( 2 \) again:
\[ \frac{2 \sin(2\theta) \sin(3\theta) \sin(5\theta)}{4 \cos(\theta)} \]

### Step 5: Use the product-to-sum identities
Using the identity \( 2 \sin(a) \sin(b) = \cos(a-b) - \cos(a+b) \):
\[ 2 \sin(2\theta) \sin(5\theta) = \cos(3\theta) - \cos(7\theta) \]
So we can rewrite the expression as:
\[ \frac{\cos(3\theta) - \cos(7\theta)}{4 \cos(\theta)} \]

### Step 6: Substitute back for \( \theta \)
Now substituting back \( \theta = \frac{\pi}{14} \):
\[ \frac{\cos\left(\frac{3\pi}{14}\right) - \cos\left(\frac{7\pi}{14}\right)}{4 \cos\left(\frac{\pi}{14}\right)} \]
Since \( \cos\left(\frac{7\pi}{14}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \), we have:
\[ \frac{\cos\left(\frac{3\pi}{14}\right)}{4 \cos\left(\frac{\pi}{14}\right)} \]

### Step 7: Evaluate \( \cos\left(\frac{3\pi}{14}\right) \)
Using the sine and cosine values, we can find that:
\[ \sin\left(\frac{7\pi}{14}\right) = 1 \quad \text{and} \quad \cos\left(\frac{3\pi}{14}\right) = \sin\left(\frac{11\pi}{14}\right) \]

### Step 8: Final evaluation
Thus, we have:
\[ \sin\left(\frac{7\pi}{14}\right) = 1 \quad \text{and} \quad \cos\left(\frac{3\pi}{14}\right) = 0 \]
This leads us to:
\[ \frac{1}{8} \]

### Final Answer
Thus, the value of \( \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \) is:
\[ \frac{1}{8} \]