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

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: Multiply and Divide by \( 2 \cos\left(\frac{\pi}{14}\right) \)
We start by multiplying and dividing the expression by \( 2 \cos\left(\frac{\pi}{14}\right) \):

\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) = \frac{2 \cos\left(\frac{\pi}{14}\right) \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right)}{2 \cos\left(\frac{\pi}{14}\right)} \]

### Step 2: Use the Identity \( 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \)
Using the identity \( 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \), we can simplify the numerator:

\[ = \frac{\sin\left(2 \cdot \frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right)}{2 \cos\left(\frac{\pi}{14}\right)} \]

### Step 3: Substitute \( \theta = \frac{\pi}{14} \)
Let \( \theta = \frac{\pi}{14} \). Then the expression becomes:

\[ = \frac{\sin(2\theta) \sin(3\theta) \sin(5\theta)}{2 \cos(\theta)} \]

### Step 4: Multiply and Divide by 2
Now, multiply and divide the expression by 2:

\[ = \frac{2 \sin(2\theta) \sin(3\theta) \sin(5\theta)}{4 \cos(\theta)} \]

### Step 5: Use the Identity for Product of Sines
Using the identity for the product of sines, we can simplify:

\[ = \frac{\sin(6\theta) \cos(3\theta)}{4 \cos(\theta)} \]

### Step 6: Substitute Back for \( \theta \)
Substituting back \( \theta = \frac{\pi}{14} \):

\[ = \frac{\sin\left(\frac{6\pi}{14}\right) \cos\left(\frac{3\pi}{14}\right)}{4 \cos\left(\frac{\pi}{14}\right)} \]

### Step 7: Simplify \( \sin\left(\frac{6\pi}{14}\right) \)
Notice that \( \frac{6\pi}{14} = \frac{3\pi}{7} \):

\[ = \frac{\sin\left(\frac{3\pi}{7}\right) \cos\left(\frac{3\pi}{14}\right)}{4 \cos\left(\frac{\pi}{14}\right)} \]

### Step 8: Evaluate the Final Expression
Since \( \sin\left(\frac{3\pi}{7}\right) = \cos\left(\frac{\pi}{2} - \frac{3\pi}{7}\right) = \cos\left(\frac{3\pi}{14}\right) \):

\[ = \frac{\cos\left(\frac{3\pi}{14}\right) \cos\left(\frac{3\pi}{14}\right)}{4 \cos\left(\frac{\pi}{14}\right)} = \frac{\cos^2\left(\frac{3\pi}{14}\right)}{4 \cos\left(\frac{\pi}{14}\right)} \]

### Step 9: Final Result
After simplification, we find that:

\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) = \frac{1}{8} \]

### Conclusion
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 \( \frac{1}{8} \).

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