If `K=sin(pi/(18))sin((5pi)/(18))sin((7pi)/(18)),` then the numerical value of `K` is_____

Explanation

To find the value of \( K = \sin\left(\frac{\pi}{18}\right) \sin\left(\frac{5\pi}{18}\right) \sin\left(\frac{7\pi}{18}\right) \), we can use some trigonometric identities and properties. Let's solve it step by step.

### Step 1: Use the product-to-sum identities
We will use the identity:
\[ \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \]
to simplify the product of two sine functions.
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