Let $K=\sin \frac{\pi }{18}\sin \frac{\mathrm{5\pi }}{18}\sin \frac{\mathrm{7\pi }}{18}$ then the value of

$\sin \left(10K\frac{\pi }{3}\right)$ is :

To find the value of the given expression, we will solve it in two main steps:
1. Evaluate the product expression for $K$.
2. Substitute the value of $K$ into the expression $\sin \left(10K\frac{\pi }{3}\right)$ and simplify.

Step 1: Simplify the expression for $K$

The given value of $K$ is:

$K=\sin \frac{\pi }{18}\sin \frac{5\pi }{18}\sin \frac{7\pi }{18}$

Converting the angles from radians to degrees (where $\pi =180°$):

$\frac{\pi }{18}=10°$

$\frac{5\pi }{18}=50°$

$\frac{7\pi }{18}=70°$

Therefore, we can rewrite $K$ as:

$K=\sin (10°)\sin (50°)\sin (70°)$

We can use the standard trigonometric identity:
$\sin \left(\theta \right)\sin (60°-\theta )\sin (60°+\theta )=\frac{1}{4}\sin \left(3\theta \right)$

Substituting $\theta =10°$ into the identity, we get:

$\sin (10°)\sin (60°-10°)\sin (60°+10°)=\frac{1}{4}\sin (3\times 10°)$

$\sin (10°)\sin (50°)\sin (70°)=\frac{1}{4}\sin (30°)$

Since we know that $\sin (30°)=\frac{1}{2}$, we have:

$K=\frac{1}{4}\times \frac{1}{2}=\frac{1}{8}$

Step 2: Calculate the value of the target expression

Now, we substitute $K=\frac{1}{8}$ into the expression:

$\sin \left(10K\frac{\pi }{3}\right)=\sin (10\times \frac{1}{8}\times \frac{\pi }{3})$

$=\sin \left(\frac{5\pi }{12}\right)$

Converting $\frac{5\pi }{12}$ to degrees:

$\frac{5\times 180°}{12}=5\times 15°=75°$

Thus, we need to find $\sin (75°)$. We can evaluate this using the sum formula $\sin (A+B)=\sin A\cos B+\cos A\sin B$:

$\sin (75°)=\sin (45°+30°)$

$=\sin (45°)\cos (30°)+\cos (45°)\sin (30°)$

$=\left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right)+\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right)$

$=\frac{\sqrt{3}+1}{2\sqrt{2}}$

Therefore, the correct answer is:

$\frac{\sqrt{3}+1}{2\sqrt{2}}$