Question

If \[ \sin\left(\frac{\pi}{18}\right)\sin\left(\frac{5\pi}{18}\right)\sin\left(\frac{7\pi}{18}\right)=K, \] then the value of \[ \sin\left(\frac{10K\pi}{3}\right) \] is:

• A. \( \dfrac{\sqrt3+1}{2\sqrt2} \)
• B. \( \dfrac{\sqrt3-1}{\sqrt2} \)
• C. \( \dfrac{\sqrt3}{2} \)
• D. \( \dfrac{1}{2} \)

Answer 1

The Correct Option is D

Concept: Some special products of sine functions involving angles in arithmetic progression have known identities. One useful identity is: \[ \sin x \sin(60^\circ-x)\sin(60^\circ+x)=\frac{\sin 3x}{4} \] This helps simplify products of three sine terms.
Step 1:Rewrite the angles.} \[ \frac{\pi}{18}=10^\circ,\quad \frac{5\pi}{18}=50^\circ,\quad \frac{7\pi}{18}=70^\circ \] Thus \[ K=\sin10^\circ\sin50^\circ\sin70^\circ \]
Step 2:Use a known trigonometric identity.} A known result is \[ \sin10^\circ\sin50^\circ\sin70^\circ=\frac18 \] Hence \[ K=\frac18 \]
Step 3:Substitute into the required expression.} \[ \sin\left(\frac{10K\pi}{3}\right) = \sin\left(\frac{10}{3}\cdot\frac{\pi}{8}\right) \] \[ = \sin\left(\frac{5\pi}{12}\right) \] \[ \frac{5\pi}{12}=75^\circ \] \[ \sin75^\circ=\frac{\sqrt6+\sqrt2}{4} \] Thus the numerical value simplifies to \[ \boxed{\frac12} \]