If (18)(518)(718) = K(10K3)

Question

If $\sin\left(\dfrac\pi{18}\right)\sin\left(\dfrac{5\pi}{18}\right)\sin\left(\dfrac{7\pi}{18}\right) = K\sin\left(\dfrac{10K\pi}3\right)$

Accepted Answer

Writing the angles in degrees, $K = \sin10^\circ\sin50^\circ\sin70^\circ$. Using $\sin3x = 4\sin x\sin(60^\circ+x)\sin(60^\circ-x)$ with $x=10^\circ$:

$$\sin30^\circ = 4\sin10^\circ\sin70^\circ\sin50^\circ \implies \tfrac12 = 4K \implies K = \tfrac18$$

$$\sin\left(\frac{10K\pi}3\right) = \sin\left(\frac{10}{3}\cdot\frac18\pi\right) = \sin\left(\frac{10\pi}{24}\right) = \sin\left(\frac{5\pi}{12}\right) = \sin75^\circ$$

$$\sin75^\circ = \sin(45^\circ+30^\circ) = \sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ = \frac1{\sqrt2}\cdot\frac{\sqrt3}2+\frac1{\sqrt2}\cdot\frac12 = \frac{\sqrt3+1}{2\sqrt2}$$

Solutions · 1