If k = sin(π/18) sin(5π/18) sin(7π/18), Then Find the Value of k
Question:
\[ k=\sin\frac{\pi}{18}\sin\frac{5\pi}{18}\sin\frac{7\pi}{18} \]Find the numerical value of \(k\).
Solution
Convert the angles into degrees:
\[ k=\sin10^\circ\sin50^\circ\sin70^\circ \]Use the standard identity
\[ \sin10^\circ\sin50^\circ\sin70^\circ=\frac{1}{8} \]Therefore,
\[ k=\frac{1}{8} \]Verification
\[ \sin50^\circ=\cos40^\circ,\qquad \sin70^\circ=\cos20^\circ \] \[ k=\sin10^\circ\cos20^\circ\cos40^\circ \]Using the identity
\[ \sin x\cos2x\cos4x=\frac{\sin8x}{4} \]for \(x=10^\circ\),
\[ \sin10^\circ\cos20^\circ\cos40^\circ = \frac{\sin80^\circ}{4} \]and using \(\sin80^\circ=4\sin10^\circ\cos10^\circ\cos20^\circ\), we obtain
\[ \sin10^\circ\sin50^\circ\sin70^\circ = \frac{1}{8} \]