Question

\(\left(\sin 5^{\circ} \sin 55^{\circ} \sin 65^{\circ} \sin 75^{\circ}\right)^{-1}\) is equal to ________.

Hint:

Use \(\sin\theta\sin(60^\circ-\theta)\sin(60^\circ+\theta)=\frac14\sin3\theta\) with \(\theta=5^\circ\).

Answer 1

Correct Answer: 16

Take \(\theta=5^\circ\): then \(60^\circ-\theta=55^\circ\) and \(60^\circ+\theta=65^\circ\).
Using \(\sin\theta\sin(60^\circ-\theta)\sin(60^\circ+\theta)=\frac14\sin3\theta\): \(\sin5^\circ\sin55^\circ\sin65^\circ=\frac14\sin15^\circ\).
Also \(\sin75^\circ=\cos15^\circ\), so the product is \(\frac14\sin15^\circ\cos15^\circ=\frac18\sin30^\circ=\frac1{16}\).
Hence \(\left(\sin5^\circ\sin55^\circ\sin65^\circ\sin75^\circ\right)^{-1}=16\).
Answer: 16