Trigonometrical Ratios and Identities 1 Question 1
1. The value of $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}$ is
(a) $\frac{1}{36}$
(b) $\frac{1}{32}$
(c) $\frac{1}{16}$
(d) $\frac{1}{18}$
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Answer:
Correct Answer: 1. (c)
Solution:
- We have, $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}$ $=\sin \left(30^{\circ}\right)\left[\sin \left(10^{\circ}\right) \sin \left(50^{\circ}\right) \sin \left(70^{\circ}\right)\right]$
$$ \begin{aligned} & =\frac{1}{2}\left[\sin \left(10^{\circ}\right) \sin \left(60^{\circ}-10^{\circ}\right) \sin \left(60^{\circ}+10^{\circ}\right)\right] \\ & =\frac{1}{2} \frac{1}{4} \sin \left(3\left(10^{\circ}\right)\right) \end{aligned} $$
$$ \left[\because \sin \theta \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right)=\frac{1}{4} \sin 3 \theta\right] $$
$$ =\frac{1}{8} \sin 30^{\circ}=\frac{1}{8} \times \frac{1}{2}=\frac{1}{16} $$
