Question
If $\theta = 30^\circ , $verify that$: \sin 3\theta = 4\sin\theta . \sin(60^\circ - \theta ) \sin(60^\circ + \theta )$
✓
Answer
Given$: \theta=30^{\circ}$
$ \sin 3 \theta$
$ =\sin 3 \times 30^{\circ}$
$ =\sin 90^{\circ}$
$ =1$
$ 4 \sin \theta \cdot \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right)$
$ =4 \sin 30^{\circ} \times \sin \left(60^{\circ}-30^{\circ}\right) \times \sin \left(60^{\circ}+30^{\circ}\right)$
$ =4 \sin 30^{\circ} \times \sin 30^{\circ} \times \sin 90^{\circ}$
$ =4 \times \frac{1}{2} \times \frac{1}{2} \times 1$
$ =1$
$ \Rightarrow \sin 3 \theta=4 \sin \theta \cdot \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right) .$
$ \sin 3 \theta$
$ =\sin 3 \times 30^{\circ}$
$ =\sin 90^{\circ}$
$ =1$
$ 4 \sin \theta \cdot \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right)$
$ =4 \sin 30^{\circ} \times \sin \left(60^{\circ}-30^{\circ}\right) \times \sin \left(60^{\circ}+30^{\circ}\right)$
$ =4 \sin 30^{\circ} \times \sin 30^{\circ} \times \sin 90^{\circ}$
$ =4 \times \frac{1}{2} \times \frac{1}{2} \times 1$
$ =1$
$ \Rightarrow \sin 3 \theta=4 \sin \theta \cdot \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right) .$
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