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Question

Mathematics Question on Trigonometric Functions

Value of sin⁡12∘.sin⁡ 48∘.sin⁡ 54∘\sin 12^{\circ}.\sin \, 48^{\circ}.\sin \, 54^{\circ} is:

A

−18\frac{-1}{8}

B

8

C

18\frac{1}{8}

D

-8

Answer

18\frac{1}{8}

Explanation

Solution

Let θ=12∘\theta = 12^{\circ} Consider given expression =1sin⁡72∘sin⁡12∘sin⁡48∘sin⁡54∘sin⁡72∘= \frac{1}{\sin 72^{\circ}} \sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ} \sin 72^{\circ} =12sin⁡3(12∘)sin⁡54∘sin⁡72∘ = \frac{1}{2} \frac{\sin 3(12^{\circ}) \sin 54^{\circ}}{\sin 72^{\circ}} =sin⁡36∘sin⁡54∘8sin⁡36∘cos⁡36∘ = \frac{\sin36^{\circ} \sin54^{\circ}}{8\sin36^{\circ} \cos36^{\circ}} =cos⁡36∘8cos⁡36∘= \frac{\cos36^{\circ}}{8 \cos36^{\circ}} =18= \frac{1}{8}