Question

# The value of $$\sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ}$$ is equal to.

A
23
B
12
C
18
D
13

Solution

## The correct option is C $$\dfrac {1}{8}$$Consider, $$\sin 12^{o}\sin 48^{o}\sin 54^{o}$$                $$=(\sin 12^{o}\sin 48^{o})\sin 54^{o}$$                $$=\dfrac{1}{2}(\cos(12-48)-\cos(12+48))\sin(90^{o}-36^{o})$$ ..... [Using $$\sin a \sin b$$ formula]                $$=\dfrac{1}{2}(\cos 36^{o}-\cos 60^{o})\cos 36^{o}$$ ........ $$[\because \cos(-x)=\cos x \text{ and } \sin(90^{o}-x)=\cos x]$$                $$=\dfrac{1}{2}\left(\dfrac{\sqrt{5}+1}{4}-\dfrac{1}{2}\right)\left(\dfrac{\sqrt{5}+1}{4}\right)$$    $$\left[\cos 36^o=\dfrac{\sqrt5 +1}{4},\sin 36^o=\dfrac{\sqrt5 -1}{4} \right]$$                $$=\dfrac{1}{2\times 4\times 4}\left({\sqrt{5}+1}-2 \right)\left({\sqrt{5}+1}\right)$$                $$=\dfrac{1}{32}\left(\sqrt{5}-1 \right)\left({\sqrt{5}+1}\right)$$                $$=\dfrac{1}{32}(5-1)$$                $$=\dfrac{4}{32}=\dfrac{1}{8}$$Hence,  $$\sin 12^{o}\sin 48^{o}\sin 54^{o}=\dfrac{1}{8}$$Mathematics

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