CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

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


A
23
loader
B
12
loader
C
18
loader
D
13
loader

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

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More



footer-image