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Prove that $$\sin(12^\circ)\sin(48^\circ)\sin(54^\circ)=\frac18.$$ Without using a calculator. I tried all identities I know but I have no idea how to proceed. I always get stuck on finding $\sin36^\circ$.

Tunk-Fey
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Kanishk
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    This is false as stated. Please, do mention that you measure angles in degrees, not in radians. The overwhelming majority of mathematicians will by default treat the argument of $\sin$ as being in radians. – Dan Shved Jul 26 '14 at 06:06
  • Thankyou for the suggestion – Kanishk Jul 26 '14 at 06:08
  • We have $$ \sin12^\circ\sin48^\circ\sin54^\circ=\frac18\tag1 $$ Multiplying $(1)$ by $2$ yields $$ 2\sin12^\circ\sin48^\circ\sin54^\circ=\frac14\tag2 $$ Using identities: $2\sin a\sin b=\cos(a-b)-\cos(a+b)$ and $\sin(90^\circ-x)=\cos x$ yields $$ \left(\cos36^\circ-\frac12\right)\cos36^\circ=\frac14\tag3 $$ Now, setting $x=\cos36^\circ$ and multiplying $(3)$ by $4$ yields $$ 4x^2-2x-1=0\tag4 $$ Solving $(4)$ then use identity $\sin\theta=\sqrt{1-\cos^2\theta}$. – Tunk-Fey Jul 26 '14 at 07:23
  • @Tunk-Fey, $$(3)\implies\cos36^\circ-\cos72^\circ=\frac12$$ (http://math.stackexchange.com/questions/827540/proving-trigonometric-equation-cos36-circ-cos72-circ-1-2) – lab bhattacharjee Jul 26 '14 at 08:08

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Use the following: $$\sin\theta \sin\left(\frac{\pi}{3}-\theta\right)\sin\left(\frac{\pi}{3}+\theta\right)=\frac{1}{4}\sin(3\theta)$$ Then you have: $$\sin(12^{\circ})\sin(60^{\circ}-12^{\circ})\sin(60^{\circ}+12^{\circ})=\frac{1}{4}\sin(36^{\circ})$$ $$\Rightarrow \sin(12^{\circ})\sin(48^{\circ})=\frac{1}{4}\frac{\sin(36^{\circ})}{\sin(72^{\circ})}$$ $$\Rightarrow \sin(12^{\circ})\sin(48^{\circ})\sin(54^{\circ})=\frac{1}{4}\frac{\sin(36^{\circ})}{\sin(72^{\circ})}\sin(54^{\circ})=\frac{1}{4}\frac{\sin(36^{\circ})}{\sin(72^{\circ})}\cos(36^{\circ})$$ $$\Rightarrow \sin(12^{\circ})\sin(48^{\circ})\sin(54^{\circ})=\boxed{\dfrac{1}{8}}$$

Pranav Arora
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$2\sin 12\cdot \sin 48 = \cos (48 - 12) - \cos (48 + 12) = \cos 36 - \dfrac{1}{2}$. So the problem is to find $\cos 36$. Use $1 - 2x^2 = 3x - 4x^3$ to solve for $\sin 18$ ( not hard ), then find $\cos 36$, and $\sin 54$. In fact, the equation is: $4x^3 - 2x^2 - 3x + 1 = 0 \to (x-1)(4x^2 + 2x - 1) = 0$, so $\sin 18 = x = \dfrac{\sqrt{5} - 1}{4}$. You can take it from here.

DeepSea
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