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Friday, November 6, 2015

Prove the equality : 33163sin80=1+8sin10

Prove the equality :
33163sin80=1+8sin10

First, note that

cos220=cos20(212cos20)=cos20(2cos60cos20)=cos20((cos(60+20)+cos(6020))=cos20(cos80+cos40)=cos20cos80+cos20cos40=12(cos(20+80)+cos(8020))+12(cos(20+40)+cos(4020))=12(cos100+cos60)+12(cos60+cos20)=12(cos(90+10)+12)+12(12+cos20)=12(sin10)+14+14+12cos20=12(cos20sin10+1)



Next, we work from the LHS of the given equality and see that

\begin{align*}\sqrt{33 − 16\sqrt{3}\sin80^\circ}&=\sqrt{33-32\left(\dfrac{\sqrt{3}}{2}\right)\sin 80^\circ}\\&=\sqrt{33-32\sin 60^\circ\sin 80^\circ}\\&=\sqrt{33-16(2\sin 60^\circ\sin 80^\circ)}\\&=\sqrt{33-16\left(-\cos( 60^\circ+80^\circ)+\cos( 80^\circ-60^\circ)\right)}\\&=\sqrt{33-16\left(-\cos(140^\circ)+\cos(20^\circ)\right)}\\&=\sqrt{33-16\left(-\cos(180^\circ-40^\circ)+\cos(20^\circ)\right)}\\&=\sqrt{33-16\left(\cos(40^\circ)+\cos(20^\circ)\right)}\\&=\sqrt{33-16\cos(40^\circ)-16\cos(20^\circ)}\\&=\sqrt{33-16(2\cos^2(20^\circ)-1)-16\cos(20^\circ)}\\&=\sqrt{33-16(\cos20^\circ-\sin10^\circ+1-1)-16\cos(20^\circ)}\\&=\sqrt{33-32\cos20^\circ+16\sin10^\circ}\\&=\sqrt{33-32(1-2\sin^2 10^\circ)+16\sin10^\circ}\\&=\sqrt{1+16\sin10^\circ+64\sin^2 10^\circ)}\\&=\sqrt{(1+8\sin10^\circ)^2}\\&=1+8\sin10^\circ\end{align*}

and we're hence done.


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