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Compute \( \sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{7\pi}{12} + \sin \frac{9\pi}{12} + \sin \frac{11\pi}{12}. \)

 May 19, 2020
 #2
avatar+25543 
+2

Compute

\( \sin \dfrac{\pi}{12} + \sin \dfrac{3\pi}{12} + \sin \dfrac{5\pi}{12} + \sin \dfrac{7\pi}{12} + \sin \dfrac{9\pi}{12} + \sin \dfrac{11\pi}{12}\).

 

Formula: \(\sin(x)+\sin(y) = 2 \sin \left( \dfrac{x+y}{2} \right) \cos \left( \dfrac{x-y}{2} \right) \\ \cos(x)+\cos(y) = 2 \cos \left( \dfrac{x+y}{2} \right) \cos \left( \dfrac{x-y}{2} \right)\)

 

\(\small{ \begin{array}{|rcll|} \hline && \mathbf{\sin \left( \dfrac{\pi}{12} \right) + \sin \left( \dfrac{3\pi}{12} \right) + \sin \left( \dfrac{5\pi}{12} \right) + \sin \left( \dfrac{7\pi}{12} \right) + \sin \left( \dfrac{9\pi}{12} \right) + \sin \left( \dfrac{11\pi}{12} \right)} \\\\ &=& \Bigg( \sin \left( \dfrac{11\pi}{12} \right) + \sin \left( \dfrac{\pi}{12} \right) \Bigg) + \Bigg( \sin \left( \dfrac{9\pi}{12} \right) + \sin \left( \dfrac{3\pi}{12} \right)\Bigg) + \Bigg( \sin \left( \dfrac{7\pi}{12} \right) + \sin \left( \dfrac{5\pi}{12} \right) \Bigg)\\\\ &=& 2 \sin \left( \dfrac{ \dfrac{11\pi}{12}+\dfrac{\pi}{12}}{2} \right) \cos \left( \dfrac{ \dfrac{11\pi}{12}-\dfrac{\pi}{12}}{2} \right) +2 \sin \left( \dfrac{ \dfrac{9\pi}{12}+\dfrac{\pi}{3}}{2} \right) \cos \left( \dfrac{ \dfrac{11\pi}{9}-\dfrac{\pi}{3}}{2} \right) +2 \sin \left( \dfrac{ \dfrac{7\pi}{12}+\dfrac{\pi}{5}}{2} \right) \cos \left( \dfrac{ \dfrac{7\pi}{12}-\dfrac{\pi}{5}}{2} \right) \\\\ &=& 2 \sin \left( \dfrac{\pi}{2} \right) \cos \left( \dfrac{5\pi}{12} \right) +2 \sin \left( \dfrac{\pi}{2} \right) \cos \left( \dfrac{3\pi}{12} \right) +2 \sin \left( \dfrac{\pi}{2} \right) \cos \left( \dfrac{\pi}{12} \right) \\\\ &=& 2 \sin \left( \dfrac{\pi}{2} \right) \left( \cos \left( \dfrac{5\pi}{12} \right) +\cos \left( \dfrac{3\pi}{12} \right)+ \cos \left( \dfrac{\pi}{12} \right)\right) \quad | \quad \sin \left( \dfrac{\pi}{2} \right) = 1 \\\\ &=& 2 \left( \cos \left( \dfrac{5\pi}{12} \right) +\cos \left( \dfrac{3\pi}{12} \right)+ \cos \left( \dfrac{\pi}{12} \right)\right) \\\\ &=& 2 \left( \cos \left( \dfrac{5\pi}{12} \right) +\cos \left( \dfrac{\pi}{12} \right) + \cos \left( \dfrac{3\pi}{12} \right)\right) \\\\ &=& 2 \left( 2 \cos \left( \dfrac{ \dfrac{5\pi}{12}+\dfrac{\pi}{12}}{2} \right) \cos \left( \dfrac{ \dfrac{5\pi}{12}-\dfrac{\pi}{12}}{2} \right) + \cos \left( \dfrac{3\pi}{12} \right)\right) \\\\ &=& 2 \left( 2 \cos \left( \dfrac{3\pi}{12} \right) \cos \left( \dfrac{\pi}{6} \right) + \cos \left( \dfrac{3\pi}{12} \right)\right) \\\\ &=& 2 \cos \left( \dfrac{3\pi}{12} \right) \left( 2 \cos \left( \dfrac{\pi}{6} \right) + 1 \right) \\\\ &=& 2 \cos \left( \dfrac{\pi}{4} \right) \left( 2 \cos \left( \dfrac{\pi}{6} \right) + 1 \right) \quad | \quad \cos \left( \dfrac{\pi}{4} \right) = \dfrac{\sqrt{2}}{2},\ \cos \left( \dfrac{\pi}{6} \right) = \dfrac{\sqrt{3}}{2} \\\\ &=& 2* \dfrac{\sqrt{2}}{2} \left( 2* \dfrac{\sqrt{3}}{2} + 1 \right) \\\\ &=& \sqrt{2}(\sqrt{3} + 1 ) \\\\ &=& \mathbf{\sqrt{2}+\sqrt{6}} \\ \hline \end{array} }\)

 

 

laugh

 May 19, 2020
 #3
avatar+110715 
0

Thanks Heureka,

 

Here is another approach.    laugh

 

\(\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{7\pi}{12} + \sin \frac{9\pi}{12} + \sin \frac{11\pi}{12}\\ =\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{\pi}{12}\\ =2(\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} )\\ =2(\sin \frac{\pi}{12} + \sin \frac{\pi}{4} + \cos \frac{\pi}{12} )\\ =2(\sin \frac{\pi}{12} + \cos \frac{\pi}{12} )+2\sin \frac{\pi}{4}\\ =2\sqrt{(\sin \frac{\pi}{12} + \cos \frac{\pi}{12} )^2}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(\sin^2 \frac{\pi}{12} + \cos^2 \frac{\pi}{12} +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(1 +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(1 +\sin \frac{\pi}{6})}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(1 +\frac{1}{2})}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(\frac{3}{2})}+ \frac{2}{\sqrt{2}}\\ =2(\frac{\sqrt3}{\sqrt2})+ \frac{2}{\sqrt{2}}\\ =\sqrt6+\sqrt2\)

 

 

 

 

LaTex

\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{7\pi}{12} + \sin \frac{9\pi}{12} + \sin \frac{11\pi}{12}\\

=\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{\pi}{12}\\

=2(\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} )\\

=2(\sin \frac{\pi}{12} + \sin \frac{\pi}{4} + \cos \frac{\pi}{12} )\\

=2(\sin \frac{\pi}{12}   + \cos \frac{\pi}{12} )+2\sin \frac{\pi}{4}\\

=2\sqrt{(\sin \frac{\pi}{12}   + \cos \frac{\pi}{12} )^2}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(\sin^2 \frac{\pi}{12}   + \cos^2 \frac{\pi}{12} +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(1 +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(1 +\sin \frac{\pi}{6})}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(1 +\frac{1}{2})}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(\frac{3}{2})}+ \frac{2}{\sqrt{2}}\\

=2(\frac{\sqrt3}{\sqrt2})+ \frac{2}{\sqrt{2}}\\

=\sqrt6+\sqrt2

 May 19, 2020

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