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Find the maximum value of \( \cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1,\)
over all real numbers \(\theta_1, \theta_2, \theta_3,\theta_4,\) and \(\theta_5\).

 Dec 4, 2019
 #1
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Help with AM-GM
Find the maximum value of
\(\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1\),
over all real numbers \(\theta_1, \theta_2, \theta_3,\theta_4\), and \(\theta_5\).

 

\(\text{Let $x = \cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1$}\)

 

\(\mathbf{\huge{AM \geq GM }}\)

\(\small{ \begin{array}{|rcll|} \hline \dfrac{\cos^2\theta_1 + \sin^2 \theta_2}{2} &\geq& \sqrt{\cos^2 \theta_1 \sin^2 \theta_2} = \cos \theta_1 \sin \theta_2 \\ \dfrac{\cos^2\theta_2 + \sin^2 \theta_3}{2} &\geq& \cos \theta_2 \sin \theta_3 \\ \dfrac{\cos^2\theta_3 + \sin^2 \theta_4}{2} &\geq& \cos \theta_3 \sin \theta_4 \\ \dfrac{\cos^2\theta_4 + \sin^2 \theta_5}{2} &\geq& \cos \theta_4 \sin \theta_5 \\ \dfrac{\cos^2\theta_5 + \sin^2 \theta_1}{2} &\geq& \cos \theta_5 \sin \theta_1 \\\\ \dfrac{1}{2}\left(\cos^2\theta_1 + \sin^2 \theta_2+ \cos^2\theta_2 + \sin^2 \theta_3+ \cos^2\theta_3 + \sin^2 \theta_4 +\cos^2\theta_4 + \sin^2 \theta_5 + \cos^2\theta_5 + \sin^2 \theta_1 \right) &\geq& x \\ \dfrac{1}{2}\left(\underbrace{\cos^2 \theta_1+ \sin^2 \theta_1}_{=1} + \underbrace{\sin^2 \theta_2+ \cos^2\theta_2}_{=1} + \underbrace{\sin^2 \theta_3+ \cos^2\theta_3}_{=1} + \underbrace{\sin^2 \theta_4+ \cos^2\theta_4}_{=1} + \underbrace{\sin^2 \theta_5+ \cos^2\theta_5}_{=1} \right) &\geq& x \\ \dfrac{5}{2}&\geq& x \\ x &\leq& \dfrac{5}{2} \\ \mathbf{ \cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1 }&\leq& \mathbf{\dfrac{5}{2}} \\ \hline \end{array} }\)

 

The maximum value of \( \cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1 \)is \(\mathbf{\dfrac{5}{2}}\)

 

laugh

 Dec 4, 2019

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