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Compute \( \large \displaystyle\sin^{6}3.25^\circ + 3\sin^{2}3.25^\circ\cos^{2}3.25^\circ + \cos^{6}3.25^\circ\)

 Jul 22, 2020
 #1
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compute

\(\sin^{6}(3.25^\circ) + 3\sin^{2}(3.25^\circ)\cos^{2}(3.25^\circ) + \cos^{6}(3.25^\circ)\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{\sin^{6}(3.25^\circ) + 3\sin^{2}(3.25^\circ)\cos^{2}(3.25^\circ) + \cos^{6}(3.25^\circ) \quad | \quad \cos^{2}(3.25^\circ)=1- \sin^{2}(3.25^\circ)} \\ &=& \sin^{6}(3.25^\circ) + 3\sin^{2}(3.25^\circ)\Big(1- \sin^{2}(3.25^\circ)\Big) + \Big(1- \sin^{2}(3.25^\circ)\Big)^3 \\ &=& \sin^{6}(3.25^\circ) + 3\sin^{2}(3.25^\circ)- 3\sin^{4}(3.25^\circ) + \Big(1- \sin^{2}(3.25^\circ)\Big)^3 \\ &=& \sin^{6}(3.25^\circ) + 3\sin^{2}(3.25^\circ)- 3\sin^{4}(3.25^\circ) + 1- 3\sin^{2}(3.25^\circ)+ 3\sin^{4}(3.25^\circ)-\sin^{6}(3.25^\circ) \\ &=& \mathbf{1} \\ \hline \end{array} \)

 

laugh

 Jul 22, 2020
edited by heureka  Jul 22, 2020
edited by heureka  Jul 22, 2020

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