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Question Number 145852 by mathdanisur last updated on 08/Jul/21

sin(π/(24))∙cos(π/(24))∙cos(π/(12))=?

$${sin}\frac{\pi}{\mathrm{24}}\centerdot{cos}\frac{\pi}{\mathrm{24}}\centerdot{cos}\frac{\pi}{\mathrm{12}}=? \\ $$

Answered by waiphyoemaung last updated on 08/Jul/21

solution  sin(π/(24)).cos(π/(24)).cos(π/(12))  =(1/2).(2sin(π/(24))cos(π/(24)))cos(π/(12))  =(1/2).sin(π/(12)).cos(π/(12))  (∵sin 2α=2sin α cos α)  =(1/4).2sin(π/(12)).cos(π/(12))  =(1/4).sin(π/6)  =(1/4)×(1/2)   (∵sin(π/6)=(1/2))  =(1/8)

$$\mathrm{solution} \\ $$$$\mathrm{sin}\frac{\pi}{\mathrm{24}}.\mathrm{cos}\frac{\pi}{\mathrm{24}}.\mathrm{cos}\frac{\pi}{\mathrm{12}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}.\left(\mathrm{2sin}\frac{\pi}{\mathrm{24}}\mathrm{cos}\frac{\pi}{\mathrm{24}}\right)\mathrm{cos}\frac{\pi}{\mathrm{12}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}.\mathrm{sin}\frac{\pi}{\mathrm{12}}.\mathrm{cos}\frac{\pi}{\mathrm{12}}\:\:\left(\because\mathrm{sin}\:\mathrm{2}\alpha=\mathrm{2sin}\:\alpha\:\mathrm{cos}\:\alpha\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}.\mathrm{2sin}\frac{\pi}{\mathrm{12}}.\mathrm{cos}\frac{\pi}{\mathrm{12}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}.\mathrm{sin}\frac{\pi}{\mathrm{6}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}×\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\left(\because\mathrm{sin}\frac{\pi}{\mathrm{6}}=\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{8}} \\ $$

Commented by mathdanisur last updated on 08/Jul/21

Thanks Ser cool

$${Thanks}\:{Ser}\:{cool} \\ $$

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