Menu Close

I-would-like-that-you-help-me-to-show-this-equality-16cos-24-cos-5-24-cos-7-24-cos-11-24-1-




Question Number 75048 by mathocean1 last updated on 06/Dec/19
I would like that you help me to   show this equality:  16cos (Π/(24))cos((5Π)/(24))cos((7Π)/(24))cos((11Π)/(24))=1
$$\mathrm{I}\:\mathrm{would}\:\mathrm{like}\:\mathrm{that}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\: \\ $$$$\mathrm{show}\:\mathrm{this}\:\mathrm{equality}: \\ $$$$\mathrm{16cos}\:\frac{\Pi}{\mathrm{24}}\mathrm{cos}\frac{\mathrm{5}\Pi}{\mathrm{24}}\mathrm{cos}\frac{\mathrm{7}\Pi}{\mathrm{24}}\mathrm{cos}\frac{\mathrm{11}\Pi}{\mathrm{24}}=\mathrm{1} \\ $$
Commented by mind is power last updated on 06/Dec/19
identite used  sin(a)=cos((π/2)−a)  sin((π/2)+α)=cos(α),  sin(2x)=2sin(x)cos(x)
$$\mathrm{identite}\:\mathrm{used} \\ $$$$\mathrm{sin}\left(\mathrm{a}\right)=\mathrm{cos}\left(\frac{\pi}{\mathrm{2}}−\mathrm{a}\right) \\ $$$$\mathrm{sin}\left(\frac{\pi}{\mathrm{2}}+\alpha\right)=\mathrm{cos}\left(\alpha\right),\:\:\mathrm{sin}\left(\mathrm{2x}\right)=\mathrm{2sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right) \\ $$
Answered by mind is power last updated on 06/Dec/19
cos(((11π)/(24)))=sin((π/(24)))  cos(((5π)/(24)))=sin(((7π)/(24)))  16cos((π/(24)))cos(((5π)/(24)))cos(((7π)/(24)))cos(((11π)/(24)))=16cos((π/(24)))sin((π/(24)))cos(((7π)/(24)))sin(((7π)/(24)))  =4sin((π/(12)))sin(((7π)/(12)))  =4sin((π/(12)))sin((π/2)+(π/(12)))=2.2sin((π/(12)))cos((π/(12)))  =2sin((π/6))=2.(1/2)=1
$$\mathrm{cos}\left(\frac{\mathrm{11}\pi}{\mathrm{24}}\right)=\mathrm{sin}\left(\frac{\pi}{\mathrm{24}}\right) \\ $$$$\mathrm{cos}\left(\frac{\mathrm{5}\pi}{\mathrm{24}}\right)=\mathrm{sin}\left(\frac{\mathrm{7}\pi}{\mathrm{24}}\right) \\ $$$$\mathrm{16cos}\left(\frac{\pi}{\mathrm{24}}\right)\mathrm{cos}\left(\frac{\mathrm{5}\pi}{\mathrm{24}}\right)\mathrm{cos}\left(\frac{\mathrm{7}\pi}{\mathrm{24}}\right)\mathrm{cos}\left(\frac{\mathrm{11}\pi}{\mathrm{24}}\right)=\mathrm{16cos}\left(\frac{\pi}{\mathrm{24}}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{24}}\right)\mathrm{cos}\left(\frac{\mathrm{7}\pi}{\mathrm{24}}\right)\mathrm{sin}\left(\frac{\mathrm{7}\pi}{\mathrm{24}}\right) \\ $$$$=\mathrm{4sin}\left(\frac{\pi}{\mathrm{12}}\right)\mathrm{sin}\left(\frac{\mathrm{7}\pi}{\mathrm{12}}\right) \\ $$$$=\mathrm{4sin}\left(\frac{\pi}{\mathrm{12}}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{2}}+\frac{\pi}{\mathrm{12}}\right)=\mathrm{2}.\mathrm{2sin}\left(\frac{\pi}{\mathrm{12}}\right)\mathrm{cos}\left(\frac{\pi}{\mathrm{12}}\right) \\ $$$$=\mathrm{2sin}\left(\frac{\pi}{\mathrm{6}}\right)=\mathrm{2}.\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{1} \\ $$
Commented by mathocean1 last updated on 06/Dec/19
thank you sir!
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}! \\ $$
Commented by peter frank last updated on 06/Dec/19
thank you
$${thank}\:{you} \\ $$
Commented by peter frank last updated on 06/Dec/19
thank you
$${thank}\:{you} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *