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Category: Algebra

Find-0-6-sin-x-sin-x-pi-3-sin-x-2pi-3-sin-3x-cos-3x-dx-Answer-pi-48-

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Question Number 159528 by HongKing last updated on 18/Nov/21 $$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{6}}} {\int}}\frac{\mathrm{sin}\left(\mathrm{x}\right)\centerdot\mathrm{sin}\left(\mathrm{x}\:+\:\frac{\pi}{\mathrm{3}}\right)\centerdot\mathrm{sin}\left(\mathrm{x}\:+\:\frac{\mathrm{2}\pi}{\mathrm{3}}\right)}{\mathrm{sin}\left(\mathrm{3x}\right)\:+\:\mathrm{cos}\left(\mathrm{3x}\right)}\:\mathrm{dx} \\ $$$$\mathrm{Answer}:\:\:\frac{\pi}{\mathrm{48}} \\ $$ Answered by mnjuly1970 last updated on 18/Nov/21…

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let-give-the-polynomial-p-x-x-1-n-x-1-n-with-n-from-N-1-give-the-factorisation-of-p-x-inside-C-x-2-prove-that-k-0-n-1-cotan-kpi-2p-1-1-2p-1-

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Question Number 28434 by abdo imad last updated on 25/Jan/18 $${let}\:{give}\:{the}\:{polynomial}\:{p}\left({x}\right)=\left({x}+\mathrm{1}\right)^{{n}} −\left({x}−\mathrm{1}\right)^{{n}} {with}\:{n} \\ $$$${from}\:{N}^{\ast} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{the}\:{factorisation}\:{of}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cotan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{p}+\mathrm{1}}} \\ $$ Terms of…

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let-put-w-e-i-2pi-n-calculate-S-n-k-0-n-1-1-x-w-k-and-W-n-k-0-n-1-1-x-w-k-2-

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Question Number 28432 by abdo imad last updated on 25/Jan/18 $${let}\:{put}\:{w}={e}^{{i}\frac{\mathrm{2}\pi}{{n}}} \:{calculate}\:\:{S}_{{n}} =\:\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{\mathrm{1}}{{x}−{w}^{{k}} }\:\:{and} \\ $$$${W}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{\mathrm{1}}{\left({x}−{w}^{{k}} \right)^{\mathrm{2}} }\:. \\ $$…

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