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we-have-z-e-2pi-7-i-a-z-z-2-z-4-and-b-z-3-z-5-z-6-we-know-a-b-1-and-1-b-a-find-S-cos-2pi-7-cos-4pi-7-cos-8pi-7-thanks-for-help-

Question Number 145162 by Skabetix last updated on 02/Jul/21 $${we}\:{have}\:{z}\:=\:{e}^{\frac{\mathrm{2}{pi}}{\mathrm{7}}{i}} \: \\ $$$${a}\:=\:{z}+\:{z}^{\mathrm{2}} \:+\:{z}^{\mathrm{4}} \:\:{and}\:{b}\:=\:{z}^{\mathrm{3}} \:+\:{z}^{\mathrm{5}} \:+{z}^{\mathrm{6}} \\ $$$${we}\:{know}\:\:{a}\:+\:{b}\:=\:−\mathrm{1}\:{and}\:\mathrm{1}−{b}={a} \\ $$$${find}\:{S}\:=\:{cos}\left(\frac{\mathrm{2}{pi}}{\mathrm{7}}\right)+\:{cos}\left(\frac{\mathrm{4}{pi}}{\mathrm{7}}\right)\:+\:{cos}\left(\frac{\mathrm{8}{pi}}{\mathrm{7}}\right) \\ $$$${thanks}\:{for}\:{help} \\ $$…

find-k-if-y-2-y-k-is-a-perfect-square-

Question Number 145057 by 7770 last updated on 02/Jul/21 $$\boldsymbol{{find}}\:\:\boldsymbol{{k}}\:\:\boldsymbol{{if}}\:\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{y}}+\boldsymbol{{k}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{perfect}} \\ $$$$\boldsymbol{{square}} \\ $$ Commented by mr W last updated on 02/Jul/21 $${y}^{\mathrm{2}} +{y}+\frac{\mathrm{1}}{\mathrm{4}}=\left({y}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}}…

Question-144875

Question Number 144875 by lapache last updated on 30/Jun/21 Answered by Dwaipayan Shikari last updated on 30/Jun/21 $$\frac{\mathrm{1}}{{n}!}\int_{\mathrm{0}} ^{\infty} {x}^{{n}} \mathrm{sin}\:\left({x}\right){e}^{−{x}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{in}!}\int_{\mathrm{0}} ^{\infty}…