Question Number 76872 by john santu last updated on 31/Dec/19 $${the}\:{product}\:{of}\:\mathrm{3}\:{integer}\:{x},{y},{z}\:{is}\:\mathrm{192} \\ $$$$.\:{z}\:=\:\mathrm{4}\:{and}\:{t}\:{is}\:{equal}\:{to}\:{average}? \\ $$$${of}\:{x}\:{and}\:{y}\:.\:{what}\:{is}\:{the}\:{minimum}\: \\ $$$${posible}\:{value}\:{of}\:{t}? \\ $$ Commented by mr W last updated…
Question Number 76830 by peter frank last updated on 30/Dec/19 Commented by mathmax by abdo last updated on 05/Jan/20 $${let}\:{remember}\:{that}\:{arctanz}\:=\frac{\mathrm{1}}{\mathrm{2}{i}}{ln}\left(\frac{\mathrm{1}+{iz}}{\mathrm{1}−{iz}}\right)\:\left({result}\:{proved}\right) \\ $$$${iln}\left(\frac{{a}−{ib}}{{a}+{ib}}\right)\:=−{iln}\left(\frac{{a}+{ib}}{{a}−{ib}}\right)\:=−{iln}\left(\frac{\mathrm{1}+{i}\frac{{b}}{{a}}}{\mathrm{1}−{i}\frac{{b}}{{a}}}\right)\:=−{i}\left(\mathrm{2}{i}\right){arctan}\left(\frac{{b}}{{a}}\right) \\ $$$$=\mathrm{2}\:{arctan}\left(\frac{{b}}{{a}}\right)\:\Rightarrow{tan}\left({iln}\left(\frac{{a}−{ib}}{{a}+{ib}}\right)\right)={tan}\left(\mathrm{2}{arctan}\left(\frac{{b}}{{a}}\right)\right) \\…
Question Number 76718 by peter frank last updated on 29/Dec/19 $${For}\:{n}\:\in\:{N}\:{prove}\:{by}\:{mathematical} \\ $$$${induction}\:{that} \\ $$$$\mathrm{cos}\:\alpha+\mathrm{cos}\:\left(\alpha+\beta\right)+\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]+…\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]= \\ $$$$\frac{\mathrm{cos}\:\left[\alpha+\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)\beta\right]\mathrm{sin}\:\frac{{n}\beta}{\mathrm{2}}}{\mathrm{sin}\:\frac{{n}}{\mathrm{2}}} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 76716 by peter frank last updated on 29/Dec/19 $${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{x}\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}=\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$ Commented by abdomathmax last updated on…
Question Number 76717 by peter frank last updated on 29/Dec/19 $${A}\:{triangle}\:{is}\:{formed}\:{by} \\ $$$${the}\:{three}\:{straight}\:{line} \\ $$$${y}={m}_{\mathrm{1}} {x}+\frac{{a}}{{m}_{\mathrm{1}} } \\ $$$${y}={m}_{\mathrm{2}} {x}+\frac{{a}}{{m}_{\mathrm{2}} } \\ $$$${y}={m}_{\mathrm{3}} {x}+\frac{{a}}{{m}_{\mathrm{3}} }…
Question Number 76715 by peter frank last updated on 29/Dec/19 $${If}\:{u}={arc}\mathrm{sin}\:\frac{{x}}{{y}}+{arc}\mathrm{tan}\:\frac{{y}}{{x}} \\ $$$${show}\:{that}\: \\ $$$${x}\frac{\partial{u}}{{dx}}+{y}\frac{\partial{u}}{{dy}}=\mathrm{0} \\ $$ Commented by abdomathmax last updated on 30/Dec/19 $$\frac{\partial{u}}{\partial{x}}\:=\frac{\mathrm{1}}{{y}\sqrt{\mathrm{1}−\frac{{x}^{\mathrm{2}}…
Question Number 142150 by iloveisrael last updated on 27/May/21 $$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)\:=?\: \\ $$ Commented by liberty last updated on 27/May/21 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}}…
Question Number 76580 by Rio Michael last updated on 28/Dec/19 $$\mathrm{how}\:\mathrm{many}\:\mathrm{divisors}\:\mathrm{does}\:\mathrm{180045}\:\mathrm{has}? \\ $$ Commented by mr W last updated on 28/Dec/19 $$\frac{\mathrm{180045}}{\mathrm{5}}=\mathrm{36009} \\ $$$$\frac{\mathrm{36009}}{\mathrm{3}}=\mathrm{12003} \\…
Question Number 76576 by Rio Michael last updated on 28/Dec/19 $${Compute}\:{the}\:{Gcd}\:\left({greatest}\:{common}\:{divisor}\right) \\ $$$${of}\:\:\mathrm{45}\:{and}\:\mathrm{1287}\: \\ $$ Answered by john santu last updated on 28/Dec/19 $$\mathrm{45}\:=\:\mathrm{3}^{\mathrm{2}} ×\mathrm{5}…
Question Number 11005 by ajfour last updated on 06/Mar/17 $${prove}\:{that}\:\frac{\mathrm{9}}{\mathrm{4}}\:<\:\left(\mathrm{log}\:_{\mathrm{2}} \mathrm{3}\right)^{\mathrm{2}} \:<\:\frac{\mathrm{25}}{\mathrm{9}}\:\:. \\ $$ Answered by mrW1 last updated on 06/Mar/17 $$\mathrm{2log}_{\mathrm{2}} \:\mathrm{3}=\mathrm{log}_{\mathrm{2}} \:\mathrm{3}^{\mathrm{2}} =\mathrm{log}_{\mathrm{2}}…