Question Number 210311 by Spillover last updated on 05/Aug/24 $${Let}\:{a}_{\mathrm{1}} =\mathrm{1}\:\:\:{a}_{\mathrm{2}} =\mathrm{2}^{\mathrm{1}} \:\:\:\:{a}_{\mathrm{3}} =\mathrm{3}^{\left(\mathrm{2}^{\mathrm{1}} \right)} \:\:{a}_{\mathrm{4}} =\mathrm{4}^{\left(\mathrm{3}^{\left(\mathrm{2}^{\mathrm{1}} \right)} \right)} \\ $$$${find}\:{the}\:{last}\:{two}\:{digits}\:{of}\:{a}_{\mathrm{23}} \:{and}\:{so}\:{on} \\ $$ Answered…
Question Number 210310 by Spillover last updated on 05/Aug/24 $${Let}\:{a}\:{be}\:{the}\:{unique}\:{real}\:{zero}\:{of}\:{x}^{\mathrm{3}} +{x}+\mathrm{1}. \\ $$$${find}\:{the}\:{simplest}\:{possible}\:{way}\:{to}\:{write}\: \\ $$$$\frac{\mathrm{18}}{\left({a}^{\mathrm{2}} +{a}+\mathrm{1}\right)^{\mathrm{2}} }\:\:{as}\:{polynomial}\:{expression}\:{in}\:\:{a} \\ $$$${with}\:{ratio}\:{coefficients} \\ $$ Commented by Frix last…
Question Number 210307 by Spillover last updated on 05/Aug/24 $$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }\:\right)} \\ $$$$ \\ $$ Answered by Spillover last updated on…
Question Number 210261 by klipto last updated on 04/Aug/24 $$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{\mathrm{sinAcosA}}−\boldsymbol{\mathrm{sinBcosB}}}{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{A}}−\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{B}}}=\boldsymbol{\mathrm{tan}}\left(\boldsymbol{\mathrm{A}}−\boldsymbol{\mathrm{B}}\right) \\ $$ Answered by efronzo1 last updated on 04/Aug/24 $$\:\:\:\frac{\mathrm{sin}\:\mathrm{A}\:\mathrm{cos}\:\mathrm{A}−\mathrm{sin}\:\mathrm{B}\:\mathrm{cos}\:\mathrm{B}}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{sin}\:^{\mathrm{2}}…
Question Number 210263 by Spillover last updated on 04/Aug/24 Answered by Frix last updated on 04/Aug/24 $$\mathrm{With}\:{t}=\mathrm{tan}\:\frac{{x}^{\mathrm{3}} }{\mathrm{2}}\:\mathrm{and}\:\alpha=\mathrm{ln}\:\mathrm{12}\:\mathrm{we}\:\mathrm{get} \\ $$$$−\frac{\mathrm{4}}{\mathrm{3}}\int\frac{{t}}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)\left({t}^{\mathrm{2}} \mathrm{sin}\:\alpha\:−\mathrm{2}{t}\left(\mathrm{1}−\mathrm{cos}\:\alpha\right)−\mathrm{sin}\:\alpha\right)}{dt}= \\ $$$$\mathrm{Let}\:{u}=\mathrm{cos}\:\alpha\:\wedge\:{v}=\mathrm{sin}\:\alpha \\…
Question Number 210235 by peter frank last updated on 03/Aug/24 Commented by Pnk2024 last updated on 03/Aug/24 $$\bigtriangleup{ADB}\:\sim\bigtriangleup{EFB}\:\:\:\:….\:\left({A}−{A}\:{test}\right) \\ $$$$\Rightarrow\:\frac{{y}}{{x}}\:=\:\frac{{AB}}{{EB}}\:\:\:…….\:\left({C}.{S}.{S}.{T}\right) \\ $$$${again} \\ $$$$\bigtriangleup{BCA}\sim\bigtriangleup{EFA} \\…
Question Number 210234 by peter frank last updated on 03/Aug/24 Commented by Pnk2024 last updated on 03/Aug/24 $$\:{we}\:{know}\:{that} \\ $$$$\:\:{sin}^{\mathrm{2}} \theta+{cos}^{\mathrm{2}} \theta=\mathrm{1} \\ $$$$\Rightarrow\:\:{cos}^{\mathrm{2}} \theta=\mathrm{1}−{sin}^{\mathrm{2}}…
Question Number 210229 by mnjuly1970 last updated on 03/Aug/24 Answered by a.lgnaoui last updated on 03/Aug/24 $$\mathrm{8}=\boldsymbol{\mathrm{a}}^{\mathrm{3}} +\boldsymbol{\mathrm{b}}^{\mathrm{3}} +\mathrm{6}\boldsymbol{\mathrm{ab}} \\ $$$$ \\ $$$$\mathrm{16}=\left(\boldsymbol{\mathrm{a}}^{\mathrm{3}} +\mathrm{4}\right)+\left(\boldsymbol{\mathrm{b}}^{\mathrm{3}} +\mathrm{4}\right)+\mathrm{6}\boldsymbol{\mathrm{ab}}\:\:\:…
Question Number 210231 by a.lgnaoui last updated on 03/Aug/24 $$\mathrm{Resoudre}\:\boldsymbol{\mathrm{dans}}\:\mathbb{R} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{a}}\mathrm{cos}\:\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{b}}\mathrm{sin}\:\boldsymbol{\mathrm{x}}=\boldsymbol{\mathrm{c}}\:\:\:\:\:\left(\boldsymbol{\mathrm{x}}\neq\mathrm{0}\right)}\\{\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{sin}\:\boldsymbol{\mathrm{x}}}\right)\:\:\:\:\:\:\:\:\:=\boldsymbol{\mathrm{d}}\:\:\:\:\left(−\mathrm{1}\leqslant\boldsymbol{\mathrm{d}}\leqslant+\mathrm{1}\right)}\end{cases} \\ $$$$ \\ $$ Commented by mr W last updated on 04/Aug/24 $${they}\:{are}\:{two}\:{different}\:{equations}\:{for}…
Question Number 210206 by Spillover last updated on 02/Aug/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\frac{\mathrm{1}}{{e}}} ^{{e}} \frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}\mathrm{log}\:^{\mathrm{7}} {x}\right)} \\ $$$$ \\ $$ Terms of Service Privacy Policy…