Question Number 220995 by hardmath last updated on 21/May/25 Answered by Frix last updated on 22/May/25 $$\mathrm{Let}\:\mathrm{the}\:\mathrm{red}\:\mathrm{line}\:=\mathrm{1} \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{30}°}=\frac{{a}}{\mathrm{sin}\:\mathrm{15}°}\:\Rightarrow \\ $$$$\:\:\:\:\:{a}=\frac{\sqrt{\mathrm{6}}−\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\frac{{a}}{\mathrm{sin}\:{x}}=\frac{\mathrm{1}}{\mathrm{sin}\:\left(\mathrm{135}°−{x}\right)}=\frac{\sqrt{\mathrm{2}}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}\Rightarrow \\ $$$$\:\:\:\:\:{a}=\frac{\sqrt{\mathrm{2}}\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}=\frac{\sqrt{\mathrm{2}}\mathrm{tan}\:{x}}{\mathrm{1}+\mathrm{tan}\:{x}}…
Question Number 220863 by fantastic last updated on 20/May/25 $$\left(\mathrm{211}\right) \\ $$$$\:\: \\ $$$${Find}\:{the}\:{derivative}\:{of}\:\Delta{x},\:{where} \\ $$$$\Delta{x}=\begin{vmatrix}{{f}_{\mathrm{1}} \left({x}\right)}&{\phi_{\mathrm{1}} \left({x}\right)}&{\Psi_{\mathrm{1}} \left({x}\right)}\\{{f}_{\mathrm{2}} \left({x}\right)}&{\phi_{\mathrm{2}} \left({x}\right)}&{\Psi_{\mathrm{2}} \left({x}\right)}\\{{f}_{\mathrm{3}} \left({x}\right)}&{\phi_{\mathrm{3}} \left({x}\right)}&{\Psi_{\mathrm{3}} \left({x}\right)}\end{vmatrix}…
Question Number 220857 by fantastic last updated on 20/May/25 $${Prove}\:{that}\:\mathrm{tan}\:\mathrm{20}^{\mathrm{0}} \mathrm{tan40}^{\mathrm{0}} \:\mathrm{tan}\:\mathrm{80}^{\mathrm{0}} =\mathrm{tan}\:\mathrm{60}^{\mathrm{0}} \\ $$ Answered by fantastic last updated on 20/May/25 $$\mathrm{tan}\:\mathrm{20}^{\mathrm{0}} \mathrm{tan40}^{\mathrm{0}} \:\mathrm{tan}\:\mathrm{80}^{\mathrm{0}}…
Question Number 220858 by Rojarani last updated on 20/May/25 Commented by mr W last updated on 21/May/25 $$\mathrm{65}? \\ $$ Commented by Rojarani last updated…
Question Number 220852 by fantastic last updated on 20/May/25 $$\underset{{x}\rightarrow\mathrm{0}} {{Lim}}\left\{\frac{{xe}^{{x}} −{log}\left(\mathrm{1}+{x}\right)}{{x}^{\mathrm{2}} }\right\} \\ $$ Answered by SdC355 last updated on 20/May/25 $$\frac{\frac{\mathrm{d}\:\:}{\mathrm{d}{x}}\left({xe}^{{x}} −\mathrm{ln}\left({x}+\mathrm{1}\right)\right)}{\frac{\mathrm{d}\:}{\mathrm{d}{x}}\:{x}^{\mathrm{2}} }=\frac{\left({x}+\mathrm{1}\right){e}^{{x}}…
Question Number 220853 by fantastic last updated on 20/May/25 $${The}\:{two}\:{solutions}\:{of}\:{the}\:{equation}\:{are}\:{the}\:{same} \\ $$$${a}\left({b}−{c}\right){x}^{\mathrm{2}\:} +{b}\left({c}−{a}\right){x}+{c}\left({a}−{b}\right)=\mathrm{0} \\ $$$${Prove}\:{that}\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{c}}=\frac{\mathrm{2}}{{b}} \\ $$ Answered by fantastic last updated on 20/May/25 $${In}\:{any}\:{quadratic}\:{equation}\:\alpha{x}^{\mathrm{2}}…
Question Number 220854 by fantastic last updated on 20/May/25 $${Solve}\:{for}\:{x}\:\:\:{and}\:\:\:\:{y} \\ $$$$\mathrm{3}^{{x}} +\mathrm{3}^{{y}} =\mathrm{4},\:\:\mathrm{3}^{−{x}} +\mathrm{3}^{−{y}\:} =\frac{\mathrm{4}}{\mathrm{3}} \\ $$ Answered by SdC355 last updated on 20/May/25…
Question Number 220855 by fantastic last updated on 20/May/25 $${If}\:\:{b}\:\mathrm{cos}\left(\theta+\mathrm{120}^{\mathrm{0}} \right)={c}\:\mathrm{cos}\:\left(\theta+\mathrm{240}^{\mathrm{0}} \right)\:{then}\:{prove}\:{that} \\ $$$${b}−{c}=−\left({b}+{c}\right)\sqrt{\mathrm{3}}\:\mathrm{tan}\:\theta \\ $$ Answered by golsendro last updated on 20/May/25 $$\:\mathrm{b}\:\mathrm{cos}\:\left(\mathrm{180}−\left(\mathrm{60}−\theta\right)\right)\:=\:\mathrm{c}\:\mathrm{cos}\:\left(\mathrm{360}−\left(\mathrm{120}−\theta\right)\right) \\…
Question Number 220848 by Nicholas666 last updated on 20/May/25 $$ \\ $$$$\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{2}}\:\sqrt{\frac{\left(\mathrm{2}\:−\:\mathrm{6}{x}\:+\:\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{2}} +\:\mathrm{4}\left(\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} \right)}{\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} }}\:{dx}\:\:\:\:\:\:\: \\ $$$$ \\ $$ Commented…
Question Number 220913 by SdC355 last updated on 20/May/25 $$\mathrm{Is}\:\mathrm{there}\:\mathrm{an}\:\mathrm{Manager}??? \\ $$$$\mathrm{pls}\:\mathrm{ban}\:\mathrm{Question}\:\mathrm{Spamming}\:\mathrm{and}… \\ $$$$\mathrm{pls}\:\mathrm{fix}\:\mathrm{invisible}\:\mathrm{line}\:\mathrm{matrix}\:\mathrm{bug} \\ $$ Commented by mr W last updated on 21/May/25 $${the}\:{developer}\:{Tinku}\:{Tara}\:{doesn}'{t}…