Question Number 211609 by MATHEMATICSAM last updated on 14/Sep/24 $$\mathrm{Prove}\:\mathrm{that},\:\mathrm{in}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{the}\:\mathrm{ratios}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sine}\:\mathrm{of}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{angles} \\ $$$$\mathrm{are}\:\mathrm{equal}.\:\mathrm{Also}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{each}\:\mathrm{ratio}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{diameter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circum}\:\mathrm{circle} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}. \\ $$ Answered by Frix last updated…
Question Number 211594 by mnjuly1970 last updated on 14/Sep/24 $$ \\ $$$$\:{If}\:,\:\:\overset{\:\:−} {{H}}_{{n}} \:=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{3}}\:−…+\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{{n}}\:\:\: \\ $$$${prove}\:{that}:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\overset{\:\:−\:} {{H}}_{{n}} −\mathrm{ln}\left(\mathrm{2}\right)}{{n}}=\mathrm{ln}^{\mathrm{2}} \left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:−−−−−−−−−−\:\:\:\:\:\: \\…
Question Number 211595 by BaliramKumar last updated on 14/Sep/24 Answered by mr W last updated on 14/Sep/24 $${m}^{\mathrm{2}} −{n}^{\mathrm{2}} =\mathrm{199} \\ $$$$\left({m}−{n}\right)\left({m}+{n}\right)=\mathrm{1}×\mathrm{199} \\ $$$${m}−{n}=\mathrm{1} \\…
Question Number 211605 by MrGaster last updated on 14/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Let}\:\boldsymbol{{a}}_{\mathrm{1}} ,\boldsymbol{{a}}_{\mathrm{2}} ,\ldots\boldsymbol{{a}}_{\boldsymbol{{n}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Is}\:\mathrm{n}\:\mathrm{real}\:\mathrm{numbers}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{All}\:\mathrm{fall}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}\:\left(−\mathrm{1},\mathrm{1}\right) \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\left(\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{1}\leq\boldsymbol{{i}},\boldsymbol{{j}}\leq\boldsymbol{{n}}} {\prod}\frac{\mathrm{1}+\boldsymbol{{a}}_{\boldsymbol{{i}}} \boldsymbol{{a}}_{\boldsymbol{{j}}} }{\mathrm{1}−\boldsymbol{{a}}_{\boldsymbol{{i}}}…
Question Number 211617 by pete last updated on 14/Sep/24 $$\mathrm{Ato}\:\:\mathrm{starts}\:\mathrm{a}\:\mathrm{business}\:\mathrm{with}\:\$\mathrm{1},\mathrm{250}.\mathrm{00}.\:\mathrm{Ama}\:\mathrm{joins}\:\mathrm{the}\:\mathrm{business} \\ $$$$\mathrm{later}\:\mathrm{with}\:\mathrm{a}\:\mathrm{capital}\:\mathrm{of}\:\mathrm{1},\mathrm{875}.\mathrm{00}.\:\mathrm{At}\:\mathrm{the}\: \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{year},\:\mathrm{profits}\:\mathrm{are}\:\mathrm{shared}\:\mathrm{equally} \\ $$$$\mathrm{between}\:\mathrm{Ato}\:\mathrm{and}\:\mathrm{Ama}.\:\mathrm{When}\:\mathrm{did}\:\mathrm{Ama} \\ $$$$\mathrm{join}\:\mathrm{the}\:\mathrm{business}? \\ $$ Answered by A5T last updated…
Question Number 211601 by MrGaster last updated on 14/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\mathrm{set}\:\boldsymbol{\alpha}\sqrt[{\mathrm{3}}]{\mathrm{2}},\mathrm{ask}\:\mathbb{Q}\left(\boldsymbol{\alpha}\right)\mathrm{Upper}\:\mathrm{irreducible}\:\mathrm{cubic}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{x}}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{All}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of} \\ $$$$ \\ $$ Terms of Service Privacy…
Question Number 211603 by MrGaster last updated on 14/Sep/24 $$\mathrm{1}.\mathrm{Given}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{tetrahedron}\:\boldsymbol{{ABCD}} \\ $$$$\mathrm{with}\:\mathrm{vertices}\:\boldsymbol{{A}}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\boldsymbol{{B}}\left(\boldsymbol{{a}},\mathrm{0},\mathrm{0}\right), \\ $$$$\boldsymbol{{C}}\left(\mathrm{0},\boldsymbol{{a}},\mathrm{0}\right),\boldsymbol{\mathrm{and}}\:\boldsymbol{{D}}\left(\mathrm{0},\mathrm{0},\boldsymbol{{a}}\right).\mathrm{Calculate}\:\mathrm{the} \\ $$$$\:\mathrm{volume}\:\boldsymbol{{V}}\:\:\mathrm{and}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{area}\:\boldsymbol{{S}}\:\boldsymbol{\mathrm{of}} \\ $$$$\mathrm{this}\:\mathrm{tetrahedron}. \\ $$ Answered by BHOOPENDRA last updated…
Question Number 211578 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{+\infty} \frac{\boldsymbol{{x}}}{\:\sqrt{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{4}} }}\boldsymbol{{dx}}. \\ $$$$ \\ $$ Answered by lepuissantcedricjunior last updated on…
Question Number 211579 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{1}}{\left(\mathrm{1}−\boldsymbol{{x}}^{\mathrm{4}} \right)\sqrt{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }}\boldsymbol{{dx}}. \\ $$$$ \\ $$ Answered by lepuissantcedricjunior last updated on 13/Sep/24…
Question Number 211574 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{25}}\\{\boldsymbol{{x}}+\mathrm{2}\boldsymbol{\mathrm{y}}−\mathrm{3}=\mathrm{0}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$ Answered by Rasheed.Sindhi…