Question Number 19321 by ajfour last updated on 09/Aug/17 $$\mathrm{Parallel}\:\mathrm{tangents}\:\mathrm{to}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{at}\:\mathrm{A} \\ $$$$\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{cut}\:\mathrm{in}\:\mathrm{the}\:\mathrm{points}\:\mathrm{C}\:\mathrm{and}\:\mathrm{D} \\ $$$$\mathrm{by}\:\mathrm{a}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{at}\:\mathrm{E}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{AD},\:\mathrm{BC}\:\mathrm{and}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{joining}\:\mathrm{the}\:\mathrm{middle}\:\mathrm{points}\:\mathrm{of}\:\mathrm{AE} \\ $$$$\mathrm{and}\:\mathrm{BE}\:\mathrm{are}\:\mathrm{concurrent}. \\ $$ Commented by ajfour…
Question Number 19313 by Tinkutara last updated on 09/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mid{z}_{\mathrm{1}} \:\pm\:{z}_{\mathrm{2}} \mid^{\mathrm{2}} \:=\:\mid{z}_{\mathrm{2}} \mid^{\mathrm{2}} \:+\:\mid{z}_{\mathrm{1}} \mid^{\mathrm{2}} \:\pm \\ $$$$\mathrm{2Re}\left({z}_{\mathrm{1}} \bar {{z}}_{\mathrm{2}} \right)\:=\:\mid{z}_{\mathrm{1}} \mid^{\mathrm{2}} \:+\:\mid{z}_{\mathrm{2}} \mid^{\mathrm{2}}…
Question Number 84849 by M±th+et£s last updated on 16/Mar/20 $${ABC}\:{is}\:{a}\:{triangle}\: \\ $$$${prove}\:{that} \\ $$$${sinA}+{sinB}+{sinC}>{sinA}\:{sinB}\:{sinC} \\ $$ Commented by ajfour last updated on 16/Mar/20 $$\mathrm{0}<\:\mathrm{sin}\:{A}\:,\:\mathrm{sin}\:{B},\:\mathrm{sin}\:{C}\:\leqslant\:\mathrm{1} \\…
Question Number 19312 by Tinkutara last updated on 09/Aug/17 $$\mathrm{Product}\:\mathrm{of}\:{n},\:{n}^{\mathrm{th}} \:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity} \\ $$$$=\:\mathrm{1}.\alpha.\alpha^{\mathrm{2}} .\alpha^{\mathrm{3}} \:…..\:\alpha^{{n}−\mathrm{1}} \:=\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \\ $$$$\mathrm{Why}?\:\mathrm{How}\:\mathrm{to}\:\mathrm{get}\:\mathrm{RHS}? \\ $$ Answered by ajfour last updated…
Question Number 84845 by Rio Michael last updated on 16/Mar/20 Commented by Rio Michael last updated on 16/Mar/20 $$\mathrm{The}\:\mathrm{circuit}\:\mathrm{above}\:\mathrm{shows}\:\mathrm{how}\:\mathrm{resistors}\:\mathrm{and}\:\mathrm{cells}\:\mathrm{can}\:\mathrm{be}\:\mathrm{connected} \\ $$$$\mathrm{given}\:\mathrm{that}\:{a},{b}\:\mathrm{and}\:{c}\:\mathrm{represent}\:\mathrm{currents}.\:\mathrm{Using}\:\mathrm{kirchoffs}\:\mathrm{laws}, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:{a},{b}\:\mathrm{and}\:{c}.\:\left[\mathrm{these}\:\mathrm{are}\:\mathrm{not}\:\mathrm{usual}\:\mathrm{symbols}\right] \\ $$$$\left.\mathrm{b}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{pd}\:\left(\mathrm{potential}\:\mathrm{difference}\right)\:\mathrm{between}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}.…
Question Number 150376 by cherokeesay last updated on 11/Aug/21 Commented by liberty last updated on 12/Aug/21 $$\left(\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{r}\right)^{\mathrm{2}} =\left(\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{r}\right)^{\mathrm{2}} +\left(\mathrm{2}\sqrt{\mathrm{3}}−\mathrm{r}\right)^{\mathrm{2}} \\ $$$$\mathrm{let}\:\mathrm{2}\sqrt{\mathrm{3}}\:=\:{a} \\ $$$$\Rightarrow\left({a}+{r}\right)^{\mathrm{2}} −\left({a}−{r}\right)^{\mathrm{2}} =\left(\mathrm{2}{a}−{r}\right)^{\mathrm{2}}…
Question Number 84843 by M±th+et£s last updated on 16/Mar/20 $$\int\frac{{sin}\left(\mathrm{7}{x}\right)}{{cos}\left(\mathrm{3}{x}\right)}\:{dx} \\ $$ Commented by jagoll last updated on 17/Mar/20 $$\mathrm{sin}\:\mathrm{7x}\:=\:\mathrm{sin}\:\left(\mathrm{4x}+\mathrm{3x}\right)\: \\ $$$$=\:\mathrm{sin}\:\mathrm{4x}\:\mathrm{cos}\:\mathrm{3x}\:+\:\mathrm{cos}\:\mathrm{4x}\:\mathrm{sin}\:\mathrm{3x} \\ $$$$\int\:\frac{\mathrm{sin}\:\mathrm{7x}}{\mathrm{cos}\:\mathrm{3x}}\:\mathrm{dx}\:=\:\int\:\left(\mathrm{sin}\:\mathrm{4x}\:+\:\mathrm{cos}\:\mathrm{4x}\:\mathrm{tan}\:\mathrm{3x}\right)\mathrm{dx} \\…
Question Number 150372 by tabata last updated on 11/Aug/21 Commented by mathdanisur last updated on 11/Aug/21 $$\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}{f}\left({x}\right){dx}=\mathrm{2}\:\:\Rightarrow\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}\left(\frac{\mathrm{1}}{\mathrm{1}+{x}}\:+\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{3}} }\:+\:…\:+\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{{n}} }\right){dx}\:=\:\mathrm{2}\:\:\left(\mathrm{1}\right) \\…
Question Number 19301 by saa last updated on 09/Aug/17 $${e}^{{i}\pi} +\mathrm{1}=\mathrm{0} \\ $$ Answered by ajfour last updated on 09/Aug/17 $$\mathrm{e}^{\mathrm{i}\pi} =−\mathrm{1}\:. \\ $$ Terms…
Question Number 150374 by mathdanisur last updated on 11/Aug/21 $$\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2}^{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{5}^{\boldsymbol{\mathrm{k}}} }\:\:=\:? \\ $$ Answered by eman_64 last updated on 11/Aug/21 $$\:\:\:=\:\sum_{\mathrm{k}=\mathrm{0}}…