Question Number 3950 by Rasheed Soomro last updated on 25/Dec/15 $$\mathcal{W}{e}\:{can}\:{make}\:{a}\:\boldsymbol{{cyllinder}}\:{from}\:{a} \\ $$$$\boldsymbol{{rectangle}}\:{by}\:{connecting}\:{its}\:{opposite} \\ $$$$\boldsymbol{{edges}}. \\ $$$${Suppose}\:{we}\:{have}\:{two}\:{copies}\:{of}\:{a} \\ $$$$\boldsymbol{{non}}−\boldsymbol{{square}}\:\boldsymbol{{rectangle}}.{From} \\ $$$${one}\:{copy}\:{we}\:{make}\:{a}\:{long}\:{cyllinder} \\ $$$${by}\:{connecting}\:{long}\:{edges}\:{of}\:{it}\: \\ $$$${whereas}\:{from}\:{other}\:{copy}\:{by}\:{connecting}…
Question Number 69482 by Henri Boucatchou last updated on 24/Sep/19 $$\underset{{n}\rightarrow\infty} {{lim}}\frac{\mathrm{2}+{cosn}}{\mathrm{4}{n}+{sinn}}\:=\:? \\ $$ Commented by Tony Lin last updated on 24/Sep/19 $$−\mathrm{1}\leqslant{cosn}\leqslant\mathrm{1} \\ $$$$\mathrm{1}\leqslant\mathrm{2}+{cosn}\leqslant\mathrm{3}…
Question Number 135019 by faysal last updated on 09/Mar/21 Answered by Dwaipayan Shikari last updated on 09/Mar/21 $${sin}\mathrm{6}°{sin}\mathrm{42}°{sin}\mathrm{66}°{sin}\mathrm{78}° \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left({cos}\mathrm{60}°−{cos}\mathrm{72}°\right)\left({cos}\mathrm{36}°−{cos}\mathrm{120}°\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\right)\left(\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{3}−\sqrt{\mathrm{5}}}{\mathrm{4}}\right)\left(\frac{\sqrt{\mathrm{5}}+\mathrm{3}}{\mathrm{4}}\right)=\frac{\mathrm{1}}{\mathrm{16}}=\Lambda \\ $$$${cos}\mathrm{6}°{cos}\mathrm{42}°{cos}\mathrm{66}°{cos}\mathrm{78}° \\…
Question Number 3944 by Rasheed Soomro last updated on 25/Dec/15 $$\mathcal{P}{rove}\:{that},\:{inside}\:\:{a}\:{given}\:{square},\:{a}\:{semicircle}\: \\ $$$${of}\:{the}\:{largest}\:{possible}\:{area},\:{can}\:{be}\:{constructed}\: \\ $$$${using}\:{ruler}\:{and}\:{compass}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 69478 by Henri Boucatchou last updated on 24/Sep/19 $$\:\:\underset{\boldsymbol{{n}}\rightarrow\infty} {\boldsymbol{{lim}sin}}\left(\boldsymbol{{n}\pi}\right)\:=\:? \\ $$ Commented by mathmax by abdo last updated on 24/Sep/19 $${sin}\left({n}\pi\right)=\mathrm{0}\:\Rightarrow{lim}_{{n}\rightarrow+\infty} {sin}\left({n}\pi\right)=\mathrm{0}…
Question Number 3943 by Rasheed Soomro last updated on 25/Dec/15 $$\mathcal{W}{hat}\:{is}\:{the}\:{area}\:\:{of}\:\:{overlapping}\:{region} \\ $$$${of}\:{three}\:{circles}\:{of}\:{radii}\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} \:{with}\:{their} \\ $$$${respective}\:{centres}\:\boldsymbol{\mathrm{C}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{C}}_{\mathrm{2}} \:{and}\:\boldsymbol{\mathrm{C}}_{\mathrm{3}} \:{when} \\ $$$$\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{1}}…
Question Number 69479 by MJS last updated on 24/Sep/19 $$\mathrm{to}\:\mathrm{Sir}\:\mathrm{Aifour}: \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{construct}\:\mathrm{polynomes}\:\mathrm{of}\:\mathrm{both}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{and} \\ $$$$\mathrm{4}^{\mathrm{th}} \:\mathrm{degree}\:\mathrm{in}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{the}\:\mathrm{constants}\:\mathrm{are} \\ $$$$\in\mathbb{Z}\:\mathrm{or}\:\in\mathbb{Q}\:\mathrm{and}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{are}\:\mathrm{not}\:\mathrm{trivial} \\ $$$$\mathrm{i}.\mathrm{e}. \\ $$$$\left({t}−\alpha\right)\left({t}+\frac{\alpha}{\mathrm{2}}−\sqrt{\beta}\right)\left({t}+\frac{\alpha}{\mathrm{2}}+\sqrt{\beta}\right)=\mathrm{0}\wedge{t}={x}+\frac{\gamma}{\mathrm{3}} \\ $$$$\Leftrightarrow \\…
Question Number 135004 by bramlexs22 last updated on 09/Mar/21 $$ \\ $$the closest distance from the point on the curve y = x ^ 3-1…
Question Number 135000 by bobhans last updated on 09/Mar/21 $$ \\ $$Find the equation of the circle through the points of intersection of x^2+y^2−1=0,x^2+y^2−2x−4y+1=0 and touching the…
Question Number 3930 by Filup last updated on 25/Dec/15 $${y}=\frac{\mathrm{log}_{{e}} \left(\frac{{x}}{{m}}−{sa}\right)}{{r}^{\mathrm{2}} } \\ $$$${yr}^{\mathrm{2}} =\mathrm{log}_{\mathrm{e}} \left(\frac{{x}}{{m}}−{sa}\right) \\ $$$${e}^{{yr}^{\mathrm{2}} } =\frac{{x}}{{m}}−{sa} \\ $$$${me}^{{rry}} ={x}−{mas} \\ $$$$…