Question Number 153033 by ZiYangLee last updated on 04/Sep/21 $${y}=\mathrm{log}\:\left(\mathrm{1}+\mathrm{cos}\:{x}\right) \\ $$$$\frac{{dy}}{{dx}}= \\ $$ Answered by puissant last updated on 04/Sep/21 $${y}={ln}\left(\mathrm{1}+{cosx}\right) \\ $$$$\frac{{dy}}{{dx}}=\frac{−{sinx}}{\mathrm{1}+{cosx}} \\…
Question Number 21962 by Tinkutara last updated on 07/Oct/17 $$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{16}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{property}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{numbers}\:\mathrm{of}\:{A}\:\mathrm{will} \\ $$$$\mathrm{not}\:\mathrm{exceed}\:\mathrm{1994}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are} \\ $$$$\mathrm{two}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}\:\mathrm{in}\:{A}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{relatively}\:\mathrm{prime}. \\ $$ Commented by Rasheed.Sindhi…
Question Number 87497 by Rio Michael last updated on 04/Apr/20 $$\mathrm{A}\:\mathrm{complex}\:\mathrm{number}\:{z}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{z}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\theta\:+\:{i}\mathrm{sin}\:\theta\right),\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{z}^{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\:\left(\mathrm{cos}\:{n}\theta\:+\:{i}\mathrm{sin}\:{n}\theta\right) \\ $$$$\mathrm{Using}\:\mathrm{De}\:\mathrm{Moivre}'\mathrm{s}\:\mathrm{theorem},\mathrm{or}\:\mathrm{otherwise},\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\left(\mathrm{i}\right)\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{4}^{{r}} }\:\mathrm{sin}\:\mathrm{2r}\theta\:\mathrm{is}\:\mathrm{a}\:\mathrm{convergent}\:\mathrm{geometic}\:\mathrm{progression}. \\ $$$$\left(\mathrm{ii}\right)\:\underset{{r}=\mathrm{0}} {\overset{\infty}…
Question Number 87492 by unknown last updated on 04/Apr/20 $$\frac{\mathrm{2}+\mathrm{3}^{\mathrm{2}} }{\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+\mathrm{4}!}+\frac{\mathrm{3}+\mathrm{4}^{\mathrm{2}} }{\mathrm{2}!+\mathrm{3}!+\mathrm{4}!+\mathrm{5}!}+…+\frac{\mathrm{2013}+\mathrm{2014}^{\mathrm{2}} }{\mathrm{2012}!+\mathrm{2013}!+\mathrm{2014}!+\mathrm{2015}!} \\ $$ Answered by mind is power last updated on 04/Apr/20 $$\underset{{k}\geqslant\mathrm{2}}…
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Question Number 87488 by jagoll last updated on 04/Apr/20 $$\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}\:+\:\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{x}+\frac{\pi}{\mathrm{4}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{x}\:\in\:\left[\:\mathrm{0},\mathrm{2}\pi\:\right]\: \\ $$ Commented by john santu last updated on 05/Apr/20 $$\left(\mathrm{2sin}^{\mathrm{2}}…
Question Number 153016 by joki last updated on 04/Sep/21 $$\int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{xe}^{\mathrm{4}−\mathrm{x}^{\mathrm{2}} } \mathrm{dx} \\ $$ Commented by mr W last updated on 04/Sep/21 $$=−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}}…
Question Number 153019 by mathdanisur last updated on 04/Sep/21 $$\mathrm{Determine}\:\mathrm{all}\:\mathrm{functions}\:\:\mathrm{f}:\mathbb{R}\rightarrow\left(\mathrm{1};+\infty\right) \\ $$$$\mathrm{continuous}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{f}\left(\mathrm{4x}\right)\:\centerdot\:\mathrm{f}\left(\mathrm{3x}\right)\:=\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:\:;\:\:\forall\mathrm{x}\in\mathbb{R} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 153012 by bobhans last updated on 04/Sep/21 $$\:{Find}\:{all}\:{ordered}\:{pairs}\:{of}\:{real}\: \\ $$$$\:{numbers}\:\left({x},{y}\right)\:{for}\:{which} \\ $$$$\:\:\begin{cases}{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}\right)=\mathrm{1}+{y}^{\mathrm{7}} }\\{\left(\mathrm{1}+{y}^{\mathrm{4}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}\right)=\mathrm{1}+{x}^{\mathrm{7}} }\end{cases} \\ $$ Commented by mr…
Question Number 21940 by Tinkutara last updated on 07/Oct/17 $$\mathrm{A}\:\mathrm{polynomial}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{satisfies} \\ $$$${f}\left({x}\right){f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{2}{f}\left({x}\right)\:+\:\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right);\:{x}\:\neq\:\mathrm{0}\:\mathrm{and} \\ $$$${f}\left(\mathrm{2}\right)\:=\:\mathrm{18},\:\mathrm{then}\:{f}\left(\mathrm{3}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$ Commented by ajfour last updated on 07/Nov/18 $${No}\:{one}\:{solved}\:{this},\:{if}\:{correct} \\…