Question Number 205432 by hardmath last updated on 21/Mar/24 $$\mathrm{Find}:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{2}\boldsymbol{\pi}} \:\mathrm{ln}\:\left(\mathrm{sinx}\:+\:\sqrt{\mathrm{1}\:+\:\mathrm{sin}^{\mathrm{2}} \:\mathrm{x}}\right)\:\mathrm{dx} \\ $$ Answered by MathedUp last updated on 21/Mar/24 $$\mathrm{0} \\ $$$$\mathrm{cauz}\:−{f}\left({z}\right)={f}\left(−{z}\right)\:,\:{f}\left({z}+\pi\right)=−{f}\left({z}\right)\:,\:{f}\left({z}+\mathrm{2}\pi\right)={f}\left({z}\right)…
Question Number 205460 by hardmath last updated on 21/Mar/24 $$\mathrm{If}\:\:\mathrm{3cosx}\:=\:\mathrm{8sin}\left(\mathrm{30}°\:−\:\mathrm{x}\right) \\ $$$$\mathrm{Find}:\:\:\mathrm{tanx}\:=\:? \\ $$ Answered by MM42 last updated on 21/Mar/24 $$\mathrm{3}{cosx}=\mathrm{4}{cosx}−\mathrm{4}\sqrt{\mathrm{3}}{sinx} \\ $$$$\Rightarrow{cosx}=\mathrm{4}\sqrt{\mathrm{3}}{sinx}\Rightarrow{tanx}=\frac{\sqrt{\mathrm{3}}}{\mathrm{12}}\:\:\checkmark \\…
Question Number 205430 by hardmath last updated on 21/Mar/24 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\:\bigtriangleup\mathrm{ABC} \\ $$$$\frac{\mathrm{1}}{\mathrm{sinA}}\:+\:\frac{\mathrm{1}}{\mathrm{sinB}}\:+\:\frac{\mathrm{1}}{\mathrm{sinC}}\:\leqslant\:\frac{\mathrm{2}}{\mathrm{3}}\:\left(\mathrm{cot}\frac{\mathrm{A}}{\mathrm{2}}\:+\:\mathrm{cot}\frac{\mathrm{B}}{\mathrm{2}}\:+\:\mathrm{cot}\frac{\mathrm{C}}{\mathrm{2}}\right) \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 205431 by hardmath last updated on 21/Mar/24 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\:\bigtriangleup\mathrm{ABC} \\ $$$$\frac{\mathrm{cotA}\:\mathrm{cotB}\:\mathrm{cotC}}{\mathrm{sinA}\:\mathrm{sinB}\:\mathrm{sinC}}\:\leqslant\:\frac{\mathrm{8}}{\mathrm{27}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 205420 by hardmath last updated on 20/Mar/24 $$\mathrm{If} \\ $$$$\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{Then}: \\ $$$$\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:\geqslant\:\mathrm{2} \\ $$ Answered…
Question Number 205422 by hardmath last updated on 20/Mar/24 $$\mathrm{If} \\ $$$$\left(\mathrm{a}\:+\:\mathrm{1}\right)\left(\mathrm{b}\:+\:\mathrm{1}\right)\left(\mathrm{c}\:+\:\mathrm{1}\right)\:=\:\mathrm{8} \\ $$$$\mathrm{Then}: \\ $$$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:\geqslant\:\mathrm{3} \\ $$ Answered by A5T last…
Question Number 205421 by hardmath last updated on 20/Mar/24 $$\mathrm{If} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{abc}=\mathrm{1} \\ $$$$\mathrm{Then}: \\ $$$$\frac{\mathrm{a}}{\mathrm{b}^{\mathrm{2024}} }\:+\:\frac{\mathrm{b}}{\mathrm{c}^{\mathrm{2024}} }\:+\:\frac{\mathrm{c}}{\mathrm{a}^{\mathrm{2024}} }\:\geqslant\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c} \\ $$ Terms of Service Privacy…
Question Number 205423 by hardmath last updated on 20/Mar/24 $$\mathrm{If} \\ $$$$\mathrm{a}\:,\:\mathrm{b}\:\in\:\mathbb{R} \\ $$$$\mathrm{Then}: \\ $$$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:\geqslant\:\mathrm{ab}\:+\:\sqrt{\frac{\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} }{\mathrm{2}}} \\ $$ Answered by Berbere…
Question Number 205380 by cortano12 last updated on 19/Mar/24 Commented by Rasheed.Sindhi last updated on 19/Mar/24 $$\boldsymbol{{please}}\:{don}'{t}\:{write}\:{your}\:{solution} \\ $$$${as}\:{a}\:{comment}!\:{Instead}\:{write}\:{it}\:{as} \\ $$$${answer}\:\boldsymbol{{in}}\:\boldsymbol{{order}}\:\boldsymbol{{to}}\:\boldsymbol{{prserve}}\:\boldsymbol{{chronological}} \\ $$$$\boldsymbol{{order}}\:\boldsymbol{{of}}\:\boldsymbol{{answers}}! \\ $$…
Question Number 205367 by hardmath last updated on 18/Mar/24 Answered by Ghisom last updated on 18/Mar/24 $$\left(\frac{{n}^{\mathrm{3}/\mathrm{2}} }{\mathrm{3}^{{n}} }+\frac{{n}^{\mathrm{2}} }{\mathrm{4}^{{n}} }+\frac{{n}^{\mathrm{5}/\mathrm{2}} }{\mathrm{5}^{{n}} }\right)^{\mathrm{1}/{n}} = \\…