Question Number 145763 by gsk2684 last updated on 07/Jul/21 $${if}\:{z}=\left(\frac{\mathrm{3}+{i}\:\mathrm{sin}\:\theta}{\mathrm{4}−{i}\:\mathrm{cos}\:\theta}\right){is}\:{purely}\:{real}\:{and}\: \\ $$$$\frac{\Pi}{\mathrm{2}}<\theta<\Pi\:{then}\:{find}\:{arg}\left(\mathrm{sin}\:\theta\:+{i}\:\mathrm{cos}\:\theta\right)? \\ $$ Answered by mathmax by abdo last updated on 08/Jul/21 $$\mathrm{z}\:\mathrm{real}\:\Rightarrow\mathrm{z}=\overset{−} {\mathrm{z}}\:\Rightarrow\frac{\mathrm{3}+\mathrm{isin}\theta}{\mathrm{4}−\mathrm{icos}\theta}=\frac{\mathrm{3}−\mathrm{isin}\theta}{\mathrm{4}+\mathrm{icos}\theta}\:\Rightarrow…
Question Number 145759 by mathdanisur last updated on 07/Jul/21 Commented by mr W last updated on 07/Jul/21 $$\sqrt{{ab}}\leqslant\frac{{a}+{b}}{\mathrm{2}}={x} \\ $$$$\Rightarrow{ab}\leqslant{x}^{\mathrm{2}} ={maximum} \\ $$ Commented by…
Question Number 80219 by Power last updated on 01/Feb/20 Answered by som(math1967) last updated on 01/Feb/20 $${let}\frac{\pi}{\mathrm{7}}=\theta\: \\ $$$$\therefore{cos}\mathrm{2}\theta+{cos}\mathrm{4}\theta+{cos}\mathrm{6}\theta \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{sin}\theta}\left(\mathrm{2}{cos}\mathrm{2}\theta{sin}\theta+\mathrm{2}{sin}\theta{cos}\mathrm{4}\theta\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{2}{sin}\theta{cos}\mathrm{6}\theta\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{sin}\theta}\left({sin}\mathrm{3}\theta−{sin}\theta+{sin}\mathrm{5}\theta−{sin}\mathrm{3}\theta\right.…
Question Number 145755 by loveineq last updated on 07/Jul/21 $$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:{abc}\:=\:\mathrm{1}.\:\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{8}\left({a}^{\mathrm{2}} +\mathrm{1}\right)\left({b}^{\mathrm{2}} +\mathrm{1}\right)\left({c}^{\mathrm{2}} +\mathrm{1}\right)}{\left({a}+\mathrm{1}\right)\left({b}+\mathrm{1}\right)\left({c}+\mathrm{1}\right)}\:\leqslant\:\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{a}\right) \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 80209 by peter frank last updated on 01/Feb/20 Commented by mathmax by abdo last updated on 01/Feb/20 $${z}+\frac{\mathrm{1}}{{z}}\:=−\mathrm{1}\:\Rightarrow{z}^{\mathrm{2}} \:+\mathrm{1}=−{z}\:\Rightarrow{z}^{\mathrm{2}} \:+{z}+\mathrm{1}=\mathrm{0} \\ $$$$\Delta=\mathrm{1}−\mathrm{4}=−\mathrm{3}\:\Rightarrow{z}_{\mathrm{1}} =\frac{−\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{2}}={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}}…
Question Number 80208 by peter frank last updated on 01/Feb/20 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 80204 by jagoll last updated on 01/Feb/20 Commented by Tony Lin last updated on 01/Feb/20 $$\mathrm{28}\left({x}+{y}+{z}\right)=\mathrm{363}+\left({x}−{z}\right) \\ $$$${if}\:{x}+{y}+{z}\geqslant\mathrm{14} \\ $$$${x}−{z}\geqslant\mathrm{29}>{x}+{y}+{z}\left({false}\right) \\ $$$${if}\:{x}+{y}+{z}=\mathrm{13} \\…
Question Number 80199 by peter frank last updated on 31/Jan/20 Commented by mr W last updated on 08/Feb/20 $${all}\:{are}\:{clear}\:{if}\:{you}\:{have}\:{a}\:{look}\:{at} \\ $$$${the}\:{graph}\:{of}\:{the}\:{functions}. \\ $$ Answered by…
Question Number 145721 by mathdanisur last updated on 07/Jul/21 $${if}\:\:{a};{b}\in\mathbb{N}\:\:;\:\:{a}\neq{b}\:\:{and}\:\:{a}+{b}=\mathrm{2}{x} \\ $$$${find}\:\:\left({a}\centerdot{b}\right)_{\boldsymbol{{m}}{ax}} =? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 80178 by Power last updated on 31/Jan/20 Commented by mr W last updated on 31/Jan/20 $${no}\:{exact}\:{value}\:{possible}! \\ $$$${x}^{\mathrm{25}} \underset{{k}=\mathrm{1}} {\overset{\mathrm{25}} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}^{{k}} }\right)=\mathrm{1} \\…