Question Number 148951 by EDWIN88 last updated on 01/Aug/21 $${Let}\:{complex}\:{number}\:{z}=\left({a}+\mathrm{cos}\:\theta\right)+\left(\mathrm{2}{a}−\mathrm{sin}\:\theta\right){i}\:. \\ $$$${If}\:\mid{z}\mid\:\leqslant\mathrm{2}\:{for}\:{any}\:\theta\in{R}\:{then}\:{the} \\ $$$${range}\:{of}\:{real}\:{number}\:{a}\:{is}\:\_\_\_ \\ $$ Answered by iloveisrael last updated on 01/Aug/21 Answered by…
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Question Number 17782 by Mr easymsn last updated on 10/Jul/17 $${let}\:{a},{b},{c},{x},{y}\:{and}\:{z}\:{be}\:{complex}\:{numbers} \\ $$$${such}\:{that}\:: \\ $$$${a}=\frac{{b}+{c}}{{x}−\mathrm{2}},\:{b}=\frac{{c}+{a}}{{y}−\mathrm{2}},\:{c}=\frac{{a}+{b}}{{z}−\mathrm{2}} \\ $$$${if}\:{xy}+{yz}+{zx}=\mathrm{1000}\:{and}\:{x}+{y}+{z}=\mathrm{2016}, \\ $$$${find}\:{the}\:{value}\:{of}\:{xyz} \\ $$ Commented by ajfour last…
Question Number 148314 by Tawa11 last updated on 27/Jul/21 $$\mathrm{solve}:\:\:\:\:\:\:\mathrm{x}^{\mathrm{x}} \:\:\:=\:\:\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{How}\:\mathrm{can}\:\mathrm{we}\:\mathrm{get}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{solution} \\ $$ Commented by Tawa11 last updated on 27/Jul/21 $$\mathrm{what}\:\mathrm{is}\:\:\:\:\:\:\mathrm{W}_{\mathrm{n}} \left(−\:\mathrm{ln}\:\mathrm{4}\right) \\…
Question Number 147569 by Sozan last updated on 21/Jul/21 $${find}\:{the}\:{taylor}\:{series}\:{of}\:{f}\left({z}\right)={sinz}\:,{z}=\frac{\pi}{\mathrm{4}}\:{in}\:{complex}\:{number} \\ $$ Answered by mathmax by abdo last updated on 22/Jul/21 $$\mathrm{f}\left(\mathrm{z}\right)=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\frac{\pi}{\mathrm{4}}\right)}{\mathrm{n}!}\left(\mathrm{z}−\frac{\pi}{\mathrm{4}}\right)^{\mathrm{n}}…
Question Number 15759 by prakash jain last updated on 13/Jun/17 $$\mathrm{Let}\:\mathrm{us}\:\mathrm{call}\:\mathrm{complex}\:\mathrm{triangle}\:\mathrm{which} \\ $$$$\mathrm{has}\:\mathrm{either}\:\mathrm{sides}\:\mathrm{or}\:\mathrm{angles}\:\mathrm{are} \\ $$$$\mathrm{complex}\:\mathrm{numbers}. \\ $$$$\mathrm{Let}\:{a},{b},{c}\:\in\mathbb{R}\:\mathrm{which}\:\mathrm{are}\:\mathrm{sides}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{complex}\:\mathrm{triangle}\:\mathrm{which}\:\mathrm{need} \\ $$$$\mathrm{not}\:\mathrm{satisfy}\:\mathrm{triangle}\:\mathrm{inequality}. \\ $$$$\mathrm{say}\:{a}=\mathrm{1},{b}=\mathrm{2}\:\mathrm{and}\:{c}=\mathrm{4}. \\ $$$$\mathrm{Prove}\:\left(\mathrm{or}\:\mathrm{counter}\:\mathrm{example}\right)…
Question Number 146782 by mathdanisur last updated on 15/Jul/21 $${Find}\:{the}\:{modulus}\:{of}\:{a}\:{complex} \\ $$$${number}: \\ $$$${Z}\:=\:{cos}\:\mathrm{40}\:+\:{i}\:{sin}\:\mathrm{40}\:+\mathrm{1}\:=\:? \\ $$ Answered by Ar Brandon last updated on 15/Jul/21 $$\mathrm{z}=\mathrm{e}^{\mathrm{40i}}…
Question Number 146778 by mathdanisur last updated on 15/Jul/21 $${Find}\:{the}\:{modulus}\:{of}\:{a}\:{complex} \\ $$$${number}: \\ $$$${Z}\:=\:{cos}\:\mathrm{40}\:+\:{i}\:{sin}\:\mathrm{20}\:+\:\mathrm{1}\:=\:? \\ $$ Answered by mathmax by abdo last updated on 15/Jul/21…
Question Number 145524 by physicstutes last updated on 05/Jul/21 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{argument}\:\mathrm{of}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{below} \\ $$$$\left(\mathrm{i}\right)\:{z}\:=\:\mathrm{1}+{e}^{\frac{\pi}{\mathrm{6}}{i}} \\ $$$$\left(\mathrm{ii}\right)\:{z}\:=\:\mathrm{1}\:−{e}^{\frac{\pi}{\mathrm{6}}{i}} \\ $$ Answered by Olaf_Thorendsen last updated on 05/Jul/21 $$\left({i}\right)\:{z}\:=\:\mathrm{1}+{e}^{{i}\frac{\pi}{\mathrm{6}}} \\…
Question Number 230 by ssahoo last updated on 25/Jan/15 $$\mathrm{Let}\:{z}\:\mathrm{and}\:{w}\:\mathrm{be}\:\mathrm{two}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\mid{z}\mid\leqslant\mathrm{1}\:,\:\mid{w}\mid\leqslant\mathrm{1}\:\mathrm{and}\:\mid{z}+{iw}\mid=\mid{z}−{i}\overline {{w}}\mid=\mathrm{2}, \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{z}. \\ $$ Commented by 123456 last updated on 16/Dec/14 $$\mid{z}+{iw}\mid\leqslant\mid{z}\mid+\mid{w}\mid\leqslant\mathrm{2}…
Question Number 143970 by ZiYangLee last updated on 20/Jun/21 $$\mathrm{Given}\:\mathrm{that}\:\omega\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number}, \\ $$$$\omega^{\mathrm{7}} =\mathrm{1},\:\omega\neq\mathrm{1},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\omega^{\mathrm{1}} +\omega^{\mathrm{2}} +\omega^{\mathrm{3}} +\omega^{\mathrm{4}} +\omega^{\mathrm{5}} +\omega^{\mathrm{6}} . \\ $$ Answered by…