Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 114409 by bemath last updated on 19/Sep/20

find solution set of equation sin x = 2 , x ∈ C .

$${find}\:{solution}\:{set}\:{of}\:{equation} \\ $$$$\:\mathrm{sin}\:{x}\:=\:\mathrm{2}\:,\:{x}\:\in\:\mathbb{C}\:. \\ $$

Answered by bobhans last updated on 19/Sep/20

sin x = ((e^(ix) −e^(−ix) )/(2i)) = 2 e^(ix) −e^(−ix) = 4i , let e^(ix) = z z−z^(−1) = 4i ; z^2 −4iz−1 = 0 z = ((4i±(√(−16+4)))/2) = (2±(√3))i then e^(ix) = (2±(√3))i = (2±(√3))e^(i.(π/2)) ix = ln (2±(√3)) +i((π/2)+2nπ); n∈Z x=(4n+1)(π/2)±i ln (2+(√3)) ; n∈Z

$$\mathrm{sin}\:{x}\:=\:\frac{{e}^{{ix}} −{e}^{−{ix}} }{\mathrm{2}{i}}\:=\:\mathrm{2}\: \\ $$$${e}^{{ix}} −{e}^{−{ix}} \:=\:\mathrm{4}{i}\:,\:{let}\:{e}^{{ix}} \:=\:{z}\: \\ $$$${z}−{z}^{−\mathrm{1}} \:=\:\mathrm{4}{i}\:;\:{z}^{\mathrm{2}} −\mathrm{4}{iz}−\mathrm{1}\:=\:\mathrm{0} \\ $$$${z}\:=\:\frac{\mathrm{4}{i}\pm\sqrt{−\mathrm{16}+\mathrm{4}}}{\mathrm{2}}\:=\:\left(\mathrm{2}\pm\sqrt{\mathrm{3}}\right){i}\: \\ $$$${then}\:{e}^{{ix}} \:=\:\left(\mathrm{2}\pm\sqrt{\mathrm{3}}\right){i}\:=\:\left(\mathrm{2}\pm\sqrt{\mathrm{3}}\right){e}^{{i}.\frac{\pi}{\mathrm{2}}} \\ $$$${ix}\:=\:\mathrm{ln}\:\left(\mathrm{2}\pm\sqrt{\mathrm{3}}\right)\:+{i}\left(\frac{\pi}{\mathrm{2}}+\mathrm{2}{n}\pi\right);\:{n}\in\mathbb{Z} \\ $$$${x}=\left(\mathrm{4}{n}+\mathrm{1}\right)\frac{\pi}{\mathrm{2}}\pm{i}\:\mathrm{ln}\:\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)\:;\:{n}\in\mathbb{Z} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com