Question

determine as raízes quadradas de z=4+3i

Answer 1

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Raízes enésimas de um número complexo

$\small\boxed{\boxed{\boxed{\boxed{\sf w_k=\sqrt[\sf n]{\sf \rho}\bigg[cos\bigg(\dfrac{\theta+2k\pi}{n}\bigg)+i~sen\bigg(\dfrac{\theta+2k\pi}{n}\bigg)\bigg]}}}}}$

$\sf z=4+3i\\\sf\rho=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\\\sf tg(\theta)=\dfrac{3}{4}\implies \theta=arc tg\bigg(\dfrac{3}{4}\bigg)\\\sf\theta=36,86^\circ\approx\dfrac{1843}{9000}\pi~rad$

$\sf w_k=\sqrt{5}\bigg[cos\bigg(\dfrac{\frac{1843}{9000}\pi+2k\pi}{2}\bigg)+i~sen\bigg(\dfrac{\frac{1843}{9000}\pi+2k\pi}{2}\bigg)\bigg]\\\sf w_k=\sqrt{5}\bigg[cos\bigg(\dfrac{1843}{18000}\pi+k\pi\bigg)+i~sen\bigg(\dfrac{1843}{18000}\pi+k\pi\bigg)\bigg]$

$\sf a~partir~daqui~atribui-se~valores~a~k\in[0,1]\\\sf para~se~obter~ra\acute izes~en\acute esimas~desejadas.$