Resposta:

[tex]z=\sqrt{2}\ cos\left(\frac{\pi}{4}\right)+i\sqrt{2}\ sen\left(\frac{\pi}{4}\right)[/tex]

Explicação passo a passo:

[tex]z = a +ib\\z = (rcos\ \theta)+i(rsen\ \theta)\\\\r = |z| = \sqrt{a^2+b^2} \\\theta = arctg\left(\dfrac{b}{a}\right)[/tex]

Para [tex]z = 1 + i[/tex] :

[tex]a = 1\\b=1\\r = \sqrt{1^2+1^2}=\sqrt{2} \\\theta = arctg\left(\dfrac{1}{1}\right)=\dfrac{\pi}{4} =45^\circ[/tex]

[tex]\boxed{z=\sqrt{2}\ cos\left(\frac{\pi}{4}\right)+i\sqrt{2}\ sen\left(\frac{\pi}{4}\right) }[/tex]