Question

Qual a forma trigonométrica e a forma polar do número complexo z = 1 + i:

Answer 1

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$\boxed{\sf{\underline{M\acute{o}dulo~de~um~n\acute{u}mero~complexo~z=a+bi}}}\\\huge\boxed{\boxed{\boxed{\boxed{\boxed{\sf{\rho=\sqrt{a^2+b^2}}}}}}}$

$\boxed{\sf{\underline{Argumento~de~um~n\acute{u}mero~complexo}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{cos(\theta)=\dfrac{a}{\rho}~~~sen(\theta)=\dfrac{b}{\rho}}}}}}$

$\boxed{\sf{\underline{Forma~trigonom\acute{e}trica~de~um~n\acute{u}mero~complexo}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{Z=\rho\left[cos(\theta)+i~sen(\theta)\right]}}}}}$

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$\sf{z=1+i}\\\sf{\rho=\sqrt{1^2+1^2}=\sqrt{2}}\\\sf{cos(\theta)=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}}\\\sf{sen(\theta)=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}}\\\sf{\theta=\dfrac{\pi}{4}}\\\huge\boxed{\boxed{\boxed{\boxed{\boxed{\sf{z=\sqrt{2}\left[cos\left(\right\dfrac{\pi}{4})+i~sen\left(\dfrac{\pi}{4}\right)\right]}}}}}}$