Question

1- Dado o número complexo Z = -3 -6i, o calculo do modulo de Z é: *

10 pontos

45

18

raiz de 45

raiz de 18

2- O argumento do número complexo; Z = 1 + i *

10 pontos

30 graus

45 graus

60 graus

120 graus

3- A forma trigonométrica do número complexo Z = 1+i e: *

10 pontos

z = √2. (cos /4 + i sen /4)

z =√2. (cos 3/4 + i sen 3/4)

z = (cos /4 + i sen /4)

z = (cos 3/4 + i sen 3/4)

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Answer 1

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$\large\boxed{\sf{\underline{m\acute{o}dulo~de~um~n\acute{u}mero~complexo}}}$ $\boxed{\boxed{\boxed{\boxed{\sf{\rho=\sqrt{a^2+b^2}}}}}}$ $\large\boxed{\sf{\underline{argumento~de~um~n\acute{u}mero~complexo}}}$$\sf{\acute{e}~o~\hat{a}ngulo~\theta~tal~que}\\\boxed{\boxed{\boxed{\sf{cos(\theta)=\dfrac{a}{\rho}~~e~~sen(\theta)=\dfrac{b}{\rho}}}}}$ $\large\boxed{\sf{\underline{Forma~trigonom\acute{e}trica~de~um~n\acute{u}mero~complexo}}}$ $\huge\boxed{\boxed{\boxed{\boxed{\sf{z=\rho\left[cos(\theta)+i\,sen(\theta)\right]}}}}}$$\dotfill$

1)

$\sf{z=-3-6i}\\\sf{\rho=\sqrt{(-3)^2+(-6)^2}=\sqrt{9+36}=\sqrt{45}}\\\sf{\rho=\sqrt{3^{\diagdown\!\!\!\!2}\cdot5}}\\\large\boxed{\boxed{\boxed{\boxed{\sf{\rho=3\sqrt{5}}}}}}$

2)

$\sf{z=1+i}\\\sf{\rho=\sqrt{1^2+1^2}=\sqrt{2}}\\\sf{cos(\theta)=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}}\\\sf{sen(\theta)=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}} \\\large\boxed{\boxed{\boxed{\boxed{\sf{\theta=45^{\circ}}}}}}$

3)

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