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⠀⠀☞ Após termos simplificado cada um dos números complexos Z1 e Z2 pudemos mais facilmente identificar que: a) Z1 × Z2 = √2 × (-3 + 3i); b) Z1 ÷ Z2 = √2/3 × (1 - i).✅
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⚡ " -O que é i?"
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⠀⠀"i" na álgebra representa o número resultante da raiz quadrada de -1, ou seja, é um número que não está definido no conjunto dos Reais. Temos que:
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[tex]\blue{\Large\text{$\sf~~$}\begin{cases}\text{$\sf~i = \sqrt{-1} $}\\\\ \text{$\sf~i^2 = (\sqrt{-1})^2 = -1 $}\end{cases}}[/tex]
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⚡ " -Quanto vale sen(π/4), cos(π/4), sen(π/2) e cos(π/2)?"
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[tex]\blue{\Large\text{$\sf~~$}\begin{cases}\text{$\sf~sen(\pi /2) = sen(90) = 1$}\\\\ \text{$\sf~cos(\pi /2) = cos(90) = 0$}\\\\\text{$\sf~sen(\pi /4) = sen(45) = \dfrac{\sqrt{2}}{2}$}\\\\\text{$\sf~cos(\pi /4) = cos(45) = \dfrac{\sqrt{2}}{2}$}\end{cases}}[/tex]
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⠀⠀Ou seja, nossos números complexos Z1 e Z2 simplificados são:
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[tex]\blue{\text{$\sf~~$}\begin{cases}\text{$\sf~Z1 = 2 \cdot (\dfrac{\sqrt{2}}{2} + i \cdot \dfrac{\sqrt{2}}{2}) = \boxed{\sf ~\sqrt{2} \cdot (1 + i)~} $}\\\\ \text{$\sf~Z2 = 3 \cdot (0 + i \cdot 1) = \boxed{\sf ~3i~}$}\end{cases}}[/tex]
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[tex]\LARGE\blue{\text{$\sf Z1 \cdot Z2 = \sqrt{2} \cdot (1 + i) \cdot (3i)$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \sqrt{2} \cdot (3i + 3i^2)$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \sqrt{2} \cdot (3i - 3)$}}[/tex]
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[tex]\Large\green{\boxed{\rm~~~\gray{Z1 \cdot Z2}~\pink{=}~\blue{ \sqrt{2} \cdot (-3 + 3i) }~~~}}[/tex] ✅
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[tex]\Large\blue{\text{$\sf Z1 \div Z2 = \sqrt{2} \cdot (1 + i) \div (3i)$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \dfrac{\sqrt{2} + \sqrt{2}i}{3i}$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \dfrac{\sqrt{2} + \sqrt{2}i}{3i} \cdot \dfrac{3i}{3i}$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \dfrac{-\sqrt{2} - \sqrt{2}i}{9} \cdot 3i$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \dfrac{-3\sqrt{2}i - 3\sqrt{2}i^2}{9}$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \dfrac{-3\sqrt{2}i + 3\sqrt{2}}{9}$}}[/tex]
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[tex]\LARGE\blue{\text{$\sf = \dfrac{\sqrt{2} - \sqrt{2}i}{3}$}}[/tex]
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[tex]\Large\green{\boxed{\rm~~~\gray{Z1 \div Z2}~\pink{=}~\blue{ \dfrac{\sqrt{2} - \sqrt{2}i}{3}}~~~}}[/tex] ✅
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[tex]\bf\large\red{\underline{\quad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]
⠀⠀☀️ Veja outra divisão de números complexos
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✈ https://brainly.com.br/tarefa/36921125
[tex]\bf\large\red{\underline{\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad}}[/tex]✍
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[tex]\bf\large\red{\underline{\quad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]☁
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