Question

Marque a alternativa correta, com base no enunciado abaixo.
Calculando na forma trigonométrica, o produto de z1.z2 dos números
complexos dados abaixo é
$z 1 = 5 \times( \cos( \frac{2\pi}{5} ) +i \sin( \frac{2\pi}{5} ) ) \: \\ e \: z2 = 2 \times ( \cos( \frac{3\pi}{5} ) + i \sin( \frac{3\pi}{5} ) )$
a) -10
b) 5/2
c) 10
d) 0​

Answer 1

Resposta:

a) -10

Explicação passo-a-passo:

$z_1z_2=5\cdot \:2\left(\cos \left(\frac{2\pi }{5}+\frac{3\pi }{5}\right)+i\sin \left(\frac{2\pi }{5}+\frac{3\pi }{5}\right)\right)$

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$\\\\\cos \left(\frac{2\pi }{5}+\frac{3\pi }{5}\right)\\\\\frac{2\pi }{5}+\frac{3\pi }{5}\\\\\mathrm{Ja\:que\:os\:denominadores\:sao\:iguais,\:combinar\:as\:fracoes}:\quad \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}\\\\=\frac{2\pi +3\pi }{5}$

$\mathrm{Somar\:elementos\:similares:}\:2\pi +3\pi =5\pi \\\\=\frac{5\pi }{5}\\\\\mathrm{Dividir:}\:\frac{5}{5}=1\\\\=\pi \\\\=\cos \left(\pi \right)\\\\\mathrm{Utilizar\:a\:seguinte\:identidade\:trivial}:\quad \cos \left(\pi \right)=\left(-1\right)\\\\=-1$

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$z_1z_2=5\cdot \:2\left(-1+i\sin \left(\frac{2\pi }{5}+\frac{3\pi }{5}\right)\right)$

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$i\sin \left(\frac{2\pi }{5}+\frac{3\pi }{5}\right)$

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$sin \left(\frac{2\pi }{5}+\frac{3\pi }{5}\right)\\\\\mathrm{Simplificar}\:\frac{2\pi }{5}+\frac{3\pi }{5}\:\mathrm{em\:uma\:fracao}:\quad \pi \\\\\frac{2\pi }{5}+\frac{3\pi }{5}\\\\\mathrm{Ja\:que\:os\:denominadores\:sao\:iguais,\:combinar\:as\:fracoes}:\quad \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}$

$=\frac{2\pi +3\pi }{5}\\\\\mathrm{Somar\:elementos\:similares:}\:2\pi +3\pi =5\pi \\\\=\frac{5\pi }{5}\\\\\mathrm{Dividir:}\:\frac{5}{5}=1\\\\=\pi \\\\=\sin \left(\pi \right)\\\\\mathrm{Utilizar\:a\:seguinte\:identidade\:trivial}:\quad \sin \left(\pi \right)=0\\\\=0$

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$=0i\\\\\mathrm{Aplicar\:a\:regra}\:0\cdot \:a=0\\\\=0$

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$z_1z_2=5\cdot \:2\left(0-1\right)\\\\\mathrm{Subtrair:}\:0-1=-1\\\\z_1z_2=5\cdot \:2\left(-1\right)\\\\\mathrm{Remover\:os\:parenteses}:\quad \left(-a\right)=-a\\\\z_1z_2=-5\cdot \:2\cdot \:1\\\\\mathrm{Multiplicar\:os\:numeros:}\:5\cdot \:2\cdot \:1=10\\\\z_1z_2=-10$