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Popular Trigonometria >

2(cos(pi/6)+isin(pi/6))*5(cos(pi/3)+isin(pi/3))

Solução

2 (cos (\frac{π}{6}) + i sin (\frac{π}{6})) \cdot 5 (cos (\frac{π}{3}) + i sin (\frac{π}{3}))

Solução

10 i

Passos da solução

2 (cos (\frac{π}{6}) + i sin (\frac{π}{6})) \cdot 5 (cos (\frac{π}{3}) + i sin (\frac{π}{3}))

Utilizar a seguinte identidade trivial:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

Utilizar a seguinte identidade trivial:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

Utilizar a seguinte identidade trivial:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Utilizar a seguinte identidade trivial:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= 2 (\frac{3}{2} + i \frac{1}{2}) \cdot 5 (\frac{1}{2} + i \frac{3}{2})

Simplificar

2 (\frac{3}{2} + i \frac{1}{2}) \cdot 5 (\frac{1}{2} + i \frac{3}{2}) : 10 i

2 (\frac{3}{2} + i \frac{1}{2}) \cdot 5 (\frac{1}{2} + i \frac{3}{2})

i \frac{1}{2} = \frac{i}{2}

i \frac{1}{2}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 i}{2}

Multiplicar:

1 i = i

= \frac{i}{2}

= 2 \cdot 5 (\frac{3}{2} + \frac{i}{2}) (\frac{1}{2} + i \frac{3}{2})

Multiplicar

i \frac{3}{2} : \frac{3 i}{2}

i \frac{3}{2}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 i}{2}

= 2 \cdot 5 (\frac{3}{2} + \frac{i}{2}) (\frac{3 i}{2} + \frac{1}{2})

Simplificar

\frac{3}{2} + \frac{i}{2} : \frac{3 + i}{2}

\frac{3}{2} + \frac{i}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 + i}{2}

= 2 \cdot 5 \cdot \frac{3 + i}{2} (\frac{3 i}{2} + \frac{1}{2})

Combinar as frações usando o mínimo múltiplo comum:

\frac{1 + 3 i}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 3 i}{2}

= 2 \cdot 5 \cdot \frac{3 + i}{2} (\frac{1 + 3 i}{2})

Aplicar a regra da aritmética complexa:

(a + bi) (c + d i) = (a c - b d) + (a d + b c) i

a = 53, b = 5, c = \frac{1}{2}, d = \frac{3}{2}

= (53 \frac{1}{2} - 5 \cdot \frac{3}{2}) + (53 \frac{3}{2} + 5 \cdot \frac{1}{2}) i

53 \frac{1}{2} - 5 \cdot \frac{3}{2} = 0

53 \frac{1}{2} - 5 \cdot \frac{3}{2}

53 \frac{1}{2} = \frac{5 3}{2}

53 \frac{1}{2}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 5 3}{2}

Multiplicar os números:

1 \cdot 5 = 5

= \frac{5 3}{2}

5 \cdot \frac{3}{2} = \frac{5 3}{2}

5 \cdot \frac{3}{2}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 5}{2}

= \frac{5 3}{2} - \frac{5 3}{2}

Somar elementos similares:

\frac{5 3}{2} - \frac{5 3}{2} = 0

= 0

53 \frac{3}{2} + 5 \cdot \frac{1}{2} = 10

53 \frac{3}{2} + 5 \cdot \frac{1}{2}

53 \frac{3}{2} = \frac{15}{2}

53 \frac{3}{2}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 5 3}{2}

3 \cdot 53 = 15

3 \cdot 53

Aplicar as propriedades dos radicais:

a a = a

33 = 3

= 5 \cdot 3

Multiplicar os números:

5 \cdot 3 = 15

= 15

= \frac{15}{2}

5 \cdot \frac{1}{2} = \frac{5}{2}

5 \cdot \frac{1}{2}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 5}{2}

Multiplicar os números:

1 \cdot 5 = 5

= \frac{5}{2}

= \frac{15}{2} + \frac{5}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{15 + 5}{2}

Somar:

15 + 5 = 20

= \frac{20}{2}

Dividir:

\frac{20}{2} = 10

= 10

= 0 + 10 i

Simplificar

= 10 i

= 10 i

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