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5(cos(135)+isin(135))

Solução

5 (cos (13 5^{\circ}) + i sin (13 5^{\circ}))

Solução

- \frac{5 2}{2} + i \frac{5 2}{2}

Passos da solução

5 (cos (13 5^{\circ}) + i sin (13 5^{\circ}))

Utilizar a seguinte identidade trivial:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

Utilizar a seguinte identidade trivial:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= 5 (- \frac{2}{2} + i \frac{2}{2})

Simplificar

5 (- \frac{2}{2} + i \frac{2}{2}) : - \frac{5 2}{2} + i \frac{5 2}{2}

5 (- \frac{2}{2} + i \frac{2}{2})

Multiplicar

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

i \frac{2}{2}

Multiplicar frações:

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

= \frac{2 i}{2}

= 5 (- \frac{2}{2} + \frac{2 i}{2})

Simplificar

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

- \frac{2}{2} + \frac{2 i}{2}

Aplicar a regra

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

= \frac{- 2 + 2 i}{2}

= 5 \cdot \frac{- 2 + 2 i}{2}

Multiplicar frações:

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

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

Fatorar

- 2 + 2 i : 2 (- 1 + i)

- 2 + 2 i

Fatorar o termo comum

2

= 2 (- 1 + i)

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

Cancelar

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

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

Aplicar as propriedades dos radicais:

2 = 2^{\frac{1}{2}}

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

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5 ( - 1 + i )}{2 ^{1 - \frac{1}{2}}}

Subtrair:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{5 ( - 1 + i )}{2 ^{\frac{1}{2}}}

Aplicar as propriedades dos radicais:

2^{\frac{1}{2}} = 2

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

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

Racionalizar

\frac{5 ( - 1 + i )}{2} : \frac{5 2 ( - 1 + i )}{2}

\frac{5 ( - 1 + i )}{2}

Multiplicar pelo conjugado

\frac{2}{2}

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

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

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

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

Reescrever

\frac{5 2 ( - 1 + i )}{2}

na forma complexa padrão:

- \frac{5 2}{2} + \frac{5 2}{2} i

\frac{5 2 ( - 1 + i )}{2}

Aplicar as propriedades dos radicais:

2 = 2^{\frac{1}{2}}

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

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5 ( - 1 + i )}{2 ^{1 - \frac{1}{2}}}

Subtrair:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{5 ( - 1 + i )}{2 ^{\frac{1}{2}}}

Aplicar as propriedades dos radicais:

2^{\frac{1}{2}} = 2

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

Expandir

5 (- 1 + i) : - 5 + 5 i

5 (- 1 + i)

Colocar os parênteses utilizando:

a (b + c) = ab + a c

a = 5, b = - 1, c = i

= 5 (- 1) + 5 i

Aplicar as regras dos sinais

+ (- a) = - a

= - 5 \cdot 1 + 5 i

Multiplicar os números:

5 \cdot 1 = 5

= - 5 + 5 i

= \frac{- 5 + 5 i}{2}

Aplicar as propriedades das frações:

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

\frac{- 5 + 5 i}{2} = - \frac{5}{2} + \frac{5 i}{2}

= - \frac{5}{2} + \frac{5 i}{2}

\frac{5}{2} = \frac{5 2}{2}

\frac{5}{2}

Multiplicar pelo conjugado

\frac{2}{2}

= \frac{5 2}{2 2}

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

= \frac{5 2}{2}

= - \frac{5}{2} + \frac{5 2}{2} i

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

- \frac{5}{2}

Multiplicar pelo conjugado

\frac{2}{2}

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

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

= - \frac{5 2}{2}

= - \frac{5 2}{2} + \frac{5 2}{2} i

= - \frac{5 2}{2} + \frac{5 2}{2} i

= - \frac{5 2}{2} + i \frac{5 2}{2}

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