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Popular Trigonometria >

3(cos(120)+isin(120))

Solução

3 (cos (12 0^{\circ}) + i sin (12 0^{\circ}))

Solução

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

Passos da solução

3 (cos (12 0^{\circ}) + i sin (12 0^{\circ}))

Utilizar a seguinte identidade trivial:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 0^{\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{1}{2}

Utilizar a seguinte identidade trivial:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\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{3}{2}

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

Simplificar

3 (- \frac{1}{2} + i \frac{3}{2}) : - \frac{3}{2} + i \frac{3 3}{2}

3 (- \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}

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

Simplificar

- \frac{1}{2} + \frac{3 i}{2} : \frac{- 1 + 3 i}{2}

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

Aplicar a regra

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

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

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

Multiplicar frações:

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

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

Reescrever

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

na forma complexa padrão:

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

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

Expandir

(- 1 + 3 i) \cdot 3 : - 3 + 33 i

(- 1 + 3 i) \cdot 3

= 3 (- 1 + 3 i)

Colocar os parênteses utilizando:

a (b + c) = ab + a c

a = 3, b = - 1, c = 3 i

= 3 (- 1) + 33 i

Aplicar as regras dos sinais

+ (- a) = - a

= - 3 \cdot 1 + 33 i

Multiplicar os números:

3 \cdot 1 = 3

= - 3 + 33 i

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

Aplicar as propriedades das frações:

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

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

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

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

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

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