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Popular Trigonometria >

5(cos(pi/6)+isin(pi/6))

Solução

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

Solução

\frac{5 3}{2} + i \frac{5}{2}

Passos da solução

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

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}

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

Simplificar

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

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

= 5 (\frac{3}{2} + \frac{i}{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}

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

Multiplicar frações:

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

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

Reescrever

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

na forma complexa padrão:

\frac{5 3}{2} + \frac{5}{2} i

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

Expandir

(3 + i) \cdot 5 : 53 + 5 i

(3 + i) \cdot 5

= 5 (3 + i)

Colocar os parênteses utilizando:

a (b + c) = ab + a c

a = 5, b = 3, c = i

= 53 + 5 i

= \frac{5 3 + 5 i}{2}

Aplicar as propriedades das frações:

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

\frac{5 3 + 5 i}{2} = \frac{5 3}{2} + \frac{5 i}{2}

= \frac{5 3}{2} + \frac{5 i}{2}

= \frac{5 3}{2} + \frac{5}{2} i

= \frac{5 3}{2} + i \frac{5}{2}

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