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Popular Trigonometria >

cos(pi/2)-sin((5pi)/3)+tan((9pi)/4)-cos((5pi)/6)+tan((7pi)/6)

Solução

cos (\frac{π}{2}) - sin (\frac{5 π}{3}) + tan (\frac{9 π}{4}) - cos (\frac{5 π}{6}) + tan (\frac{7 π}{6})

Solução

\frac{3}{3} + 1 + 3

Decimal

3.30940 \dots

Passos da solução

cos (\frac{π}{2}) - sin (\frac{5 π}{3}) + tan (\frac{9 π}{4}) - cos (\frac{5 π}{6}) + tan (\frac{7 π}{6})

tan (\frac{9 π}{4}) = tan (\frac{π}{4})

tan (\frac{9 π}{4})

Reescrever

\frac{9 π}{4}

como

π \cdot 2 + \frac{π}{4}

= tan (π 2 + \frac{π}{4})

Utilizar a periodicidade de

tan

tan (x + π \cdot k) = tan (x)

tan (π \cdot 2 + \frac{π}{4}) = tan (\frac{π}{4})

= tan (\frac{π}{4})

= cos (\frac{π}{2}) - sin (\frac{5 π}{3}) + tan (\frac{π}{4}) - cos (\frac{5 π}{6}) + tan (\frac{7 π}{6})

tan (\frac{7 π}{6}) = tan (\frac{π}{6})

tan (\frac{7 π}{6})

Reescrever

\frac{7 π}{6}

como

π + \frac{π}{6}

= tan (π + \frac{π}{6})

Utilizar a periodicidade de

tan

tan (x + π) = tan (x)

tan (π + \frac{π}{6}) = tan (\frac{π}{6})

= tan (\frac{π}{6})

= cos (\frac{π}{2}) - sin (\frac{5 π}{3}) + tan (\frac{π}{4}) - cos (\frac{5 π}{6}) + tan (\frac{π}{6})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

Reeecreva usando identidades trigonométricas:

sin (\frac{5 π}{3}) = - \frac{3}{2}

sin (\frac{5 π}{3})

Reeecreva usando identidades trigonométricas:

sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

sin (\frac{5 π}{3})

Escrever

sin (\frac{5 π}{3})

como

sin (π + \frac{2 π}{3})

= sin (π + \frac{2 π}{3})

Use a identidade de soma de ângulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

= sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

Utilizar a seguinte identidade trivial:

sin (π) = 0

sin (π)

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}

= 0

Utilizar a seguinte identidade trivial:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 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:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

Utilizar a seguinte identidade trivial:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 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}

= 0 \cdot (- \frac{1}{2}) + (- 1) \frac{3}{2}

Simplificar

= - \frac{3}{2}

Utilizar a seguinte identidade trivial:

tan (\frac{π}{4}) = 1

tan (\frac{π}{4})

tan (x)

tabela de periodicidade com ciclo de

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

Utilizar a seguinte identidade trivial:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 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:

tan (\frac{π}{6}) = \frac{3}{3}

tan (\frac{π}{6})

tan (x)

tabela de periodicidade com ciclo de

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

= 0 - (- \frac{3}{2}) + 1 - (- \frac{3}{2}) + \frac{3}{3}

Simplificar

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

0 - (- \frac{3}{2}) + 1 - (- \frac{3}{2}) + \frac{3}{3}

Aplicar a regra

- (- a) = a

= 0 + \frac{3}{2} + 1 + \frac{3}{2} + \frac{3}{3}

Agrupar termos semelhantes

= \frac{3}{2} + \frac{3}{2} + \frac{3}{3} + 0 + 1

Somar elementos similares:

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

= 2 \cdot \frac{3}{2} + \frac{3}{3} + 0 + 1

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

2 \cdot \frac{3}{2}

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 3

= 3 + \frac{3}{3} + 0 + 1

Somar:

0 + 1 = 1

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

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

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sin (0.245)