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Popular Trigonometria >

cos((x-60)/2)<0

Solução

cos (\frac{x - 60}{2}) < 0

Solução

π + 60 + 4 πn < x < 3 π + 60 + 4 πn

Notação de intervalo

(π + 60 + 4 πn, 3 π + 60 + 4 πn)

Decimal

63.14159 \dots + 4 πn < x < 69.42477 \dots + 4 πn

Passos da solução

cos (\frac{x - 60}{2}) < 0

Para

cos (x) < a

, se

- 1 < a \leq 1

então

arccos (a) + 2 πn < x < 2 π - arccos (a) + 2 πn

arccos (0) + 2 πn < \frac{x - 60}{2} < 2 π - arccos (0) + 2 πn

a < u < b

então

a < u and u < b

arccos (0) + 2 πn < \frac{x - 60}{2} and \frac{x - 60}{2} < 2 π - arccos (0) + 2 πn

arccos (0) + 2 πn < \frac{x - 60}{2} : x > π + 4 πn + 60

arccos (0) + 2 πn < \frac{x - 60}{2}

Trocar lados

\frac{x - 60}{2} > arccos (0) + 2 πn

Simplificar

arccos (0) + 2 πn : \frac{π}{2} + 2 πn

arccos (0) + 2 πn

Utilizar a seguinte identidade trivial:

arccos (0) = \frac{π}{2}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{2} + 2 πn

\frac{x - 60}{2} > \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x - 60}{2} > \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 ( x - 60 )}{2} > 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 ( x - 60 )}{2} > 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 ( x - 60 )}{2} : x - 60

\frac{2 ( x - 60 )}{2}

Dividir:

\frac{2}{2} = 1

= x - 60

Simplificar

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiplicar frações:

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

= \frac{π 2}{2}

Eliminar o fator comum:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x - 60 > π + 4 πn

x - 60 > π + 4 πn

x - 60 > π + 4 πn

Mova

60

para o lado direito

x - 60 > π + 4 πn

Adicionar

60

a ambos os lados

x - 60 + 60 > π + 4 πn + 60

Simplificar

x > π + 4 πn + 60

x > π + 4 πn + 60

\frac{x - 60}{2} < 2 π - arccos (0) + 2 πn : x < 3 π + 4 πn + 60

\frac{x - 60}{2} < 2 π - arccos (0) + 2 πn

Simplificar

2 π - arccos (0) + 2 πn : 2 π - \frac{π}{2} + 2 πn

2 π - arccos (0) + 2 πn

Utilizar a seguinte identidade trivial:

arccos (0) = \frac{π}{2}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{2} + 2 πn

\frac{x - 60}{2} < 2 π - \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x - 60}{2} < 2 π - \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 ( x - 60 )}{2} < 2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 ( x - 60 )}{2} < 2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 ( x - 60 )}{2} : x - 60

\frac{2 ( x - 60 )}{2}

Dividir:

\frac{2}{2} = 1

= x - 60

Simplificar

2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot 2 π = 4 π

2 \cdot 2 π

Multiplicar os números:

2 \cdot 2 = 4

= 4 π

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiplicar frações:

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

= \frac{π 2}{2}

Eliminar o fator comum:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= 4 π - π + 4 πn

Somar elementos similares:

4 π - π = 3 π

= 3 π + 4 πn

x - 60 < 3 π + 4 πn

x - 60 < 3 π + 4 πn

x - 60 < 3 π + 4 πn

Mova

60

para o lado direito

x - 60 < 3 π + 4 πn

Adicionar

60

a ambos os lados

x - 60 + 60 < 3 π + 4 πn + 60

Simplificar

x < 3 π + 4 πn + 60

x < 3 π + 4 πn + 60

Combinar os intervalos

x > π + 4 πn + 60 and x < 3 π + 4 πn + 60

Junte intervalos que se sobrepoem

π + 60 + 4 πn < x < 3 π + 60 + 4 πn

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