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Popular Trigonometria >

cos((pi*x)/2)>0

Solução

cos (\frac{π \cdot x}{2}) > 0

Solução

- 1 + 4 n < x < 1 + 4 n

Notação de intervalo

(- 1 + 4 n, 1 + 4 n)

Passos da solução

cos (\frac{π x}{2}) > 0

Para

cos (x) > a

, se

- 1 \leq a < 1

então

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

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

a < u < b

então

a < u and u < b

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

- arccos (0) + 2 πn < \frac{π x}{2} : x > - 1 + 4 n

- arccos (0) + 2 πn < \frac{π x}{2}

Trocar lados

\frac{π x}{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{π x}{2} > - \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{π x}{2} > - \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Dividir:

\frac{2}{2} = 1

= π x

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

π x > - π + 4 πn

π x > - π + 4 πn

π x > - π + 4 πn

Dividir ambos os lados por

π

π x > - π + 4 πn

Dividir ambos os lados por

π

\frac{π x}{π} > - \frac{π}{π} + \frac{4 πn}{π}

Simplificar

\frac{π x}{π} > - \frac{π}{π} + \frac{4 πn}{π}

Simplificar

\frac{π x}{π} : x

\frac{π x}{π}

Eliminar o fator comum:

π

= x

Simplificar

- \frac{π}{π} + \frac{4 πn}{π} : - 1 + 4 n

- \frac{π}{π} + \frac{4 πn}{π}

Aplicar a regra

\frac{a}{a} = 1

\frac{π}{π} = 1

= - 1 + \frac{4 πn}{π}

Cancelar

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Eliminar o fator comum:

π

= 4 n

= - 1 + 4 n

x > - 1 + 4 n

x > - 1 + 4 n

x > - 1 + 4 n

\frac{π x}{2} < arccos (0) + 2 πn : x < 1 + 4 n

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

Multiplicar ambos os lados por

2

\frac{π x}{2} < \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 π x}{2} < 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 π x}{2} < 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Dividir:

\frac{2}{2} = 1

= π x

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

π x < π + 4 πn

π x < π + 4 πn

π x < π + 4 πn

Dividir ambos os lados por

π

π x < π + 4 πn

Dividir ambos os lados por

π

\frac{π x}{π} < \frac{π}{π} + \frac{4 πn}{π}

Simplificar

\frac{π x}{π} < \frac{π}{π} + \frac{4 πn}{π}

Simplificar

\frac{π x}{π} : x

\frac{π x}{π}

Eliminar o fator comum:

π

= x

Simplificar

\frac{π}{π} + \frac{4 πn}{π} : 1 + 4 n

\frac{π}{π} + \frac{4 πn}{π}

Aplicar a regra

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{4 πn}{π}

Cancelar

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Eliminar o fator comum:

π

= 4 n

= 1 + 4 n

x < 1 + 4 n

x < 1 + 4 n

x < 1 + 4 n

Combinar os intervalos

x > - 1 + 4 n and x < 1 + 4 n

Junte intervalos que se sobrepoem

- 1 + 4 n < x < 1 + 4 n

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cos(x)(1-tan(x))<= 0

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sin (x) \geq \frac{- 2}{2}

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