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Popular Trigonometria >

2sin(x/2)+1>0

Solução

2 sin (\frac{x}{2}) + 1 > 0

Solução

- \frac{π}{3} + 4 πn < x < \frac{7 π}{3} + 4 πn

Notação de intervalo

(- \frac{π}{3} + 4 πn, \frac{7 π}{3} + 4 πn)

Decimal

- 1.04719 \dots + 4 πn < x < 7.33038 \dots + 4 πn

Passos da solução

2 sin (\frac{x}{2}) + 1 > 0

Mova

1

para o lado direito

2 sin (\frac{x}{2}) + 1 > 0

Subtrair

1

de ambos os lados

2 sin (\frac{x}{2}) + 1 - 1 > 0 - 1

Simplificar

2 sin (\frac{x}{2}) > - 1

2 sin (\frac{x}{2}) > - 1

Dividir ambos os lados por

2

2 sin (\frac{x}{2}) > - 1

Dividir ambos os lados por

2

\frac{2 sin ( \frac{x}{2} )}{2} > \frac{- 1}{2}

Simplificar

sin (\frac{x}{2}) > - \frac{1}{2}

sin (\frac{x}{2}) > - \frac{1}{2}

Para

sin (x) > a

, se

- 1 \leq a < 1

então

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

arcsin (- \frac{1}{2}) + 2 πn < \frac{x}{2} < π - arcsin (- \frac{1}{2}) + 2 πn

a < u < b

então

a < u and u < b

arcsin (- \frac{1}{2}) + 2 πn < \frac{x}{2} and \frac{x}{2} < π - arcsin (- \frac{1}{2}) + 2 πn

arcsin (- \frac{1}{2}) + 2 πn < \frac{x}{2} : x > - \frac{π}{3} + 4 πn

arcsin (- \frac{1}{2}) + 2 πn < \frac{x}{2}

Trocar lados

\frac{x}{2} > arcsin (- \frac{1}{2}) + 2 πn

Simplificar

arcsin (- \frac{1}{2}) + 2 πn : - \frac{π}{6} + 2 πn

arcsin (- \frac{1}{2}) + 2 πn

arcsin (- \frac{1}{2}) = - \frac{π}{6}

arcsin (- \frac{1}{2})

Utilizar a seguinte propriedade:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Utilizar a seguinte identidade trivial:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6}

= - \frac{π}{6}

= - \frac{π}{6} + 2 πn

\frac{x}{2} > - \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} > - \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} > - 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} > - 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

- 2 \cdot \frac{π}{6} + 2 \cdot 2 πn : - \frac{π}{3} + 4 πn

- 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiplicar frações:

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

= \frac{π 2}{6}

Eliminar o fator comum:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= - \frac{π}{3} + 4 πn

x > - \frac{π}{3} + 4 πn

x > - \frac{π}{3} + 4 πn

x > - \frac{π}{3} + 4 πn

\frac{x}{2} < π - arcsin (- \frac{1}{2}) + 2 πn : x < \frac{7 π}{3} + 4 πn

\frac{x}{2} < π - arcsin (- \frac{1}{2}) + 2 πn

Simplificar

π - arcsin (- \frac{1}{2}) + 2 πn : π + \frac{π}{6} + 2 πn

π - arcsin (- \frac{1}{2}) + 2 πn

arcsin (- \frac{1}{2}) = - \frac{π}{6}

arcsin (- \frac{1}{2})

Utilizar a seguinte propriedade:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Utilizar a seguinte identidade trivial:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6}

= - \frac{π}{6}

= π - (- \frac{π}{6}) + 2 πn

Aplicar a regra

- (- a) = a

= π + \frac{π}{6} + 2 πn

\frac{x}{2} < π + \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} < π + \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} < 2 π + 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} < 2 π + 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

2 π + 2 \cdot \frac{π}{6} + 2 \cdot 2 πn : 2 π + \frac{π}{3} + 4 πn

2 π + 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiplicar frações:

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

= \frac{π 2}{6}

Eliminar o fator comum:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= 2 π + \frac{π}{3} + 4 πn

x < 2 π + \frac{π}{3} + 4 πn

x < 2 π + \frac{π}{3} + 4 πn

Simplificar

2 π + \frac{π}{3} : \frac{7 π}{3}

2 π + \frac{π}{3}

Converter para fração:

2 π = \frac{2 π 3}{3}

= \frac{2 π 3}{3} + \frac{π}{3}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{2 π 3 + π}{3}

2 π 3 + π = 7 π

2 π 3 + π

Multiplicar os números:

2 \cdot 3 = 6

= 6 π + π

Somar elementos similares:

6 π + π = 7 π

= 7 π

= \frac{7 π}{3}

x < \frac{7 π}{3} + 4 πn

x < \frac{7 π}{3} + 4 πn

Combinar os intervalos

x > - \frac{π}{3} + 4 πn and x < \frac{7 π}{3} + 4 πn

Junte intervalos que se sobrepoem

- \frac{π}{3} + 4 πn < x < \frac{7 π}{3} + 4 πn

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