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Popular Trigonometria >

1/2 >sin(x/2)

Solução

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

Solução

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

Notação de intervalo

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

Decimal

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

Passos da solução

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

Trocar lados

sin (\frac{x}{2}) < \frac{1}{2}

Para

sin (x) < a

, se

- 1 < a \leq 1

então

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

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

a < u < b

então

a < u and u < b

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

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

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

Trocar lados

\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

Utilizar a seguinte identidade trivial:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}

= - 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}

Aplicar as propriedades das frações:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{7 π}{3}

x > - \frac{7 π}{3} + 4 πn

x > - \frac{7 π}{3} + 4 πn

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

\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

Utilizar a seguinte identidade trivial:

arcsin (\frac{1}{2}) = \frac{π}{6}

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} + 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

Combinar os intervalos

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

Junte intervalos que se sobrepoem

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

Exemplos populares

sin(x)<(-sqrt(3))/2

sin (x) < \frac{- 3}{2}

0<2-sec(x^2)

0 < 2 - sec (x^{2})

-0.25<= 0.5sin(2x),0<= x<= 360

- 0.25 \leq 0.5 sin (2 x), 0^{\circ} \leq x \leq 36 0^{\circ}

cos(x/2)>= (sqrt(2))/2

cos (\frac{x}{2}) \geq \frac{2}{2}

cos^2(x)>-2

cos^{2} (x) > - 2