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Popular Trigonometria >

sin^2(x/2)+cos(x)>0

Solução

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

Solução

π + 4 πn < x < 3 π + 4 πn

Notação de intervalo

(π + 4 πn, 3 π + 4 πn)

Decimal

3.14159 \dots + 4 πn < x < 9.42477 \dots + 4 πn

Passos da solução

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

Sea:

u = \frac{x}{2}

sin^{2} (u) + cos (2 u) > 0

sin^{2} (u) + cos (2 u) > 0 : - \frac{π}{2} + 2 πn < u < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < u < \frac{3 π}{2} + 2 πn

sin^{2} (u) + cos (2 u) > 0

Usar a seguinte identidade:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

cos^{2} (u) - sin^{2} (u) + sin^{2} (u) > 0

Simplificar

cos^{2} (u) > 0

Para

u^{n} > 0

, se

n

é par então

u < 0

u > 0

cos (u) < 0 or cos (u) > 0

cos (u) < 0 : \frac{π}{2} + 2 πn < u < \frac{3 π}{2} + 2 πn

cos (u) < 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 < u < 2 π - arccos (0) + 2 πn

Simplificar

arccos (0) : \frac{π}{2}

arccos (0)

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}

Simplificar

2 π - arccos (0) : \frac{3 π}{2}

2 π - arccos (0)

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}

Simplificar

2 π - \frac{π}{2}

Converter para fração:

2 π = \frac{2 π 2}{2}

= \frac{2 π 2}{2} - \frac{π}{2}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{2 π 2 - π}{2}

2 π 2 - π = 3 π

2 π 2 - π

Multiplicar os números:

2 \cdot 2 = 4

= 4 π - π

Somar elementos similares:

4 π - π = 3 π

= 3 π

= \frac{3 π}{2}

= \frac{3 π}{2}

\frac{π}{2} + 2 πn < u < \frac{3 π}{2} + 2 πn

cos (u) > 0 : - \frac{π}{2} + 2 πn < u < \frac{π}{2} + 2 πn

cos (u) > 0

Para

cos (x) > a

, se

- 1 \leq a < 1

então

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

- arccos (0) + 2 πn < u < arccos (0) + 2 πn

Simplificar

- arccos (0) : - \frac{π}{2}

- arccos (0)

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}

Simplificar

arccos (0) : \frac{π}{2}

arccos (0)

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}

- \frac{π}{2} + 2 πn < u < \frac{π}{2} + 2 πn

Combinar os intervalos

\frac{π}{2} + 2 πn < u < \frac{3 π}{2} + 2 πn or - \frac{π}{2} + 2 πn < u < \frac{π}{2} + 2 πn

Junte intervalos que se sobrepoem

- \frac{π}{2} + 2 πn < u < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < u < \frac{3 π}{2} + 2 πn

- \frac{π}{2} + 2 πn < u < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < u < \frac{3 π}{2} + 2 πn

Substituir na equação

\frac{x}{2} = u

- \frac{π}{2} + 2 πn < (\frac{x}{2}) < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < (\frac{x}{2}) < \frac{3 π}{2} + 2 πn

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

- \frac{π}{2} + 2 πn < (\frac{x}{2}) < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < (\frac{x}{2}) < \frac{3 π}{2} + 2 πn

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

Falso para todo

x \in R

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

a < u < b

então

a < u and u < b

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

- \frac{π}{2} + 2 πn < \frac{x}{2} : x > - π + 4 πn

- \frac{π}{2} + 2 πn < \frac{x}{2}

Trocar lados

\frac{x}{2} > - \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} > - \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

\frac{2 x}{2} > - 2 \cdot \frac{π}{2} + 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}{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 > - π + 4 πn

x > - π + 4 πn

x > - π + 4 πn

\frac{x}{2} < \frac{π}{2} + 2 πn : x < π + 4 πn

\frac{x}{2} < \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} < \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

\frac{2 x}{2} < 2 \cdot \frac{π}{2} + 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}{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 < π + 4 πn

x < π + 4 πn

x < π + 4 πn

Combinar os intervalos

x > - π + 4 πn and x < π + 4 πn

Junte intervalos que se sobrepoem

Falso para todo x \in R

\frac{π}{2} + 2 πn < \frac{x}{2} < \frac{3 π}{2} + 2 πn : π + 4 πn < x < 3 π + 4 πn

\frac{π}{2} + 2 πn < \frac{x}{2} < \frac{3 π}{2} + 2 πn

a < u < b

então

a < u and u < b

\frac{π}{2} + 2 πn < \frac{x}{2} and \frac{x}{2} < \frac{3 π}{2} + 2 πn

\frac{π}{2} + 2 πn < \frac{x}{2} : x > π + 4 πn

\frac{π}{2} + 2 πn < \frac{x}{2}

Trocar lados

\frac{x}{2} > \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} > \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

\frac{2 x}{2} > 2 \cdot \frac{π}{2} + 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}{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 > π + 4 πn

x > π + 4 πn

x > π + 4 πn

\frac{x}{2} < \frac{3 π}{2} + 2 πn : x < 3 π + 4 πn

\frac{x}{2} < \frac{3 π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} < \frac{3 π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} < 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} < 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiplicar frações:

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

= \frac{3 π 2}{2}

Eliminar o fator comum:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

x < 3 π + 4 πn

x < 3 π + 4 πn

x < 3 π + 4 πn

Combinar os intervalos

x > π + 4 πn and x < 3 π + 4 πn

Junte intervalos que se sobrepoem

π + 4 πn < x < 3 π + 4 πn

Combinar os intervalos

Falso para todo x \in R or π + 4 πn < x < 3 π + 4 πn

Junte intervalos que se sobrepoem

π + 4 πn < x < 3 π + 4 πn

π + 4 πn < x < 3 π + 4 πn

Exemplos populares

cos(x)>-(sqrt(3))/2

cos (x) > - \frac{3}{2}

pi/2-arctan(x^6)<0.0001

\frac{π}{2} - arctan (x^{6}) < 0.0001

2sin(x)cos(x)>cos(x)

2 sin (x) cos (x) > cos (x)

tan(θ)<= sqrt(3)

tan (θ) \leq 3

1-4(cos(x)sin(x/2))>= 0

1 - 4 (cos (x) sin (\frac{x}{2})) \geq 0