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Popular Trigonometria >

-1/2 <= cos(2x)

Solução

- \frac{1}{2} \leq cos (2 x)

Solução

- \frac{π}{3} + πn \leq x \leq \frac{π}{3} + πn

Notação de intervalo

[- \frac{π}{3} + πn, \frac{π}{3} + πn]

Decimal

- 1.04719 \dots + πn \leq x \leq 1.04719 \dots + πn

Passos da solução

- \frac{1}{2} \leq cos (2 x)

Trocar lados

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

Para

cos (x) \geq a

, se

- 1 < a < 1

então

- arccos (a) + 2 πn \leq x \leq arccos (a) + 2 πn

- arccos (- \frac{1}{2}) + 2 πn \leq 2 x \leq arccos (- \frac{1}{2}) + 2 πn

a \leq u \leq b

então

a \leq u and u \leq b

- arccos (- \frac{1}{2}) + 2 πn \leq 2 x and 2 x \leq arccos (- \frac{1}{2}) + 2 πn

- arccos (- \frac{1}{2}) + 2 πn \leq 2 x : x \geq - \frac{π}{3} + πn

- arccos (- \frac{1}{2}) + 2 πn \leq 2 x

Trocar lados

2 x \geq - arccos (- \frac{1}{2}) + 2 πn

Simplificar

- arccos (- \frac{1}{2}) + 2 πn : - \frac{2 π}{3} + 2 πn

- arccos (- \frac{1}{2}) + 2 πn

Utilizar a seguinte identidade trivial:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

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{2 π}{3} + 2 πn

2 x \geq - \frac{2 π}{3} + 2 πn

Dividir ambos os lados por

2

2 x \geq - \frac{2 π}{3} + 2 πn

Dividir ambos os lados por

2

\frac{2 x}{2} \geq - \frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} \geq - \frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

- \frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Aplicar as propriedades das frações:

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

= \frac{2 π}{3 \cdot 2}

Multiplicar os números:

3 \cdot 2 = 6

= \frac{2 π}{6}

Eliminar o fator comum:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= - \frac{π}{3} + πn

x \geq - \frac{π}{3} + πn

x \geq - \frac{π}{3} + πn

x \geq - \frac{π}{3} + πn

2 x \leq arccos (- \frac{1}{2}) + 2 πn : x \leq \frac{π}{3} + πn

2 x \leq arccos (- \frac{1}{2}) + 2 πn

Simplificar

arccos (- \frac{1}{2}) + 2 πn : \frac{2 π}{3} + 2 πn

arccos (- \frac{1}{2}) + 2 πn

Utilizar a seguinte identidade trivial:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

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{2 π}{3} + 2 πn

2 x \leq \frac{2 π}{3} + 2 πn

Dividir ambos os lados por

2

2 x \leq \frac{2 π}{3} + 2 πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

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

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Aplicar as propriedades das frações:

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

= \frac{2 π}{3 \cdot 2}

Multiplicar os números:

3 \cdot 2 = 6

= \frac{2 π}{6}

Eliminar o fator comum:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

x \leq \frac{π}{3} + πn

x \leq \frac{π}{3} + πn

x \leq \frac{π}{3} + πn

Combinar os intervalos

x \geq - \frac{π}{3} + πn and x \leq \frac{π}{3} + πn

Junte intervalos que se sobrepoem

- \frac{π}{3} + πn \leq x \leq \frac{π}{3} + πn

Exemplos populares

-2+3csc(2x-pi)>= 0

- 2 + 3 csc (2 x - π) \geq 0

cos(4x)> 1/2

cos (4 x) > \frac{1}{2}

tan(x)>=-(sqrt(3))/3 ,0<= x<= 2pi

tan (x) \geq - \frac{3}{3}, 0 \leq x \leq 2 π

sin(θ)<3

sin (θ) < 3

sin(θ)<2

sin (θ) < 2