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Popular Trigonometria >

4cos(x/2)+1>2sqrt(2)+1,0<= x<= 6pi

Solução

4 cos (\frac{x}{2}) + 1 > 22 + 1, 0 \leq x \leq 6 π

Solução

0 \leq x < \frac{π}{2} or 4 π - \frac{π}{2} < x < 4 π + \frac{π}{2}

Notação de intervalo

[0, \frac{π}{2}) \cup (4 π - \frac{π}{2}, 4 π + \frac{π}{2})

Decimal

0 \leq x < 1.57079 \dots or 10.99557 \dots < x < 14.13716 \dots

Passos da solução

4 cos (\frac{x}{2}) + 1 > 22 + 1, 0 \leq x \leq 6 π

Subtrair

1

de ambos os lados

4 cos (\frac{x}{2}) + 1 - 1 > 22 + 1 - 1

Simplificar

4 cos (\frac{x}{2}) > 22

Dividir ambos os lados por

4

4 cos (\frac{x}{2}) > 22

Dividir ambos os lados por

4

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

Simplificar

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

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

Para

cos (x) > a

, se

- 1 \leq a < 1

então

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

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

a < u < b

então

a < u and u < b

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

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

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

Trocar lados

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

Simplificar

- arccos (\frac{2}{2}) + 2 πn : - \frac{π}{4} + 2 πn

- arccos (\frac{2}{2}) + 2 πn

Utilizar a seguinte identidade trivial:

arccos (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

\frac{x}{2} > - \frac{π}{4} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} > - \frac{π}{4} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

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

- 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiplicar frações:

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

= \frac{π 2}{4}

Eliminar o fator comum:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= - \frac{π}{2} + 4 πn

x > - \frac{π}{2} + 4 πn

x > - \frac{π}{2} + 4 πn

x > - \frac{π}{2} + 4 πn

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

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

Simplificar

arccos (\frac{2}{2}) + 2 πn : \frac{π}{4} + 2 πn

arccos (\frac{2}{2}) + 2 πn

Utilizar a seguinte identidade trivial:

arccos (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

\frac{x}{2} < \frac{π}{4} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} < \frac{π}{4} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

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

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiplicar frações:

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

= \frac{π 2}{4}

Eliminar o fator comum:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x < \frac{π}{2} + 4 πn

x < \frac{π}{2} + 4 πn

x < \frac{π}{2} + 4 πn

Combinar os intervalos

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

Junte intervalos que se sobrepoem

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

Combinar os intervalos

- \frac{π}{2} + 4 πn < x < \frac{π}{2} + 4 πn and 0 \leq x \leq 6 π

0 \leq x < \frac{π}{2} or 4 π - \frac{π}{2} < x < 4 π + \frac{π}{2}

Gráfico

Exemplos populares

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