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Popular Trigonometria >

cos^2(x)-sin^2(x)>0

Solução

cos^{2} (x) - sin^{2} (x) > 0

Solução

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

Notação de intervalo

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

Decimal

- 0.78539 \dots + πn < x < 0.78539 \dots + πn

Passos da solução

cos^{2} (x) - sin^{2} (x) > 0

Usar a seguinte identidade:

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

cos (2 x) > 0

Para

cos (x) > a

, se

- 1 \leq a < 1

então

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

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

a < u < b

então

a < u and u < b

- arccos (0) + 2 πn < 2 x and 2 x < arccos (0) + 2 πn

- arccos (0) + 2 πn < 2 x : x > - \frac{π}{4} + πn

- arccos (0) + 2 πn < 2 x

Trocar lados

2 x > - arccos (0) + 2 πn

Simplificar

- arccos (0) + 2 πn : - \frac{π}{2} + 2 πn

- arccos (0) + 2 πn

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} + 2 πn

2 x > - \frac{π}{2} + 2 πn

Dividir ambos os lados por

2

2 x > - \frac{π}{2} + 2 πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

- \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : - \frac{π}{4} + πn

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

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Aplicar as propriedades das frações:

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

= \frac{π}{2 \cdot 2}

Multiplicar os números:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= - \frac{π}{4} + πn

x > - \frac{π}{4} + πn

x > - \frac{π}{4} + πn

x > - \frac{π}{4} + πn

2 x < arccos (0) + 2 πn : x < \frac{π}{4} + πn

2 x < arccos (0) + 2 πn

Simplificar

arccos (0) + 2 πn : \frac{π}{2} + 2 πn

arccos (0) + 2 πn

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} + 2 πn

2 x < \frac{π}{2} + 2 πn

Dividir ambos os lados por

2

2 x < \frac{π}{2} + 2 πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

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

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Aplicar as propriedades das frações:

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

= \frac{π}{2 \cdot 2}

Multiplicar os números:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x < \frac{π}{4} + πn

x < \frac{π}{4} + πn

x < \frac{π}{4} + πn

Combinar os intervalos

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

Junte intervalos que se sobrepoem

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

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sec(2x)<0

sec (2 x) < 0

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\frac{3}{2} \cdot cos (2 x) \geq 0

sin^2(x)+1/2 sin(x)>0

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

1/2 sin(x)-5<=-1/2

\frac{1}{2} sin (x) - 5 \leq - \frac{1}{2}