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4cos(x+pi/4)>= 2sqrt(2)

Solución

4 cos (x + \frac{π}{4}) \geq 22

Solución

- \frac{π}{2} + 2 πn \leq x \leq 2 πn

Notación de intervalos

[- \frac{π}{2} + 2 πn, 2 πn]

Notación decimal

- 1.57079 \dots + 2 πn \leq x \leq 2 πn

Pasos de solución

4 cos (x + \frac{π}{4}) \geq 22

Dividir ambos lados entre

4

4 cos (x + \frac{π}{4}) \geq 22

Dividir ambos lados entre

4

\frac{4 cos ( x + \frac{π}{4} )}{4} \geq \frac{2 2}{4}

Simplificar

cos (x + \frac{π}{4}) \geq \frac{2}{2}

cos (x + \frac{π}{4}) \geq \frac{2}{2}

Para

cos (x) \geq a

, si

- 1 < a < 1

entonces

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

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

a \leq u \leq b

entonces

a \leq u and u \leq b

- arccos (\frac{2}{2}) + 2 πn \leq x + \frac{π}{4} and x + \frac{π}{4} \leq arccos (\frac{2}{2}) + 2 πn

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

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

Intercambiar lados

x + \frac{π}{4} \geq - 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 la siguiente identidad 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

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

Mover

\frac{π}{4}

al lado derecho

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

Restar

\frac{π}{4}

de ambos lados

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

Simplificar

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

Simplificar

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Sumar elementos similares:

\frac{π}{4} - \frac{π}{4} \geq 0

= x

Simplificar

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

- \frac{π}{4} + 2 πn - \frac{π}{4}

Agrupar términos semejantes

= - \frac{π}{4} - \frac{π}{4} + 2 πn

Combinar las fracciones usando el mínimo común denominador:

- \frac{π}{2}

Aplicar la regla

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

= \frac{- π - π}{4}

Sumar elementos similares:

- π - π = - 2 π

= \frac{- 2 π}{4}

Aplicar las propiedades de las fracciones:

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

= - \frac{2 π}{4}

Eliminar los terminos comunes:

2

= - \frac{π}{2}

= - \frac{π}{2} + 2 πn

x \geq - \frac{π}{2} + 2 πn

x \geq - \frac{π}{2} + 2 πn

x \geq - \frac{π}{2} + 2 πn

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

x + \frac{π}{4} \leq 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 la siguiente identidad 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

x + \frac{π}{4} \leq \frac{π}{4} + 2 πn

Mover

\frac{π}{4}

al lado derecho

x + \frac{π}{4} \leq \frac{π}{4} + 2 πn

Restar

\frac{π}{4}

de ambos lados

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

Simplificar

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

Simplificar

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Sumar elementos similares:

\frac{π}{4} - \frac{π}{4} \leq 0

= x

Simplificar

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

\frac{π}{4} + 2 πn - \frac{π}{4}

Agrupar términos semejantes

= \frac{π}{4} - \frac{π}{4} + 2 πn

Sumar elementos similares:

\frac{π}{4} - \frac{π}{4} = 0

= 2 πn

x \leq 2 πn

x \leq 2 πn

x \leq 2 πn

Combinar los rangos

x \geq - \frac{π}{2} + 2 πn and x \leq 2 πn

Mezclar intervalos sobrepuestos

- \frac{π}{2} + 2 πn \leq x \leq 2 πn

Ejemplos populares

sec^2(x)<= 4

sec^{2} (x) \leq 4

cos(x)>-1/2

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

sin(θ)<0,cos(θ)>0

sin (θ) < 0, cos (θ) > 0

cot(x)<-1

cot (x) < - 1

2cos(x)<= 1,-pi<= x<= pi

2 cos (x) \leq 1, - π \leq x \leq π