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(sin(x)+cos(x))>= 1/2

Solución

(sin (x) + cos (x)) \geq \frac{1}{2}

Solución

\frac{4 arcsin ( \frac{2}{4} ) - π}{4} + 2 πn \leq x \leq \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4} + 2 πn

Notación de intervalos

\frac{4 arcsin ( \frac{2}{4} ) - π}{4} + 2 πn, \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4} + 2 πn

Notación decimal

- 0.42403 \dots + 2 πn \leq x \leq 1.99482 \dots + 2 πn

Pasos de solución

sin (x) + cos (x) \geq \frac{1}{2}

Usar la siguiente identidad:

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

2 sin (\frac{π}{4} + x) \geq \frac{1}{2}

Dividir ambos lados entre

2

2 sin (\frac{π}{4} + x) \geq \frac{1}{2}

Dividir ambos lados entre

2

\frac{2 sin ( \frac{π}{4} + x )}{2} \geq \frac{\frac{1}{2}}{2}

Simplificar

\frac{2 sin ( \frac{π}{4} + x )}{2} \geq \frac{\frac{1}{2}}{2}

Simplificar

\frac{2 sin ( \frac{π}{4} + x )}{2} : sin (\frac{π}{4} + x)

\frac{2 sin ( \frac{π}{4} + x )}{2}

Eliminar los terminos comunes:

2

= sin (\frac{π}{4} + x)

Simplificar

\frac{\frac{1}{2}}{2} : \frac{2}{4}

\frac{\frac{1}{2}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{1}{2 2}

Racionalizar

\frac{1}{2 2} : \frac{2}{4}

\frac{1}{2 2}

Multiplicar por el conjugado

\frac{2}{2}

= \frac{1 \cdot 2}{2 2 2}

1 \cdot 2 = 2

222 = 4

222

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Sumar elementos similares:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiplicar fracciones:

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

= \frac{1 \cdot 2}{2}

Eliminar los terminos comunes:

2

= 1

= 2^{1 + 1}

Sumar:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= \frac{2}{4}

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

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

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

Para

sin (x) \geq a

, si

- 1 < a < 1

entonces

arcsin (a) + 2 πn \leq x \leq π - arcsin (a) + 2 πn

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

a \leq u \leq b

entonces

a \leq u and u \leq b

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

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

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

Intercambiar lados

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

Mover

\frac{π}{4}

al lado derecho

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

Restar

\frac{π}{4}

de ambos lados

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

Simplificar

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

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

Simplificar

arcsin (\frac{2}{4}) - \frac{π}{4} : \frac{4 arcsin ( \frac{2}{4} ) - π}{4}

arcsin (\frac{2}{4}) - \frac{π}{4}

Convertir a fracción:

arcsin (\frac{2}{4}) = \frac{arcsin ( \frac{2}{4} ) 4}{4}

= \frac{arcsin ( \frac{2}{4} ) \cdot 4}{4} - \frac{π}{4}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{arcsin ( \frac{2}{4} ) \cdot 4 - π}{4}

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

\frac{π}{4} + x \leq π - arcsin (\frac{2}{4}) + 2 πn : x \leq \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4} + 2 πn

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

Mover

\frac{π}{4}

al lado derecho

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

Restar

\frac{π}{4}

de ambos lados

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

Simplificar

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

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

Simplificar

π - arcsin (\frac{2}{4}) - \frac{π}{4} : \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4}

π - arcsin (\frac{2}{4}) - \frac{π}{4}

Convertir a fracción:

π = \frac{π 4}{4}, arcsin (\frac{2}{4}) = \frac{arcsin ( \frac{2}{4} ) 4}{4}

= \frac{π 4}{4} - \frac{arcsin ( \frac{2}{4} ) \cdot 4}{4} - \frac{π}{4}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{π 4 - arcsin ( \frac{2}{4} ) \cdot 4 - π}{4}

π 4 - arcsin (\frac{2}{4}) \cdot 4 - π = 3 π - 4 arcsin (\frac{2}{4})

π 4 - arcsin (\frac{2}{4}) \cdot 4 - π

Agrupar términos semejantes

= 4 π - π - 4 arcsin (\frac{2}{4})

Sumar elementos similares:

4 π - π = 3 π

= 3 π - 4 arcsin (\frac{2}{4})

= \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4}

x \leq \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4} + 2 πn

Combinar los rangos

x \geq \frac{4 arcsin ( \frac{2}{4} ) - π}{4} + 2 πn and x \leq \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4} + 2 πn

Mezclar intervalos sobrepuestos

\frac{4 arcsin ( \frac{2}{4} ) - π}{4} + 2 πn \leq x \leq \frac{3 π - 4 arcsin ( \frac{2}{4} )}{4} + 2 πn

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