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-7.5sin(pi/6 x)+10.5>13.75

Solución

- 7.5 sin (\frac{π}{6} x) + 10.5 > 13.75

Solución

\frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n < x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Notación de intervalos

(\frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n, - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n)

Notación decimal

- 5.14402 \dots + 12 n < x < - 0.85597 \dots + 12 n

Pasos de solución

- 7.5 sin (\frac{π}{6} x) + 10.5 > 13.75

Multiplicar ambos lados por

100

- 7.5 sin (\frac{π}{6} x) + 10.5 > 13.75

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 7.5 sin (\frac{π}{6} x) \cdot 100 + 10.5 \cdot 100 > 13.75 \cdot 100

Simplificar

- 750 sin (\frac{π}{6} x) + 1050 > 1375

- 750 sin (\frac{π}{6} x) + 1050 > 1375

Mover

1050

al lado derecho

- 750 sin (\frac{π}{6} x) + 1050 > 1375

Restar

1050

de ambos lados

- 750 sin (\frac{π}{6} x) + 1050 - 1050 > 1375 - 1050

Simplificar

- 750 sin (\frac{π}{6} x) > 325

- 750 sin (\frac{π}{6} x) > 325

Multiplicar ambos lados por

- 1

- 750 sin (\frac{π}{6} x) > 325

Multiplicar ambos lados por -1 (invierte la desigualdad)

(- 750 sin (\frac{π}{6} x)) (- 1) < 325 (- 1)

Simplificar

750 sin (\frac{π}{6} x) < - 325

750 sin (\frac{π}{6} x) < - 325

Dividir ambos lados entre

750

750 sin (\frac{π}{6} x) < - 325

Dividir ambos lados entre

750

\frac{750 sin ( \frac{π}{6} x )}{750} < \frac{- 325}{750}

Simplificar

\frac{750 sin ( \frac{π}{6} x )}{750} < \frac{- 325}{750}

Simplificar

\frac{750 sin ( \frac{π}{6} x )}{750} : sin (\frac{π}{6} x)

\frac{750 sin ( \frac{π}{6} x )}{750}

Dividir:

\frac{750}{750} = 1

= sin (\frac{π}{6} x)

Simplificar

\frac{- 325}{750} : - \frac{13}{30}

\frac{- 325}{750}

Aplicar las propiedades de las fracciones:

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

= - \frac{325}{750}

Eliminar los terminos comunes:

25

= - \frac{13}{30}

sin (\frac{π}{6} x) < - \frac{13}{30}

sin (\frac{π}{6} x) < - \frac{13}{30}

sin (\frac{π}{6} x) < - \frac{13}{30}

Para

sin (x) < a

, si

- 1 < a \leq 1

entonces

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn

a < u < b

entonces

a < u and u < b

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x and \frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x : x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x

Intercambiar lados

\frac{π}{6} x > - π - arcsin (- \frac{13}{30}) + 2 πn

Simplificar

- π - arcsin (- \frac{13}{30}) + 2 πn : - π + arcsin (\frac{13}{30}) + 2 πn

- π - arcsin (- \frac{13}{30}) + 2 πn

Utilizar la siguiente propiedad:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{13}{30}) = - arcsin (\frac{13}{30})

= - π - (- arcsin (\frac{13}{30})) + 2 πn

Aplicar la regla

- (- a) = a

= - π + arcsin (\frac{13}{30}) + 2 πn

\frac{π}{6} x > - π + arcsin (\frac{13}{30}) + 2 πn

Multiplicar ambos lados por

6

\frac{π}{6} x > - π + arcsin (\frac{13}{30}) + 2 πn

Multiplicar ambos lados por

6

6 \cdot \frac{π}{6} x > - 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} x > - 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiplicar fracciones:

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

= \frac{6 π}{6} x

Eliminar los terminos comunes:

6

= x π

Simplificar

- 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn : - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

- 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Multiplicar los numeros:

6 \cdot 2 = 12

= - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

Dividir ambos lados entre

π

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

Dividir ambos lados entre

π

\frac{π x}{π} > - \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplificar

\frac{π x}{π} > - \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplificar

\frac{π x}{π} : x

\frac{π x}{π}

Eliminar los terminos comunes:

π

= x

Simplificar

- \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π} : - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

- \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Cancelar

\frac{6 π}{π} : 6

\frac{6 π}{π}

Eliminar los terminos comunes:

π

= 6

= - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Cancelar

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Eliminar los terminos comunes:

π

= 12 n

= - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

x > - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

x > - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Simplificar

- 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} : \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π}

- 6 + \frac{6 arcsin ( \frac{13}{30} )}{π}

Convertir a fracción:

6 = \frac{6 π}{π}

= - \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π}

x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n

x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n

\frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn : x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

\frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn

Simplificar

arcsin (- \frac{13}{30}) + 2 πn : - arcsin (\frac{13}{30}) + 2 πn

arcsin (- \frac{13}{30}) + 2 πn

Utilizar la siguiente propiedad:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{13}{30}) = - arcsin (\frac{13}{30})

= - arcsin (\frac{13}{30}) + 2 πn

\frac{π}{6} x < - arcsin (\frac{13}{30}) + 2 πn

Multiplicar ambos lados por

6

\frac{π}{6} x < - arcsin (\frac{13}{30}) + 2 πn

Multiplicar ambos lados por

6

6 \cdot \frac{π}{6} x < - 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} x < - 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiplicar fracciones:

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

= \frac{6 π}{6} x

Eliminar los terminos comunes:

6

= x π

Simplificar

- 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn : - 6 arcsin (\frac{13}{30}) + 12 πn

- 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Multiplicar los numeros:

6 \cdot 2 = 12

= - 6 arcsin (\frac{13}{30}) + 12 πn

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

Dividir ambos lados entre

π

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

Dividir ambos lados entre

π

\frac{π x}{π} < - \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplificar

x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Combinar los rangos

x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n and x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Mezclar intervalos sobrepuestos

\frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n < x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

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