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0<82.5-67.5cos(pi/6 t)<20

Solución

0 < 82.5 - 67.5 cos (\frac{π}{6} t) < 20

Solución

- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Notación de intervalos

(- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n, \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n)

Notación decimal

- 0.73972 \dots + 12 n < t < 0.73972 \dots + 12 n

Pasos de solución

0 < 82.5 - 67.5 cos (\frac{π}{6} t) < 20

a < u < b

entonces

a < u and u < b

0 < 82.5 - 67.5 cos (\frac{π}{6} t) and 82.5 - 67.5 cos (\frac{π}{6} t) < 20

0 < 82.5 - 67.5 cos (\frac{π}{6} t) :

Verdadero para todo

t \in R

0 < 82.5 - 67.5 cos (\frac{π}{6} t)

Intercambiar lados

82.5 - 67.5 cos (\frac{π}{6} t) > 0

Multiplicar ambos lados por

10

82.5 - 67.5 cos (\frac{π}{6} t) > 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

82.5 \cdot 10 - 67.5 cos (\frac{π}{6} t) \cdot 10 > 0 \cdot 10

Simplificar

825 - 675 cos (\frac{π}{6} t) > 0

825 - 675 cos (\frac{π}{6} t) > 0

Mover

825

al lado derecho

825 - 675 cos (\frac{π}{6} t) > 0

Restar

825

de ambos lados

825 - 675 cos (\frac{π}{6} t) - 825 > 0 - 825

Simplificar

- 675 cos (\frac{π}{6} t) > - 825

- 675 cos (\frac{π}{6} t) > - 825

Multiplicar ambos lados por

- 1

- 675 cos (\frac{π}{6} t) > - 825

Multiplicar ambos lados por -1 (invierte la desigualdad)

(- 675 cos (\frac{π}{6} t)) (- 1) < (- 825) (- 1)

Simplificar

675 cos (\frac{π}{6} t) < 825

675 cos (\frac{π}{6} t) < 825

Dividir ambos lados entre

675

675 cos (\frac{π}{6} t) < 825

Dividir ambos lados entre

675

\frac{675 cos ( \frac{π}{6} t )}{675} < \frac{825}{675}

Simplificar

cos (\frac{π}{6} t) < \frac{11}{9}

cos (\frac{π}{6} t) < \frac{11}{9}

Rango de

cos (\frac{π}{6} t) : - 1 \leq cos (\frac{π}{6} t) \leq 1

Definición de rango de función

El rango de la función basica

cos

- 1 \leq cos (\frac{π}{6} t) \leq 1

- 1 \leq cos (\frac{π}{6} t) \leq 1

cos (\frac{π}{6} t) < \frac{11}{9} and - 1 \leq cos (\frac{π}{6} t) \leq 1 : - 1 \leq cos (\frac{π}{6} t) \leq 1

Sea

y

cos (\frac{π}{6} t)

Combinar los rangos

y < \frac{11}{9} and - 1 \leq y \leq 1

Mezclar intervalos sobrepuestos

y < \frac{11}{9} and - 1 \leq y \leq 1

La intersección de dos intervalos, es el conjunto de numérico común a los dos intervalos

y < \frac{11}{9}

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

Verdadero para todo t

Verdadero para todo t \in R

82.5 - 67.5 cos (\frac{π}{6} t) < 20 : - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

82.5 - 67.5 cos (\frac{π}{6} t) < 20

Multiplicar ambos lados por

10

82.5 - 67.5 cos (\frac{π}{6} t) < 20

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

82.5 \cdot 10 - 67.5 cos (\frac{π}{6} t) \cdot 10 < 20 \cdot 10

Simplificar

825 - 675 cos (\frac{π}{6} t) < 200

825 - 675 cos (\frac{π}{6} t) < 200

Mover

825

al lado derecho

825 - 675 cos (\frac{π}{6} t) < 200

Restar

825

de ambos lados

825 - 675 cos (\frac{π}{6} t) - 825 < 200 - 825

Simplificar

- 675 cos (\frac{π}{6} t) < - 625

- 675 cos (\frac{π}{6} t) < - 625

Multiplicar ambos lados por

- 1

- 675 cos (\frac{π}{6} t) < - 625

Multiplicar ambos lados por -1 (invierte la desigualdad)

(- 675 cos (\frac{π}{6} t)) (- 1) > (- 625) (- 1)

Simplificar

675 cos (\frac{π}{6} t) > 625

675 cos (\frac{π}{6} t) > 625

Dividir ambos lados entre

675

675 cos (\frac{π}{6} t) > 625

Dividir ambos lados entre

675

\frac{675 cos ( \frac{π}{6} t )}{675} > \frac{625}{675}

Simplificar

cos (\frac{π}{6} t) > \frac{25}{27}

cos (\frac{π}{6} t) > \frac{25}{27}

Para

cos (x) > a

, si

- 1 \leq a < 1

entonces

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

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

a < u < b

entonces

a < u and u < b

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t and \frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t : t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t

Intercambiar lados

\frac{π}{6} t > - arccos (\frac{25}{27}) + 2 πn

Multiplicar ambos lados por

6

\frac{π}{6} t > - arccos (\frac{25}{27}) + 2 πn

Multiplicar ambos lados por

6

6 \cdot \frac{π}{6} t > - 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} t > - 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} t : π t

6 \cdot \frac{π}{6} t

Multiplicar fracciones:

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

= \frac{6 π}{6} t

Eliminar los terminos comunes:

6

= t π

Simplificar

- 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn : - 6 arccos (\frac{25}{27}) + 12 πn

- 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Multiplicar los numeros:

6 \cdot 2 = 12

= - 6 arccos (\frac{25}{27}) + 12 πn

π t > - 6 arccos (\frac{25}{27}) + 12 πn

π t > - 6 arccos (\frac{25}{27}) + 12 πn

π t > - 6 arccos (\frac{25}{27}) + 12 πn

Dividir ambos lados entre

π

π t > - 6 arccos (\frac{25}{27}) + 12 πn

Dividir ambos lados entre

π

\frac{π t}{π} > - \frac{6 arccos ( \frac{25}{27} )}{π} + \frac{12 πn}{π}

Simplificar

t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

\frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn : t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

\frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

Multiplicar ambos lados por

6

\frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

Multiplicar ambos lados por

6

6 \cdot \frac{π}{6} t < 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} t < 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplificar

6 \cdot \frac{π}{6} t : π t

6 \cdot \frac{π}{6} t

Multiplicar fracciones:

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

= \frac{6 π}{6} t

Eliminar los terminos comunes:

6

= t π

Simplificar

6 arccos (\frac{25}{27}) + 6 \cdot 2 πn : 6 arccos (\frac{25}{27}) + 12 πn

6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Multiplicar los numeros:

6 \cdot 2 = 12

= 6 arccos (\frac{25}{27}) + 12 πn

π t < 6 arccos (\frac{25}{27}) + 12 πn

π t < 6 arccos (\frac{25}{27}) + 12 πn

π t < 6 arccos (\frac{25}{27}) + 12 πn

Dividir ambos lados entre

π

π t < 6 arccos (\frac{25}{27}) + 12 πn

Dividir ambos lados entre

π

\frac{π t}{π} < \frac{6 arccos ( \frac{25}{27} )}{π} + \frac{12 πn}{π}

Simplificar

t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Combinar los rangos

t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n and t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Mezclar intervalos sobrepuestos

- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Combinar los rangos

Verdadero para todo t \in R and - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Mezclar intervalos sobrepuestos

- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Ejemplos populares

csc(x)=4\land cot(x)<0

csc (x) = 4 and cot (x) < 0

cos(x^2)0<x<sqrt(x)

cos (x^{2}) 0 < x < x

sin(x)=-4/5 \land cos(x)<0,sin(2x)

sin (x) = - \frac{4}{5} and cos (x) < 0, sin (2 x)

cos(θ)= 3/7 \land sin(θ)>0

cos (θ) = \frac{3}{7} and sin (θ) > 0

cosh(θ)= 26/7 \land θ<0,sinh(θ)

cosh (θ) = \frac{26}{7} and θ < 0, sinh (θ)