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11.75=5sin((pit)/6-(7pi)/6)+8

Solución

11.75 = 5 sin (\frac{π t}{6} - \frac{7 π}{6}) + 8

Solución

t = 7 + 12 n + \frac{6 \cdot 0.84806 \dots}{π}, t = 13 + 12 n - \frac{6 \cdot 0.84806 \dots}{π}

Grados

t = 493.87124 \dots^{\circ} + 687.54935 \dots^{\circ} n, t = 652.04434 \dots^{\circ} + 687.54935 \dots^{\circ} n

Pasos de solución

11.75 = 5 sin (\frac{π t}{6} - \frac{7 π}{6}) + 8

Intercambiar lados

5 sin (\frac{π t}{6} - \frac{7 π}{6}) + 8 = 11.75

Mover

8

al lado derecho

5 sin (\frac{π t}{6} - \frac{7 π}{6}) + 8 = 11.75

Restar

8

de ambos lados

5 sin (\frac{π t}{6} - \frac{7 π}{6}) + 8 - 8 = 11.75 - 8

Simplificar

5 sin (\frac{π t}{6} - \frac{7 π}{6}) = 3.75

5 sin (\frac{π t}{6} - \frac{7 π}{6}) = 3.75

Dividir ambos lados entre

5

5 sin (\frac{π t}{6} - \frac{7 π}{6}) = 3.75

Dividir ambos lados entre

5

\frac{5 sin ( \frac{π t}{6} - \frac{7 π}{6} )}{5} = \frac{3.75}{5}

Simplificar

sin (\frac{π t}{6} - \frac{7 π}{6}) = 0.75

sin (\frac{π t}{6} - \frac{7 π}{6}) = 0.75

Aplicar propiedades trigonométricas inversas

sin (\frac{π t}{6} - \frac{7 π}{6}) = 0.75

Soluciones generales para

sin (\frac{π t}{6} - \frac{7 π}{6}) = 0.75

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

\frac{π t}{6} - \frac{7 π}{6} = arcsin (0.75) + 2 πn, \frac{π t}{6} - \frac{7 π}{6} = π - arcsin (0.75) + 2 πn

\frac{π t}{6} - \frac{7 π}{6} = arcsin (0.75) + 2 πn, \frac{π t}{6} - \frac{7 π}{6} = π - arcsin (0.75) + 2 πn

Resolver

\frac{π t}{6} - \frac{7 π}{6} = arcsin (0.75) + 2 πn : t = 7 + 12 n + \frac{6 arcsin ( 0.75 )}{π}

\frac{π t}{6} - \frac{7 π}{6} = arcsin (0.75) + 2 πn

Mover

\frac{7 π}{6}

al lado derecho

\frac{π t}{6} - \frac{7 π}{6} = arcsin (0.75) + 2 πn

Sumar

\frac{7 π}{6}

a ambos lados

\frac{π t}{6} - \frac{7 π}{6} + \frac{7 π}{6} = arcsin (0.75) + 2 πn + \frac{7 π}{6}

Simplificar

\frac{π t}{6} = arcsin (0.75) + 2 πn + \frac{7 π}{6}

\frac{π t}{6} = arcsin (0.75) + 2 πn + \frac{7 π}{6}

Multiplicar ambos lados por

6

\frac{π t}{6} = arcsin (0.75) + 2 πn + \frac{7 π}{6}

Multiplicar ambos lados por

6

\frac{6 π t}{6} = 6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6}

Simplificar

\frac{6 π t}{6} = 6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6}

Simplificar

\frac{6 π t}{6} : π t

\frac{6 π t}{6}

Dividir:

\frac{6}{6} = 1

= π t

Simplificar

6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6} : 6 arcsin (0.75) + 12 πn + 7 π

6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6}

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiplicar los numeros:

6 \cdot 2 = 12

= 12 πn

6 \cdot \frac{7 π}{6} = 7 π

6 \cdot \frac{7 π}{6}

Multiplicar fracciones:

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

= \frac{7 π 6}{6}

Eliminar los terminos comunes:

6

= 7 π

= 6 arcsin (0.75) + 12 πn + 7 π

π t = 6 arcsin (0.75) + 12 πn + 7 π

π t = 6 arcsin (0.75) + 12 πn + 7 π

π t = 6 arcsin (0.75) + 12 πn + 7 π

Dividir ambos lados entre

π

π t = 6 arcsin (0.75) + 12 πn + 7 π

Dividir ambos lados entre

π

\frac{π t}{π} = \frac{6 arcsin ( 0.75 )}{π} + \frac{12 πn}{π} + \frac{7 π}{π}

Simplificar

\frac{π t}{π} = \frac{6 arcsin ( 0.75 )}{π} + \frac{12 πn}{π} + \frac{7 π}{π}

Simplificar

\frac{π t}{π} : t

\frac{π t}{π}

Eliminar los terminos comunes:

π

= t

Simplificar

\frac{6 arcsin ( 0.75 )}{π} + \frac{12 πn}{π} + \frac{7 π}{π} : 7 + 12 n + \frac{6 arcsin ( 0.75 )}{π}

\frac{6 arcsin ( 0.75 )}{π} + \frac{12 πn}{π} + \frac{7 π}{π}

Agrupar términos semejantes

= \frac{7 π}{π} + \frac{12 πn}{π} + \frac{6 arcsin ( 0.75 )}{π}

Cancelar

\frac{7 π}{π} : 7

\frac{7 π}{π}

Eliminar los terminos comunes:

π

= 7

= 7 + \frac{12 πn}{π} + \frac{6 arcsin ( 0.75 )}{π}

Cancelar

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Eliminar los terminos comunes:

π

= 12 n

= 7 + 12 n + \frac{6 arcsin ( 0.75 )}{π}

t = 7 + 12 n + \frac{6 arcsin ( 0.75 )}{π}

t = 7 + 12 n + \frac{6 arcsin ( 0.75 )}{π}

t = 7 + 12 n + \frac{6 arcsin ( 0.75 )}{π}

Resolver

\frac{π t}{6} - \frac{7 π}{6} = π - arcsin (0.75) + 2 πn : t = 13 + 12 n - \frac{6 arcsin ( 0.75 )}{π}

\frac{π t}{6} - \frac{7 π}{6} = π - arcsin (0.75) + 2 πn

Mover

\frac{7 π}{6}

al lado derecho

\frac{π t}{6} - \frac{7 π}{6} = π - arcsin (0.75) + 2 πn

Sumar

\frac{7 π}{6}

a ambos lados

\frac{π t}{6} - \frac{7 π}{6} + \frac{7 π}{6} = π - arcsin (0.75) + 2 πn + \frac{7 π}{6}

Simplificar

\frac{π t}{6} = π - arcsin (0.75) + 2 πn + \frac{7 π}{6}

\frac{π t}{6} = π - arcsin (0.75) + 2 πn + \frac{7 π}{6}

Multiplicar ambos lados por

6

\frac{π t}{6} = π - arcsin (0.75) + 2 πn + \frac{7 π}{6}

Multiplicar ambos lados por

6

\frac{6 π t}{6} = 6 π - 6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6}

Simplificar

\frac{6 π t}{6} = 6 π - 6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6}

Simplificar

\frac{6 π t}{6} : π t

\frac{6 π t}{6}

Dividir:

\frac{6}{6} = 1

= π t

Simplificar

6 π - 6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6} : 13 π + 12 πn - 6 arcsin (0.75)

6 π - 6 arcsin (0.75) + 6 \cdot 2 πn + 6 \cdot \frac{7 π}{6}

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiplicar los numeros:

6 \cdot 2 = 12

= 12 πn

6 \cdot \frac{7 π}{6} = 7 π

6 \cdot \frac{7 π}{6}

Multiplicar fracciones:

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

= \frac{7 π 6}{6}

Eliminar los terminos comunes:

6

= 7 π

= 6 π - 6 arcsin (0.75) + 12 πn + 7 π

Agrupar términos semejantes

= 6 π + 7 π + 12 πn - 6 arcsin (0.75)

Sumar elementos similares:

6 π + 7 π = 13 π

= 13 π + 12 πn - 6 arcsin (0.75)

π t = 13 π + 12 πn - 6 arcsin (0.75)

π t = 13 π + 12 πn - 6 arcsin (0.75)

π t = 13 π + 12 πn - 6 arcsin (0.75)

Dividir ambos lados entre

π

π t = 13 π + 12 πn - 6 arcsin (0.75)

Dividir ambos lados entre

π

\frac{π t}{π} = \frac{13 π}{π} + \frac{12 πn}{π} - \frac{6 arcsin ( 0.75 )}{π}

Simplificar

\frac{π t}{π} = \frac{13 π}{π} + \frac{12 πn}{π} - \frac{6 arcsin ( 0.75 )}{π}

Simplificar

\frac{π t}{π} : t

\frac{π t}{π}

Eliminar los terminos comunes:

π

= t

Simplificar

\frac{13 π}{π} + \frac{12 πn}{π} - \frac{6 arcsin ( 0.75 )}{π} : 13 + 12 n - \frac{6 arcsin ( 0.75 )}{π}

\frac{13 π}{π} + \frac{12 πn}{π} - \frac{6 arcsin ( 0.75 )}{π}

Cancelar

\frac{13 π}{π} : 13

\frac{13 π}{π}

Eliminar los terminos comunes:

π

= 13

= 13 + \frac{12 πn}{π} - \frac{6 arcsin ( 0.75 )}{π}

Cancelar

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Eliminar los terminos comunes:

π

= 12 n

= 13 + 12 n - \frac{6 arcsin ( 0.75 )}{π}

t = 13 + 12 n - \frac{6 arcsin ( 0.75 )}{π}

t = 13 + 12 n - \frac{6 arcsin ( 0.75 )}{π}

t = 13 + 12 n - \frac{6 arcsin ( 0.75 )}{π}

t = 7 + 12 n + \frac{6 arcsin ( 0.75 )}{π}, t = 13 + 12 n - \frac{6 arcsin ( 0.75 )}{π}

Mostrar soluciones en forma decimal

t = 7 + 12 n + \frac{6 \cdot 0.84806 \dots}{π}, t = 13 + 12 n - \frac{6 \cdot 0.84806 \dots}{π}

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cot (a) = \frac{7}{51}

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sin (x) = \frac{39}{44}

arctan(0.5)-arctan(1/3)=arctan(x)

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