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Popular Trigonometria >

105=100+30sin((12pi)/5 t)

Solução

105 = 100 + 30 sin (\frac{12 π}{5} t)

Solução

t = \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}, t = \frac{5}{12} - \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}

Graus

t = 1.27245 \dots^{\circ} + 47.74648 \dots^{\circ} n, t = 22.60078 \dots^{\circ} + 47.74648 \dots^{\circ} n

Passos da solução

105 = 100 + 30 sin (\frac{12 π}{5} t)

Trocar lados

100 + 30 sin (\frac{12 π}{5} t) = 105

Mova

100

para o lado direito

100 + 30 sin (\frac{12 π}{5} t) = 105

Subtrair

100

de ambos os lados

100 + 30 sin (\frac{12 π}{5} t) - 100 = 105 - 100

Simplificar

30 sin (\frac{12 π}{5} t) = 5

30 sin (\frac{12 π}{5} t) = 5

Dividir ambos os lados por

30

30 sin (\frac{12 π}{5} t) = 5

Dividir ambos os lados por

30

\frac{30 sin ( \frac{12 π}{5} t )}{30} = \frac{5}{30}

Simplificar

sin (\frac{12 π}{5} t) = \frac{1}{6}

sin (\frac{12 π}{5} t) = \frac{1}{6}

Aplique as propriedades trigonométricas inversas

sin (\frac{12 π}{5} t) = \frac{1}{6}

Soluções gerais para

sin (\frac{12 π}{5} t) = \frac{1}{6}

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

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn, \frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn, \frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

Resolver

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn : t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn

Multiplicar ambos os lados por

5

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn

Multiplicar ambos os lados por

5

5 \cdot \frac{12 π}{5} t = 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplificar

5 \cdot \frac{12 π}{5} t = 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplificar

5 \cdot \frac{12 π}{5} t : 12 π t

5 \cdot \frac{12 π}{5} t

Multiplicar frações:

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

= \frac{12 \cdot 5 π}{5} t

Eliminar o fator comum:

5

= t \cdot 12 π

Simplificar

5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn : 5 arcsin (\frac{1}{6}) + 10 πn

5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Multiplicar os números:

5 \cdot 2 = 10

= 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

Dividir ambos os lados por

12 π

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

Dividir ambos os lados por

12 π

\frac{12 π t}{12 π} = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplificar

\frac{12 π t}{12 π} = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplificar

\frac{12 π t}{12 π} : t

\frac{12 π t}{12 π}

Dividir:

\frac{12}{12} = 1

= \frac{π t}{π}

Eliminar o fator comum:

π

= t

Simplificar

\frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π} : \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Cancelar

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Cancelar

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Eliminar o fator comum:

2

= \frac{5 πn}{6 π}

Eliminar o fator comum:

π

= \frac{5 n}{6}

= \frac{5 n}{6}

= \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

Resolver

\frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn : t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

Multiplicar ambos os lados por

5

\frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

Multiplicar ambos os lados por

5

5 \cdot \frac{12 π}{5} t = 5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplificar

5 \cdot \frac{12 π}{5} t = 5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplificar

5 \cdot \frac{12 π}{5} t : 12 π t

5 \cdot \frac{12 π}{5} t

Multiplicar frações:

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

= \frac{12 \cdot 5 π}{5} t

Eliminar o fator comum:

5

= t \cdot 12 π

Simplificar

5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn : 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Multiplicar os números:

5 \cdot 2 = 10

= 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

Dividir ambos os lados por

12 π

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

Dividir ambos os lados por

12 π

\frac{12 π t}{12 π} = \frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplificar

\frac{12 π t}{12 π} = \frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplificar

\frac{12 π t}{12 π} : t

\frac{12 π t}{12 π}

Dividir:

\frac{12}{12} = 1

= \frac{π t}{π}

Eliminar o fator comum:

π

= t

Simplificar

\frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π} : \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Cancelar

\frac{5 π}{12 π} : \frac{5}{12}

\frac{5 π}{12 π}

Eliminar o fator comum:

π

= \frac{5}{12}

= \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Cancelar

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Cancelar

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Eliminar o fator comum:

2

= \frac{5 πn}{6 π}

Eliminar o fator comum:

π

= \frac{5 n}{6}

= \frac{5 n}{6}

= \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}, t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

Mostrar soluções na forma decimal

t = \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}, t = \frac{5}{12} - \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}

Gráfico

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