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Popular Trigonometria >

sin(3x)= 1/(sqrt(2)),0<= x<= 2pi

Solução

sin (3 x) = \frac{1}{2}, 0 \leq x \leq 2 π

Solução

x = \frac{π}{12}, x = \frac{π}{4}, x = \frac{3 π}{4}, x = \frac{11 π}{12}, x = \frac{17 π}{12}, x = \frac{19 π}{12}

Passos da solução

sin (3 x) = \frac{1}{2}, 0 \leq x \leq 2 π

Simplificar

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

\frac{1}{2}

Multiplicar pelo conjugado

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (3 x) = \frac{2}{2}

Soluções gerais para

sin (3 x) = \frac{2}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

3 x = \frac{π}{4} + 2 πn, 3 x = \frac{3 π}{4} + 2 πn

3 x = \frac{π}{4} + 2 πn, 3 x = \frac{3 π}{4} + 2 πn

Resolver

3 x = \frac{π}{4} + 2 πn : x = \frac{π}{12} + \frac{2 πn}{3}

3 x = \frac{π}{4} + 2 πn

Dividir ambos os lados por

3

3 x = \frac{π}{4} + 2 πn

Dividir ambos os lados por

3

\frac{3 x}{3} = \frac{\frac{π}{4}}{3} + \frac{2 πn}{3}

Simplificar

\frac{3 x}{3} = \frac{\frac{π}{4}}{3} + \frac{2 πn}{3}

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{\frac{π}{4}}{3} + \frac{2 πn}{3} : \frac{π}{12} + \frac{2 πn}{3}

\frac{\frac{π}{4}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

Aplicar as propriedades das frações:

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

= \frac{π}{4 \cdot 3}

Multiplicar os números:

4 \cdot 3 = 12

= \frac{π}{12}

= \frac{π}{12} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}

Resolver

3 x = \frac{3 π}{4} + 2 πn : x = \frac{π}{4} + \frac{2 πn}{3}

3 x = \frac{3 π}{4} + 2 πn

Dividir ambos os lados por

3

3 x = \frac{3 π}{4} + 2 πn

Dividir ambos os lados por

3

\frac{3 x}{3} = \frac{\frac{3 π}{4}}{3} + \frac{2 πn}{3}

Simplificar

\frac{3 x}{3} = \frac{\frac{3 π}{4}}{3} + \frac{2 πn}{3}

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{\frac{3 π}{4}}{3} + \frac{2 πn}{3} : \frac{π}{4} + \frac{2 πn}{3}

\frac{\frac{3 π}{4}}{3} + \frac{2 πn}{3}

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

Aplicar as propriedades das frações:

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

= \frac{3 π}{4 \cdot 3}

Multiplicar os números:

4 \cdot 3 = 12

= \frac{3 π}{12}

Eliminar o fator comum:

3

= \frac{π}{4}

= \frac{π}{4} + \frac{2 πn}{3}

x = \frac{π}{4} + \frac{2 πn}{3}

x = \frac{π}{4} + \frac{2 πn}{3}

x = \frac{π}{4} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}, x = \frac{π}{4} + \frac{2 πn}{3}

Soluções para o intervalo

0 \leq x \leq 2 π

x = \frac{π}{12}, x = \frac{π}{4}, x = \frac{3 π}{4}, x = \frac{11 π}{12}, x = \frac{17 π}{12}, x = \frac{19 π}{12}

Gráfico

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