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Popular >

sin^2(4x)=1

Solução

sin^{2} (4 x) = 1

Solução

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}

Graus

x = 22. 5^{\circ} + 9 0^{\circ} n, x = 67. 5^{\circ} + 9 0^{\circ} n

Passos da solução

sin^{2} (4 x) = 1

Usando o método de substituição

sin^{2} (4 x) = 1

Sea:

sin (4 x) = u

u^{2} = 1

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

Para

x^{2} = f (a)

as soluções são

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Aplicar a regra

1 = 1

= 1

- 1 = - 1

- 1

Aplicar a regra

1 = 1

= - 1

u = 1, u = - 1

Substituir na equação

u = sin (4 x)

sin (4 x) = 1, sin (4 x) = - 1

sin (4 x) = 1, sin (4 x) = - 1

sin (4 x) = 1 : x = \frac{π}{8} + \frac{πn}{2}

sin (4 x) = 1

Soluções gerais para

sin (4 x) = 1

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}

4 x = \frac{π}{2} + 2 πn

4 x = \frac{π}{2} + 2 πn

Resolver

4 x = \frac{π}{2} + 2 πn : x = \frac{π}{8} + \frac{πn}{2}

4 x = \frac{π}{2} + 2 πn

Dividir ambos os lados por

4

4 x = \frac{π}{2} + 2 πn

Dividir ambos os lados por

4

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

Simplificar

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

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

\frac{\frac{π}{2}}{4} + \frac{2 πn}{4} : \frac{π}{8} + \frac{πn}{2}

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

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

Aplicar as propriedades das frações:

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

= \frac{π}{2 \cdot 4}

Multiplicar os números:

2 \cdot 4 = 8

= \frac{π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar o fator comum:

2

= \frac{πn}{2}

= \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

sin (4 x) = - 1 : x = \frac{3 π}{8} + \frac{πn}{2}

sin (4 x) = - 1

Soluções gerais para

sin (4 x) = - 1

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}

4 x = \frac{3 π}{2} + 2 πn

4 x = \frac{3 π}{2} + 2 πn

Resolver

4 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{8} + \frac{πn}{2}

4 x = \frac{3 π}{2} + 2 πn

Dividir ambos os lados por

4

4 x = \frac{3 π}{2} + 2 πn

Dividir ambos os lados por

4

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

Simplificar

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

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

\frac{\frac{3 π}{2}}{4} + \frac{2 πn}{4} : \frac{3 π}{8} + \frac{πn}{2}

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

\frac{\frac{3 π}{2}}{4} = \frac{3 π}{8}

\frac{\frac{3 π}{2}}{4}

Aplicar as propriedades das frações:

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

= \frac{3 π}{2 \cdot 4}

Multiplicar os números:

2 \cdot 4 = 8

= \frac{3 π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar o fator comum:

2

= \frac{πn}{2}

= \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

Combinar toda as soluções

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}

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