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Popular Trigonometria >

sin(4x)-sin(2x)=0

Solução

sin (4 x) - sin (2 x) = 0

Solução

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = 2 πn, x = π + 2 πn

Graus

x = 3 0^{\circ} + 12 0^{\circ} n, x = 9 0^{\circ} + 12 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Passos da solução

sin (4 x) - sin (2 x) = 0

Reeecreva usando identidades trigonométricas

- sin (2 x) + sin (4 x)

Use a identidade da transformação de soma em produto:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 sin (\frac{4 x - 2 x}{2}) cos (\frac{4 x + 2 x}{2})

Simplificar

2 sin (\frac{4 x - 2 x}{2}) cos (\frac{4 x + 2 x}{2}) : 2 sin (x) cos (3 x)

2 sin (\frac{4 x - 2 x}{2}) cos (\frac{4 x + 2 x}{2})

\frac{4 x - 2 x}{2} = x

\frac{4 x - 2 x}{2}

Somar elementos similares:

4 x - 2 x = 2 x

= \frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

= 2 sin (x) cos (\frac{4 x + 2 x}{2})

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

\frac{4 x + 2 x}{2}

Somar elementos similares:

4 x + 2 x = 6 x

= \frac{6 x}{2}

Dividir:

\frac{6}{2} = 3

= 3 x

= 2 sin (x) cos (3 x)

= 2 sin (x) cos (3 x)

2 cos (3 x) sin (x) = 0

Resolver cada parte separadamente

cos (3 x) = 0 or sin (x) = 0

cos (3 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

cos (3 x) = 0

Soluções gerais para

cos (3 x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Resolver

3 x = \frac{π}{2} + 2 πn : x = \frac{π}{6} + \frac{2 πn}{3}

3 x = \frac{π}{2} + 2 πn

Dividir ambos os lados por

3

3 x = \frac{π}{2} + 2 πn

Dividir ambos os lados por

3

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

Simplificar

\frac{3 x}{3} = \frac{\frac{π}{2}}{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}{2}}{3} + \frac{2 πn}{3} : \frac{π}{6} + \frac{2 πn}{3}

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

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Aplicar as propriedades das frações:

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

= \frac{π}{2 \cdot 3}

Multiplicar os números:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

Resolver

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

3 x = \frac{3 π}{2} + 2 πn

Dividir ambos os lados por

3

3 x = \frac{3 π}{2} + 2 πn

Dividir ambos os lados por

3

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

Simplificar

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

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

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

Aplicar as propriedades das frações:

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

= \frac{3 π}{2 \cdot 3}

Multiplicar os números:

2 \cdot 3 = 6

= \frac{3 π}{6}

Eliminar o fator comum:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluções gerais para

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combinar toda as soluções

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = 2 πn, x = π + 2 πn

Gráfico

Exemplos populares

4sin^2(θ)-3=0

4 sin^{2} (θ) - 3 = 0

cos(x)=-1/2

cos (x) = - \frac{1}{2}

csc((7pi)/6)

csc (\frac{7 π}{6})

cos((4pi)/3)

cos (\frac{4 π}{3})

cos(3x)=1

cos (3 x) = 1