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Popular Trigonometria >

[sin(4x)cos(x)-cos(4x)sin(x)]^2=1

Solução

[sin (4 x) cos (x) - cos (4 x) sin (x)]^{2} = 1

Solução

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

Graus

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

Passos da solução

[sin (4 x) cos (x) - cos (4 x) sin (x)]^{2} = 1

Subtrair

1

de ambos os lados

(sin (4 x) cos (x) - cos (4 x) sin (x))^{2} - 1 = 0

Fatorar

(sin (4 x) cos (x) - cos (4 x) sin (x))^{2} - 1 : (sin (4 x) cos (x) - cos (4 x) sin (x) + 1) (sin (4 x) cos (x) - cos (4 x) sin (x) - 1)

(sin (4 x) cos (x) - cos (4 x) sin (x))^{2} - 1

Reescrever

1

como

1^{2}

= (sin (4 x) cos (x) - cos (4 x) sin (x))^{2} - 1^{2}

Aplicar a regra da diferença de quadrados:

x^{2} - y^{2} = (x + y) (x - y)

(sin (4 x) cos (x) - cos (4 x) sin (x))^{2} - 1^{2} = ((sin (4 x) cos (x) - cos (4 x) sin (x)) + 1) ((sin (4 x) cos (x) - cos (4 x) sin (x)) - 1)

= ((sin (4 x) cos (x) - cos (4 x) sin (x)) + 1) ((sin (4 x) cos (x) - cos (4 x) sin (x)) - 1)

Simplificar

= (sin (4 x) cos (x) - cos (4 x) sin (x) + 1) (sin (4 x) cos (x) - cos (4 x) sin (x) - 1)

(sin (4 x) cos (x) - cos (4 x) sin (x) + 1) (sin (4 x) cos (x) - cos (4 x) sin (x) - 1) = 0

Resolver cada parte separadamente

sin (4 x) cos (x) - cos (4 x) sin (x) + 1 = 0 or sin (4 x) cos (x) - cos (4 x) sin (x) - 1 = 0

sin (4 x) cos (x) - cos (4 x) sin (x) + 1 = 0 : x = \frac{π}{2} + \frac{2 πn}{3}

sin (4 x) cos (x) - cos (4 x) sin (x) + 1 = 0

Reeecreva usando identidades trigonométricas

sin (4 x) cos (x) - cos (4 x) sin (x) + 1

Use a identidade de diferença de ângulos:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= 1 + sin (4 x - x)

1 + sin (4 x - x) = 0

Mova

1

para o lado direito

1 + sin (4 x - x) = 0

Subtrair

1

de ambos os lados

1 + sin (4 x - x) - 1 = 0 - 1

Simplificar

sin (4 x - x) = - 1

sin (4 x - x) = - 1

Soluções gerais para

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

4 x - x = \frac{3 π}{2} + 2 πn

Resolver

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

4 x - x = \frac{3 π}{2} + 2 πn

Somar elementos similares:

4 x - x = 3 x

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}{2} + \frac{2 πn}{3}

sin (4 x) cos (x) - cos (4 x) sin (x) - 1 = 0 : x = \frac{π}{6} + \frac{2 πn}{3}

sin (4 x) cos (x) - cos (4 x) sin (x) - 1 = 0

Reeecreva usando identidades trigonométricas

sin (4 x) cos (x) - cos (4 x) sin (x) - 1

Use a identidade de diferença de ângulos:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= - 1 + sin (4 x - x)

- 1 + sin (4 x - x) = 0

Mova

1

para o lado direito

- 1 + sin (4 x - x) = 0

Adicionar

1

a ambos os lados

- 1 + sin (4 x - x) + 1 = 0 + 1

Simplificar

sin (4 x - x) = 1

sin (4 x - x) = 1

Soluções gerais para

sin (4 x - 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 - x = \frac{π}{2} + 2 πn

4 x - x = \frac{π}{2} + 2 πn

Resolver

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

4 x - x = \frac{π}{2} + 2 πn

Somar elementos similares:

4 x - x = 3 x

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}

x = \frac{π}{6} + \frac{2 πn}{3}

Combinar toda as soluções

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

Gráfico

Exemplos populares

cos(θ)=(202)/(sqrt(331)*2\sqrt{33)}

cos (θ) = \frac{202}{331 \cdot 2 33}

5sin(θ)+12cos(θ)=13

5 sin (θ) + 12 cos (θ) = 13

2-2cos^2(x)+3cos(x)=0

2 - 2 cos^{2} (x) + 3 cos (x) = 0

cot(x)= 5/12 ,pi<x<(3pi)/2

cot (x) = \frac{5}{12}, π < x < \frac{3 π}{2}

tan(x)= 1/5 ,tan(x+pi/2)

tan (x) = \frac{1}{5}, tan (x + \frac{π}{2})