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Popular Trigonometria >

sin(8x)+sin(4x)-0.5sin(6x)=0

Solução

sin (8 x) + sin (4 x) - 0.5 sin (6 x) = 0

Solução

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}, x = \frac{1.31811 \dots}{2} + πn, x = π - \frac{1.31811 \dots}{2} + πn

Graus

x = 0^{\circ} + 6 0^{\circ} n, x = 3 0^{\circ} + 6 0^{\circ} n, x = 37.76124 \dots^{\circ} + 18 0^{\circ} n, x = 142.23875 \dots^{\circ} + 18 0^{\circ} n

Passos da solução

sin (8 x) + sin (4 x) - 0.5 sin (6 x) = 0

Reeecreva usando identidades trigonométricas

sin (4 x) + sin (8 x) - 0.5 sin (6 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})

= - 0.5 sin (6 x) + 2 sin (\frac{4 x + 8 x}{2}) cos (\frac{4 x - 8 x}{2})

2 sin (\frac{4 x + 8 x}{2}) cos (\frac{4 x - 8 x}{2}) = 2 cos (2 x) sin (6 x)

2 sin (\frac{4 x + 8 x}{2}) cos (\frac{4 x - 8 x}{2})

\frac{4 x + 8 x}{2} = 6 x

\frac{4 x + 8 x}{2}

Somar elementos similares:

4 x + 8 x = 12 x

= \frac{12 x}{2}

Dividir:

\frac{12}{2} = 6

= 6 x

= 2 sin (6 x) cos (\frac{4 x - 8 x}{2})

\frac{4 x - 8 x}{2} = - 2 x

\frac{4 x - 8 x}{2}

Somar elementos similares:

4 x - 8 x = - 4 x

= \frac{- 4 x}{2}

Aplicar as propriedades das frações:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4 x}{2}

Dividir:

\frac{4}{2} = 2

= - 2 x

= 2 sin (6 x) cos (- 2 x)

Use a identidade de ângulo negativo:

cos (- x) = cos (x)

= 2 cos (2 x) sin (6 x)

= - 0.5 sin (6 x) + 2 cos (2 x) sin (6 x)

- 0.5 sin (6 x) + 2 cos (2 x) sin (6 x) = 0

Fatorar

- 0.5 sin (6 x) + 2 cos (2 x) sin (6 x) : sin (6 x) (2 cos (2 x) - 0.5)

- 0.5 sin (6 x) + 2 cos (2 x) sin (6 x)

Fatorar o termo comum

sin (6 x)

= sin (6 x) (- 0.5 + 2 cos (2 x))

sin (6 x) (2 cos (2 x) - 0.5) = 0

Resolver cada parte separadamente

sin (6 x) = 0 or 2 cos (2 x) - 0.5 = 0

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

sin (6 x) = 0

Soluções gerais para

sin (6 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}

6 x = 0 + 2 πn, 6 x = π + 2 πn

6 x = 0 + 2 πn, 6 x = π + 2 πn

Resolver

6 x = 0 + 2 πn : x = \frac{πn}{3}

6 x = 0 + 2 πn

0 + 2 πn = 2 πn

6 x = 2 πn

Dividir ambos os lados por

6

6 x = 2 πn

Dividir ambos os lados por

6

\frac{6 x}{6} = \frac{2 πn}{6}

Simplificar

x = \frac{πn}{3}

x = \frac{πn}{3}

Resolver

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

6 x = π + 2 πn

Dividir ambos os lados por

6

6 x = π + 2 πn

Dividir ambos os lados por

6

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

Simplificar

x = \frac{π}{6} + \frac{πn}{3}

x = \frac{π}{6} + \frac{πn}{3}

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

2 cos (2 x) - 0.5 = 0 : x = \frac{arccos ( 0.25 )}{2} + πn, x = π - \frac{arccos ( 0.25 )}{2} + πn

2 cos (2 x) - 0.5 = 0

Mova

0.5

para o lado direito

2 cos (2 x) - 0.5 = 0

Adicionar

0.5

a ambos os lados

2 cos (2 x) - 0.5 + 0.5 = 0 + 0.5

Simplificar

2 cos (2 x) = 0.5

2 cos (2 x) = 0.5

Dividir ambos os lados por

2

2 cos (2 x) = 0.5

Dividir ambos os lados por

2

\frac{2 cos ( 2 x )}{2} = \frac{0.5}{2}

Simplificar

cos (2 x) = 0.25

cos (2 x) = 0.25

Aplique as propriedades trigonométricas inversas

cos (2 x) = 0.25

Soluções gerais para

cos (2 x) = 0.25

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 x = arccos (0.25) + 2 πn, 2 x = 2 π - arccos (0.25) + 2 πn

2 x = arccos (0.25) + 2 πn, 2 x = 2 π - arccos (0.25) + 2 πn

Resolver

2 x = arccos (0.25) + 2 πn : x = \frac{arccos ( 0.25 )}{2} + πn

2 x = arccos (0.25) + 2 πn

Dividir ambos os lados por

2

2 x = arccos (0.25) + 2 πn

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{arccos ( 0.25 )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{arccos ( 0.25 )}{2} + πn

x = \frac{arccos ( 0.25 )}{2} + πn

Resolver

2 x = 2 π - arccos (0.25) + 2 πn : x = π - \frac{arccos ( 0.25 )}{2} + πn

2 x = 2 π - arccos (0.25) + 2 πn

Dividir ambos os lados por

2

2 x = 2 π - arccos (0.25) + 2 πn

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( 0.25 )}{2} + \frac{2 πn}{2}

Simplificar

x = π - \frac{arccos ( 0.25 )}{2} + πn

x = π - \frac{arccos ( 0.25 )}{2} + πn

x = \frac{arccos ( 0.25 )}{2} + πn, x = π - \frac{arccos ( 0.25 )}{2} + πn

Combinar toda as soluções

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}, x = \frac{arccos ( 0.25 )}{2} + πn, x = π - \frac{arccos ( 0.25 )}{2} + πn

Mostrar soluções na forma decimal

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}, x = \frac{1.31811 \dots}{2} + πn, x = π - \frac{1.31811 \dots}{2} + πn

Gráfico

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