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Popular Trigonometria >

sin(4x)+sin(2x)=0,0<= x<= 180

Solução

sin (4 x) + sin (2 x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

Solução

x = 9 0^{\circ}, x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

Radianos

x = \frac{π}{2}, x = 0, x = \frac{π}{3}, x = \frac{2 π}{3}, x = π

Passos da solução

sin (4 x) + sin (2 x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

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

Simplificar

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

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

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

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

Somar elementos similares:

2 x + 4 x = 6 x

= \frac{6 x}{2}

Dividir:

\frac{6}{2} = 3

= 3 x

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

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

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

Somar elementos similares:

2 x - 4 x = - 2 x

= \frac{- 2 x}{2}

Aplicar as propriedades das frações:

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

= - \frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= - x

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

Use a identidade de ângulo negativo:

cos (- x) = cos (x)

= 2 cos (x) sin (3 x)

= 2 cos (x) sin (3 x)

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

Resolver cada parte separadamente

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

cos (x) = 0, 0 \leq x \leq 18 0^{\circ} : x = 9 0^{\circ}

cos (x) = 0, 0 \leq x \leq 18 0^{\circ}

Soluções gerais para

cos (x) = 0

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Soluções para o intervalo

0 \leq x \leq 18 0^{\circ}

x = 9 0^{\circ}

sin (3 x) = 0, 0 \leq x \leq 18 0^{\circ} : x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

sin (3 x) = 0, 0 \leq x \leq 18 0^{\circ}

Soluções gerais para

sin (3 x) = 0

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Resolver

3 x = 0 + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{3}

3 x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

3 x = 36 0^{\circ} n

Dividir ambos os lados por

3

3 x = 36 0^{\circ} n

Dividir ambos os lados por

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3}

Simplificar

x = \frac{36 0 ^{\circ} n}{3}

x = \frac{36 0 ^{\circ} n}{3}

Resolver

3 x = 18 0^{\circ} + 36 0^{\circ} n : x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

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

Dividir ambos os lados por

3

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

Dividir ambos os lados por

3

\frac{3 x}{3} = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Simplificar

x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

x = \frac{36 0 ^{\circ} n}{3}, x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Soluções para o intervalo

0 \leq x \leq 18 0^{\circ}

x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

Combinar toda as soluções

x = 9 0^{\circ}, x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

Gráfico

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(sqrt(3)tan(x))/(sec(x))-cos(x)=0

\frac{3 tan ( x )}{sec ( x )} - cos (x) = 0

2cos(2x)=2

2 cos (2 x) = 2

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

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3tan(2x)-sqrt(3)=0

3 tan (2 x) - 3 = 0