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Popular Trigonometria >

cos(3x)=cos(2x)cos(x)

Solução

cos (3 x) = cos (2 x) cos (x)

Solução

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

Graus

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Passos da solução

cos (3 x) = cos (2 x) cos (x)

Subtrair

cos (2 x) cos (x)

de ambos os lados

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

Reeecreva usando identidades trigonométricas

cos (3 x) - cos (2 x) cos (x)

cos (3 x) = 4 cos^{3} (x) - 3 cos (x)

cos (3 x)

Reeecreva usando identidades trigonométricas

cos (3 x)

Reescrever como

= cos (2 x + x)

Use a identidade de soma de ângulos:

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

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

Utilizar a identidade trigonométrica do arco duplo:

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

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

Simplificar

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x) : cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

2 sin (x) cos (x) sin (x) = 2 sin^{2} (x) cos (x)

2 sin (x) cos (x) sin (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

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

Somar:

1 + 1 = 2

= 2 cos (x) sin^{2} (x)

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

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

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

Utilizar a identidade trigonométrica do arco duplo:

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

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Expandir

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x) : 4 cos^{3} (x) - 3 cos (x)

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

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

Expandir

cos (x) (2 cos^{2} (x) - 1) : 2 cos^{3} (x) - cos (x)

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

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = cos (x), b = 2 cos^{2} (x), c = 1

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

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

Simplificar

2 cos^{2} (x) cos (x) - 1 \cdot cos (x) : 2 cos^{3} (x) - cos (x)

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

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos (x) = cos^{2 + 1} (x)

= 2 cos^{2 + 1} (x)

Somar:

2 + 1 = 3

= 2 cos^{3} (x)

1 \cdot cos (x) = cos (x)

1 cos (x)

Multiplicar:

1 \cdot cos (x) = cos (x)

= cos (x)

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

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

= 2 cos^{3} (x) - cos (x) - 2 (1 - cos^{2} (x)) cos (x)

Expandir

- 2 cos (x) (1 - cos^{2} (x)) : - 2 cos (x) + 2 cos^{3} (x)

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

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = - 2 cos (x), b = 1, c = cos^{2} (x)

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

Aplicar as regras dos sinais

- (- a) = a

= - 2 \cdot 1 cos (x) + 2 cos^{2} (x) cos (x)

Simplificar

- 2 \cdot 1 \cdot cos (x) + 2 cos^{2} (x) cos (x) : - 2 cos (x) + 2 cos^{3} (x)

- 2 \cdot 1 cos (x) + 2 cos^{2} (x) cos (x)

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 cos (x)

Multiplicar os números:

2 \cdot 1 = 2

= 2 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos (x) = cos^{2 + 1} (x)

= 2 cos^{2 + 1} (x)

Somar:

2 + 1 = 3

= 2 cos^{3} (x)

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

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

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

Simplificar

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x) : 4 cos^{3} (x) - 3 cos (x)

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

Agrupar termos semelhantes

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

Somar elementos similares:

2 cos^{3} (x) + 2 cos^{3} (x) = 4 cos^{3} (x)

= 4 cos^{3} (x) - cos (x) - 2 cos (x)

Somar elementos similares:

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

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x) - cos (2 x) cos (x)

- 3 cos (x) + 4 cos^{3} (x) - cos (2 x) cos (x) = 0

Fatorar

- 3 cos (x) + 4 cos^{3} (x) - cos (2 x) cos (x) : - cos (x) (3 - 4 cos^{2} (x) + cos (2 x))

- 3 cos (x) + 4 cos^{3} (x) - cos (2 x) cos (x)

Aplicar as propriedades dos expoentes:

a^{b + c} = a^{b} a^{c}

cos^{3} (x) = cos (x) cos^{2} (x)

= - 3 cos (x) + 4 cos (x) cos^{2} (x) - cos (x) cos (2 x)

Fatorar o termo comum

- cos (x)

= - cos (x) (3 - 4 cos^{2} (x) + cos (2 x))

- cos (x) (3 - 4 cos^{2} (x) + cos (2 x)) = 0

Resolver cada parte separadamente

cos (x) = 0 or 3 - 4 cos^{2} (x) + cos (2 x) = 0

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

cos (x) = 0

Soluções gerais para

cos (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}

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

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

3 - 4 cos^{2} (x) + cos (2 x) = 0 : x = 2 πn, x = π + 2 πn

3 - 4 cos^{2} (x) + cos (2 x) = 0

Reeecreva usando identidades trigonométricas

3 + cos (2 x) - 4 cos^{2} (x)

Utilizar a identidade trigonométrica do arco duplo:

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

= 3 + 2 cos^{2} (x) - 1 - 4 cos^{2} (x)

Simplificar

3 + 2 cos^{2} (x) - 1 - 4 cos^{2} (x) : - 2 cos^{2} (x) + 2

3 + 2 cos^{2} (x) - 1 - 4 cos^{2} (x)

Agrupar termos semelhantes

= 2 cos^{2} (x) - 4 cos^{2} (x) + 3 - 1

Somar elementos similares:

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

= - 2 cos^{2} (x) + 3 - 1

Somar/subtrair:

3 - 1 = 2

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

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

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

Usando o método de substituição

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

Sea:

cos (x) = u

2 - 2 u^{2} = 0

2 - 2 u^{2} = 0 : u = 1, u = - 1

2 - 2 u^{2} = 0

Mova

2

para o lado direito

2 - 2 u^{2} = 0

Subtrair

2

de ambos os lados

2 - 2 u^{2} - 2 = 0 - 2

Simplificar

- 2 u^{2} = - 2

- 2 u^{2} = - 2

Dividir ambos os lados por

- 2

- 2 u^{2} = - 2

Dividir ambos os lados por

- 2

\frac{- 2 u ^{2}}{- 2} = \frac{- 2}{- 2}

Simplificar

u^{2} = 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 = cos (x)

cos (x) = 1, cos (x) = - 1

cos (x) = 1, cos (x) = - 1

cos (x) = 1 : x = 2 πn

cos (x) = 1

Soluções gerais para

cos (x) = 1

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}

x = 0 + 2 πn

x = 0 + 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

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

Soluções gerais para

cos (x) = - 1

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}

x = π + 2 πn

x = π + 2 πn

Combinar toda as soluções

x = 2 πn, x = π + 2 πn

Combinar toda as soluções

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

Gráfico

Exemplos populares

3cos^2(x)+sin^2(x)+5sin(x)=0

3 cos^{2} (x) + sin^{2} (x) + 5 sin (x) = 0

2tan(x)=5sin(x)

2 tan (x) = 5 sin (x)

5cot(x)=2

5 cot (x) = 2

tan(3x+(3pi)/4)=1

tan (3 x + \frac{3 π}{4}) = 1

2sin(3x+1)=1

2 sin (3 x + 1) = 1