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Popular Trigonometría >

2cos(x)cos(2x)-2sin(x)sin(2x)=-1

Solución

2 cos (x) cos (2 x) - 2 sin (x) sin (2 x) = - 1

Solución

x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Grados

x = 4 0^{\circ} + 12 0^{\circ} n, x = 8 0^{\circ} + 12 0^{\circ} n

Pasos de solución

2 cos (x) cos (2 x) - 2 sin (x) sin (2 x) = - 1

Restar

- 1

de ambos lados

2 cos (x) cos (2 x) - 2 sin (x) sin (2 x) + 1 = 0

Re-escribir usando identidades trigonométricas

1 + 2 cos (2 x) cos (x) - 2 sin (2 x) sin (x)

Utilizar la identidad producto-suma:

cos (s) cos (t) = \frac{1}{2} (cos (s - t) + cos (s + t))

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

2 \cdot \frac{1}{2} (cos (2 x - x) + cos (2 x + x)) = cos (x) + cos (3 x)

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

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

Sumar elementos similares:

2 x - x = x

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

Sumar elementos similares:

2 x + x = 3 x

= 1 \cdot (cos (x) + cos (3 x))

Simplificar

= cos (x) + cos (3 x)

= 1 + cos (x) + cos (3 x) - 2 sin (2 x) sin (x)

Utilizar la identidad producto-suma:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= 1 + cos (3 x) + cos (x) - 2 \cdot \frac{1}{2} (cos (2 x - x) - cos (2 x + x))

Simplificar

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

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

2 \cdot \frac{1}{2} (cos (2 x - x) - cos (2 x + x)) = cos (x) - cos (3 x)

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

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

Sumar elementos similares:

2 x - x = x

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

Sumar elementos similares:

2 x + x = 3 x

= 1 \cdot (cos (x) - cos (3 x))

Simplificar

= cos (x) - cos (3 x)

= 1 + cos (3 x) + cos (x) - (cos (x) - cos (3 x))

- (cos (x) - cos (3 x)) : - cos (x) + cos (3 x)

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

Poner los parentesis

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

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

= - cos (x) + cos (3 x)

= 1 + cos (3 x) + cos (x) - cos (x) + cos (3 x)

Simplificar

1 + cos (3 x) + cos (x) - cos (x) + cos (3 x) : 2 cos (3 x) + 1

1 + cos (3 x) + cos (x) - cos (x) + cos (3 x)

Agrupar términos semejantes

= cos (3 x) + cos (x) - cos (x) + cos (3 x) + 1

Sumar elementos similares:

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

= 2 cos (3 x) + cos (x) - cos (x) + 1

Sumar elementos similares:

cos (x) - cos (x) = 0

= 2 cos (3 x) + 1

= 2 cos (3 x) + 1

= 2 cos (3 x) + 1

1 + 2 cos (3 x) = 0

Desplace

1

a la derecha

1 + 2 cos (3 x) = 0

Restar

1

de ambos lados

1 + 2 cos (3 x) - 1 = 0 - 1

Simplificar

2 cos (3 x) = - 1

2 cos (3 x) = - 1

Dividir ambos lados entre

2

2 cos (3 x) = - 1

Dividir ambos lados entre

2

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

Simplificar

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

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

Soluciones generales para

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

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

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

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

Resolver

3 x = \frac{2 π}{3} + 2 πn : x = \frac{2 π}{9} + \frac{2 πn}{3}

3 x = \frac{2 π}{3} + 2 πn

Dividir ambos lados entre

3

3 x = \frac{2 π}{3} + 2 πn

Dividir ambos lados entre

3

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

Simplificar

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

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{\frac{2 π}{3}}{3} + \frac{2 πn}{3} : \frac{2 π}{9} + \frac{2 πn}{3}

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

\frac{\frac{2 π}{3}}{3} = \frac{2 π}{9}

\frac{\frac{2 π}{3}}{3}

Aplicar las propiedades de las fracciones:

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

= \frac{2 π}{3 \cdot 3}

Multiplicar los numeros:

3 \cdot 3 = 9

= \frac{2 π}{9}

= \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

Resolver

3 x = \frac{4 π}{3} + 2 πn : x = \frac{4 π}{9} + \frac{2 πn}{3}

3 x = \frac{4 π}{3} + 2 πn

Dividir ambos lados entre

3

3 x = \frac{4 π}{3} + 2 πn

Dividir ambos lados entre

3

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

Simplificar

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

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{\frac{4 π}{3}}{3} + \frac{2 πn}{3} : \frac{4 π}{9} + \frac{2 πn}{3}

\frac{\frac{4 π}{3}}{3} + \frac{2 πn}{3}

\frac{\frac{4 π}{3}}{3} = \frac{4 π}{9}

\frac{\frac{4 π}{3}}{3}

Aplicar las propiedades de las fracciones:

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

= \frac{4 π}{3 \cdot 3}

Multiplicar los numeros:

3 \cdot 3 = 9

= \frac{4 π}{9}

= \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Gráfica

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