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cos(x+1/6 pi)cos(x-1/6 pi)=cos(2x)

Solución

cos (x + \frac{1}{6} π) cos (x - \frac{1}{6} π) = cos (2 x)

Solución

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Grados

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Pasos de solución

cos (x + \frac{1}{6} π) cos (x - \frac{1}{6} π) = cos (2 x)

Re-escribir usando identidades trigonométricas

cos (x + \frac{1}{6} π) cos (x - \frac{1}{6} π) = cos (2 x)

Re-escribir usando identidades trigonométricas

cos (x + \frac{1}{6} π)

Utilizar la identidad de suma de ángulos:

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

= cos (x) cos (\frac{1}{6} π) - sin (x) sin (\frac{1}{6} π)

Simplificar

cos (x) cos (\frac{1}{6} π) - sin (x) sin (\frac{1}{6} π) : \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

cos (x) cos (\frac{1}{6} π) - sin (x) sin (\frac{1}{6} π)

cos (x) cos (\frac{1}{6} π) = \frac{3}{2} cos (x)

cos (x) cos (\frac{1}{6} π)

Multiplicar

\frac{1}{6} π : \frac{π}{6}

\frac{1}{6} π

Multiplicar fracciones:

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

= \frac{1 π}{6}

Multiplicar:

1 π = π

= \frac{π}{6}

= cos (\frac{π}{6}) cos (x)

Simplificar

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Utilizar la siguiente identidad trivial:

cos (\frac{π}{6}) = \frac{3}{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}

= \frac{3}{2}

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

sin (x) sin (\frac{1}{6} π) = \frac{1}{2} sin (x)

sin (x) sin (\frac{1}{6} π)

Multiplicar

\frac{1}{6} π : \frac{π}{6}

\frac{1}{6} π

Multiplicar fracciones:

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

= \frac{1 π}{6}

Multiplicar:

1 π = π

= \frac{π}{6}

= sin (\frac{π}{6}) sin (x)

Simplificar

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Utilizar la siguiente identidad trivial:

sin (\frac{π}{6}) = \frac{1}{2}

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} 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}

= \frac{1}{2}

= \frac{1}{2} sin (x)

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

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

Utilizar la identidad de diferencia de ángulos:

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

= cos (x) cos (\frac{1}{6} π) + sin (x) sin (\frac{1}{6} π)

Simplificar

cos (x) cos (\frac{1}{6} π) + sin (x) sin (\frac{1}{6} π) : \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

cos (x) cos (\frac{1}{6} π) + sin (x) sin (\frac{1}{6} π)

cos (x) cos (\frac{1}{6} π) = \frac{3}{2} cos (x)

cos (x) cos (\frac{1}{6} π)

Multiplicar

\frac{1}{6} π : \frac{π}{6}

\frac{1}{6} π

Multiplicar fracciones:

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

= \frac{1 π}{6}

Multiplicar:

1 π = π

= \frac{π}{6}

= cos (\frac{π}{6}) cos (x)

Simplificar

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Utilizar la siguiente identidad trivial:

cos (\frac{π}{6}) = \frac{3}{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}

= \frac{3}{2}

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

sin (x) sin (\frac{1}{6} π) = \frac{1}{2} sin (x)

sin (x) sin (\frac{1}{6} π)

Multiplicar

\frac{1}{6} π : \frac{π}{6}

\frac{1}{6} π

Multiplicar fracciones:

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

= \frac{1 π}{6}

Multiplicar:

1 π = π

= \frac{π}{6}

= sin (\frac{π}{6}) sin (x)

Simplificar

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Utilizar la siguiente identidad trivial:

sin (\frac{π}{6}) = \frac{1}{2}

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} 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}

= \frac{1}{2}

= \frac{1}{2} sin (x)

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

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

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

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

Restar

cos (2 x)

de ambos lados

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

Simplificar

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

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

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

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

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

\frac{3}{2} cos (x)

Multiplicar fracciones:

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

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

\frac{1}{2} sin (x) = \frac{sin ( x )}{2}

\frac{1}{2} sin (x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{2}

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

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

\frac{3}{2} cos (x)

Multiplicar fracciones:

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

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

\frac{1}{2} sin (x) = \frac{sin ( x )}{2}

\frac{1}{2} sin (x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{2}

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

Simplificar

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

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

Aplicar la regla

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

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

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

Combinar las fracciones usando el mínimo común denominador:

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

Aplicar la regla

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

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

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

Quitar los parentesis:

(a) = a

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

Multiplicar fracciones:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

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

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

Convertir a fracción:

cos (2 x) = \frac{cos ( 2 x ) 4}{4}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

(3 cos (x) - sin (x)) (3 cos (x) + sin (x)) - 4 cos (2 x) = 0

Re-escribir usando identidades trigonométricas

(- sin (x) + cos (x) 3) (sin (x) + cos (x) 3) - 4 cos (2 x)

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Expandir

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

(- sin (x) + 3 cos (x)) (sin (x) + 3 cos (x))

Aplicar la siguiente regla para binomios al cuadrado:

(a - b) (a + b) = a^{2} - b^{2}

a = 3 cos (x), b = sin (x)

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

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

(3 cos (x))^{2}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

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

(3)^{2} : 3

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

= \frac{1 \cdot 2}{2}

Eliminar los terminos comunes:

2

= 1

= 3

= 3 cos^{2} (x)

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

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

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 4, b = cos^{2} (x), c = sin^{2} (x)

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

Aplicar las reglas de los signos

- (- a) = a

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

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

Simplificar

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

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

Sumar elementos similares:

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

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

Sumar elementos similares:

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

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

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

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

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

Factorizar

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

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

Reescribir

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

como

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

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

Aplicar las leyes de los exponentes:

a = (a)^{2}

3 = (3)^{2}

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

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

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

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

(3 sin (x) + cos (x)) (3 sin (x) - cos (x)) = 0

Resolver cada parte por separado

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

3 sin (x) + cos (x) = 0 : x = \frac{5 π}{6} + πn

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

Re-escribir usando identidades trigonométricas

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

Dividir ambos lados entre

cos (x), cos (x) \neq = 0

\frac{3 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

\frac{3 sin ( x )}{cos ( x )} + 1 = 0

Utilizar la identidad trigonométrica básica:

\frac{sin ( x )}{cos ( x )} = tan (x)

3 tan (x) + 1 = 0

3 tan (x) + 1 = 0

Mover

1

al lado derecho

3 tan (x) + 1 = 0

Restar

1

de ambos lados

3 tan (x) + 1 - 1 = 0 - 1

Simplificar

3 tan (x) = - 1

3 tan (x) = - 1

Dividir ambos lados entre

3

3 tan (x) = - 1

Dividir ambos lados entre

3

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

Simplificar

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

Simplificar

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Eliminar los terminos comunes:

3

= tan (x)

Simplificar

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Aplicar las propiedades de las fracciones:

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

= - \frac{1}{3}

Racionalizar

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiplicar por el conjugado

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Aplicar las leyes de los exponentes:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

Soluciones generales para

tan (x) = - \frac{3}{3}

tan (x)

tabla de valores periódicos con

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

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

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

Re-escribir usando identidades trigonométricas

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

Dividir ambos lados entre

cos (x), cos (x) \neq = 0

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

Simplificar

\frac{3 sin ( x )}{cos ( x )} - 1 = 0

Utilizar la identidad trigonométrica básica:

\frac{sin ( x )}{cos ( x )} = tan (x)

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

Mover

1

al lado derecho

3 tan (x) - 1 = 0

Sumar

1

a ambos lados

3 tan (x) - 1 + 1 = 0 + 1

Simplificar

3 tan (x) = 1

3 tan (x) = 1

Dividir ambos lados entre

3

3 tan (x) = 1

Dividir ambos lados entre

3

\frac{3 tan ( x )}{3} = \frac{1}{3}

Simplificar

\frac{3 tan ( x )}{3} = \frac{1}{3}

Simplificar

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Eliminar los terminos comunes:

3

= tan (x)

Simplificar

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiplicar por el conjugado

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Aplicar las leyes de los exponentes:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

Soluciones generales para

tan (x) = \frac{3}{3}

tan (x)

tabla de valores periódicos con

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combinar toda las soluciones

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Ejemplos populares

tan(a/2)= 1/2

tan (\frac{a}{2}) = \frac{1}{2}

sin(x)=0.731

sin (x) = 0.731

sin(x)=0.766

sin (x) = 0.766

sin(x)=0.755

sin (x) = 0.755

3tan^2(x)-5tan(x)+2=0

3 tan^{2} (x) - 5 tan (x) + 2 = 0