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cos(x+60)*cos(x-60)= 1/2

Solución

cos (x + 60) \cdot cos (x - 60) = \frac{1}{2}

Solución

x = \frac{1.38389 \dots}{2} + πn, x = π - \frac{1.38389 \dots}{2} + πn

Grados

x = 39.64555 \dots^{\circ} + 18 0^{\circ} n, x = 140.35444 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

cos (x + 60) cos (x - 60) = \frac{1}{2}

Restar

\frac{1}{2}

de ambos lados

cos (x + 60) cos (x - 60) - \frac{1}{2} = 0

Simplificar

cos (x + 60) cos (x - 60) - \frac{1}{2} : \frac{2 cos ( x + 60 ) cos ( x - 60 ) - 1}{2}

cos (x + 60) cos (x - 60) - \frac{1}{2}

Convertir a fracción:

cos (x + 60) cos (x - 60) = \frac{cos ( x + 60 ) cos ( x - 60 ) 2}{2}

= \frac{cos ( x + 60 ) cos ( x - 60 ) \cdot 2}{2} - \frac{1}{2}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{cos ( x + 60 ) cos ( x - 60 ) \cdot 2 - 1}{2}

\frac{2 cos ( x + 60 ) cos ( x - 60 ) - 1}{2} = 0

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

2 cos (x + 60) cos (x - 60) - 1 = 0

Re-escribir usando identidades trigonométricas

- 1 + 2 cos (- 60 + x) cos (60 + 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 (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x))

2 \cdot \frac{1}{2} (cos (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x)) = cos (120) + cos (2 x)

2 \cdot \frac{1}{2} (cos (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x))

Multiplicar fracciones:

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

= \frac{1 \cdot 2 ( cos ( - 60 + x - ( 60 + x ) ) + cos ( - 60 + x + 60 + x ) )}{2}

Eliminar los terminos comunes:

2

= 1 \cdot (cos (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x))

cos (- 60 + x - (60 + x)) = cos (120)

cos (- 60 + x - (60 + x))

Expandir

- 60 + x - (60 + x) : - 120

- 60 + x - (60 + x)

- (60 + x) : - 60 - x

- (60 + x)

Poner los parentesis

= - (60) - (x)

Aplicar las reglas de los signos

+ (- a) = - a

= - 60 - x

= - 60 + x - 60 - x

Simplificar

- 60 + x - 60 - x : - 120

- 60 + x - 60 - x

Agrupar términos semejantes

= x - x - 60 - 60

Sumar elementos similares:

x - x = 0

= - 60 - 60

Restar:

- 60 - 60 = - 120

= - 120

= - 120

= cos (- 120)

Utilizar la siguiente propiedad:

cos (- x) = cos (x)

cos (- 120) = cos (120)

= cos (120)

cos (- 60 + x + 60 + x) = cos (2 x)

cos (- 60 + x + 60 + x)

Simplificar

- 60 + x + 60 + x : 2 x

- 60 + x + 60 + x

Agrupar términos semejantes

= x + x - 60 + 60

Sumar elementos similares:

x + x = 2 x

= 2 x - 60 + 60

- 60 + 60 = 0

= 2 x

= cos (2 x)

= 1 \cdot (cos (2 x) + cos (120))

Simplificar

= cos (120) + cos (2 x)

= - 1 + cos (120) + cos (2 x)

- 1 + cos (120) + cos (2 x) = 0

Mover

1

al lado derecho

- 1 + cos (120) + cos (2 x) = 0

Sumar

1

a ambos lados

- 1 + cos (120) + cos (2 x) + 1 = 0 + 1

Simplificar

cos (120) + cos (2 x) = 1

cos (120) + cos (2 x) = 1

Mover

cos (120)

al lado derecho

cos (120) + cos (2 x) = 1

Restar

cos (120)

de ambos lados

cos (120) + cos (2 x) - cos (120) = 1 - cos (120)

Simplificar

cos (2 x) = 1 - cos (120)

cos (2 x) = 1 - cos (120)

Aplicar propiedades trigonométricas inversas

cos (2 x) = 1 - cos (120)

Soluciones generales para

cos (2 x) = 1 - cos (120)

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

2 x = arccos (1 - cos (120)) + 2 πn, 2 x = 2 π - arccos (1 - cos (120)) + 2 πn

2 x = arccos (1 - cos (120)) + 2 πn, 2 x = 2 π - arccos (1 - cos (120)) + 2 πn

Resolver

2 x = arccos (1 - cos (120)) + 2 πn : x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

2 x = arccos (1 - cos (120)) + 2 πn

Dividir ambos lados entre

2

2 x = arccos (1 - cos (120)) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{arccos ( 1 - cos ( 120 ) )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

Resolver

2 x = 2 π - arccos (1 - cos (120)) + 2 πn : x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

2 x = 2 π - arccos (1 - cos (120)) + 2 πn

Dividir ambos lados entre

2

2 x = 2 π - arccos (1 - cos (120)) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( 1 - cos ( 120 ) )}{2} + \frac{2 πn}{2}

Simplificar

x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn, x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

Mostrar soluciones en forma decimal

x = \frac{1.38389 \dots}{2} + πn, x = π - \frac{1.38389 \dots}{2} + πn

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