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

Solución

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

Solución

x = 0.65123 \dots + 2 πn, x = 2 π - 0.65123 \dots + 2 πn, x = 2.25063 \dots + 2 πn, x = - 2.25063 \dots + 2 πn

Grados

x = 37.31321 \dots^{\circ} + 36 0^{\circ} n, x = 322.68678 \dots^{\circ} + 36 0^{\circ} n, x = 128.95184 \dots^{\circ} + 36 0^{\circ} n, x = - 128.95184 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Simplificar

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

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

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

\frac{1}{3} cos (x)

Multiplicar fracciones:

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

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

Multiplicar:

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

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

- cos (x) + (- 1 + 2 cos^{2} (x)) \cdot 3 = 0

Usando el método de sustitución

- cos (x) + (- 1 + 2 cos^{2} (x)) \cdot 3 = 0

Sea:

cos (x) = u

- u + (- 1 + 2 u^{2}) \cdot 3 = 0

- u + (- 1 + 2 u^{2}) \cdot 3 = 0 : u = \frac{1 + 73}{12}, u = \frac{1 - 73}{12}

- u + (- 1 + 2 u^{2}) \cdot 3 = 0

Desarrollar

- u + (- 1 + 2 u^{2}) \cdot 3 : - u - 3 + 6 u^{2}

- u + (- 1 + 2 u^{2}) \cdot 3

= - u + 3 (- 1 + 2 u^{2})

Expandir

3 (- 1 + 2 u^{2}) : - 3 + 6 u^{2}

3 (- 1 + 2 u^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 3, b = - 1, c = 2 u^{2}

= 3 (- 1) + 3 \cdot 2 u^{2}

Aplicar las reglas de los signos

+ (- a) = - a

= - 3 \cdot 1 + 3 \cdot 2 u^{2}

Simplificar

- 3 \cdot 1 + 3 \cdot 2 u^{2} : - 3 + 6 u^{2}

- 3 \cdot 1 + 3 \cdot 2 u^{2}

Multiplicar los numeros:

3 \cdot 1 = 3

= - 3 + 3 \cdot 2 u^{2}

Multiplicar los numeros:

3 \cdot 2 = 6

= - 3 + 6 u^{2}

= - 3 + 6 u^{2}

= - u - 3 + 6 u^{2}

- u - 3 + 6 u^{2} = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

6 u^{2} - u - 3 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

6 u^{2} - u - 3 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 6, b = - 1, c = - 3

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 6 ( - 3 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 6 ( - 3 )}{2 \cdot 6}

(- 1)^{2} - 4 \cdot 6 (- 3) = 73

(- 1)^{2} - 4 \cdot 6 (- 3)

Aplicar la regla

- (- a) = a

= (- 1)^{2} + 4 \cdot 6 \cdot 3

(- 1)^{2} = 1

(- 1)^{2}

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

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

= 1^{2}

Aplicar la regla

1^{a} = 1

= 1

4 \cdot 6 \cdot 3 = 72

4 \cdot 6 \cdot 3

Multiplicar los numeros:

4 \cdot 6 \cdot 3 = 72

= 72

= 1 + 72

Sumar:

1 + 72 = 73

= 73

u_{1, 2} = \frac{- ( - 1 ) \pm 73}{2 \cdot 6}

Separar las soluciones

u_{1} = \frac{- ( - 1 ) + 73}{2 \cdot 6}, u_{2} = \frac{- ( - 1 ) - 73}{2 \cdot 6}

u = \frac{- ( - 1 ) + 73}{2 \cdot 6} : \frac{1 + 73}{12}

\frac{- ( - 1 ) + 73}{2 \cdot 6}

Aplicar la regla

- (- a) = a

= \frac{1 + 73}{2 \cdot 6}

Multiplicar los numeros:

2 \cdot 6 = 12

= \frac{1 + 73}{12}

u = \frac{- ( - 1 ) - 73}{2 \cdot 6} : \frac{1 - 73}{12}

\frac{- ( - 1 ) - 73}{2 \cdot 6}

Aplicar la regla

- (- a) = a

= \frac{1 - 73}{2 \cdot 6}

Multiplicar los numeros:

2 \cdot 6 = 12

= \frac{1 - 73}{12}

Las soluciones a la ecuación de segundo grado son:

u = \frac{1 + 73}{12}, u = \frac{1 - 73}{12}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{1 + 73}{12}, cos (x) = \frac{1 - 73}{12}

cos (x) = \frac{1 + 73}{12}, cos (x) = \frac{1 - 73}{12}

cos (x) = \frac{1 + 73}{12} : x = arccos (\frac{1 + 73}{12}) + 2 πn, x = 2 π - arccos (\frac{1 + 73}{12}) + 2 πn

cos (x) = \frac{1 + 73}{12}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{1 + 73}{12}

Soluciones generales para

cos (x) = \frac{1 + 73}{12}

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

x = arccos (\frac{1 + 73}{12}) + 2 πn, x = 2 π - arccos (\frac{1 + 73}{12}) + 2 πn

x = arccos (\frac{1 + 73}{12}) + 2 πn, x = 2 π - arccos (\frac{1 + 73}{12}) + 2 πn

cos (x) = \frac{1 - 73}{12} : x = arccos (\frac{1 - 73}{12}) + 2 πn, x = - arccos (\frac{1 - 73}{12}) + 2 πn

cos (x) = \frac{1 - 73}{12}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{1 - 73}{12}

Soluciones generales para

cos (x) = \frac{1 - 73}{12}

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

x = arccos (\frac{1 - 73}{12}) + 2 πn, x = - arccos (\frac{1 - 73}{12}) + 2 πn

x = arccos (\frac{1 - 73}{12}) + 2 πn, x = - arccos (\frac{1 - 73}{12}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{1 + 73}{12}) + 2 πn, x = 2 π - arccos (\frac{1 + 73}{12}) + 2 πn, x = arccos (\frac{1 - 73}{12}) + 2 πn, x = - arccos (\frac{1 - 73}{12}) + 2 πn

Mostrar soluciones en forma decimal

x = 0.65123 \dots + 2 πn, x = 2 π - 0.65123 \dots + 2 πn, x = 2.25063 \dots + 2 πn, x = - 2.25063 \dots + 2 πn

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