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Popular Trigonometría >

4cos(2x)-3cos(x)+1=0

Solución

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

Solución

x = 0.59538 \dots + 2 πn, x = 2 π - 0.59538 \dots + 2 πn, x = 2.04085 \dots + 2 πn, x = - 2.04085 \dots + 2 πn

Grados

x = 34.11286 \dots^{\circ} + 36 0^{\circ} n, x = 325.88713 \dots^{\circ} + 36 0^{\circ} n, x = 116.93210 \dots^{\circ} + 36 0^{\circ} n, x = - 116.93210 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Re-escribir usando identidades trigonométricas

1 - 3 cos (x) + 4 cos (2 x)

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

1 - 3 cos (x) + 4 (2 cos^{2} (x) - 1) : 8 cos^{2} (x) - 3 cos (x) - 3

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

Expandir

4 (2 cos^{2} (x) - 1) : 8 cos^{2} (x) - 4

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 4, b = 2 cos^{2} (x), c = 1

= 4 \cdot 2 cos^{2} (x) - 4 \cdot 1

Simplificar

4 \cdot 2 cos^{2} (x) - 4 \cdot 1 : 8 cos^{2} (x) - 4

4 \cdot 2 cos^{2} (x) - 4 \cdot 1

Multiplicar los numeros:

4 \cdot 2 = 8

= 8 cos^{2} (x) - 4 \cdot 1

Multiplicar los numeros:

4 \cdot 1 = 4

= 8 cos^{2} (x) - 4

= 8 cos^{2} (x) - 4

= 1 - 3 cos (x) + 8 cos^{2} (x) - 4

Simplificar

1 - 3 cos (x) + 8 cos^{2} (x) - 4 : 8 cos^{2} (x) - 3 cos (x) - 3

1 - 3 cos (x) + 8 cos^{2} (x) - 4

Agrupar términos semejantes

= - 3 cos (x) + 8 cos^{2} (x) + 1 - 4

Sumar/restar lo siguiente:

1 - 4 = - 3

= 8 cos^{2} (x) - 3 cos (x) - 3

= 8 cos^{2} (x) - 3 cos (x) - 3

= 8 cos^{2} (x) - 3 cos (x) - 3

- 3 - 3 cos (x) + 8 cos^{2} (x) = 0

Usando el método de sustitución

- 3 - 3 cos (x) + 8 cos^{2} (x) = 0

Sea:

cos (x) = u

- 3 - 3 u + 8 u^{2} = 0

- 3 - 3 u + 8 u^{2} = 0 : u = \frac{3 + 105}{16}, u = \frac{3 - 105}{16}

- 3 - 3 u + 8 u^{2} = 0

Escribir en la forma binómica

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

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

Resolver con la fórmula general para ecuaciones de segundo grado:

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

Formula general para ecuaciones de segundo grado:

Para

a = 8, b = - 3, c = - 3

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 8 ( - 3 )}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 8 ( - 3 )}{2 \cdot 8}

(- 3)^{2} - 4 \cdot 8 (- 3) = 105

(- 3)^{2} - 4 \cdot 8 (- 3)

Aplicar la regla

- (- a) = a

= (- 3)^{2} + 4 \cdot 8 \cdot 3

Aplicar las leyes de los exponentes:

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

n

es par

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

= 3^{2} + 4 \cdot 8 \cdot 3

Multiplicar los numeros:

4 \cdot 8 \cdot 3 = 96

= 3^{2} + 96

3^{2} = 9

= 9 + 96

Sumar:

9 + 96 = 105

= 105

u_{1, 2} = \frac{- ( - 3 ) \pm 105}{2 \cdot 8}

Separar las soluciones

u_{1} = \frac{- ( - 3 ) + 105}{2 \cdot 8}, u_{2} = \frac{- ( - 3 ) - 105}{2 \cdot 8}

u = \frac{- ( - 3 ) + 105}{2 \cdot 8} : \frac{3 + 105}{16}

\frac{- ( - 3 ) + 105}{2 \cdot 8}

Aplicar la regla

- (- a) = a

= \frac{3 + 105}{2 \cdot 8}

Multiplicar los numeros:

2 \cdot 8 = 16

= \frac{3 + 105}{16}

u = \frac{- ( - 3 ) - 105}{2 \cdot 8} : \frac{3 - 105}{16}

\frac{- ( - 3 ) - 105}{2 \cdot 8}

Aplicar la regla

- (- a) = a

= \frac{3 - 105}{2 \cdot 8}

Multiplicar los numeros:

2 \cdot 8 = 16

= \frac{3 - 105}{16}

Las soluciones a la ecuación de segundo grado son:

u = \frac{3 + 105}{16}, u = \frac{3 - 105}{16}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{3 + 105}{16}, cos (x) = \frac{3 - 105}{16}

cos (x) = \frac{3 + 105}{16}, cos (x) = \frac{3 - 105}{16}

cos (x) = \frac{3 + 105}{16} : x = arccos (\frac{3 + 105}{16}) + 2 πn, x = 2 π - arccos (\frac{3 + 105}{16}) + 2 πn

cos (x) = \frac{3 + 105}{16}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{3 + 105}{16}

Soluciones generales para

cos (x) = \frac{3 + 105}{16}

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

x = arccos (\frac{3 + 105}{16}) + 2 πn, x = 2 π - arccos (\frac{3 + 105}{16}) + 2 πn

x = arccos (\frac{3 + 105}{16}) + 2 πn, x = 2 π - arccos (\frac{3 + 105}{16}) + 2 πn

cos (x) = \frac{3 - 105}{16} : x = arccos (\frac{3 - 105}{16}) + 2 πn, x = - arccos (\frac{3 - 105}{16}) + 2 πn

cos (x) = \frac{3 - 105}{16}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{3 - 105}{16}

Soluciones generales para

cos (x) = \frac{3 - 105}{16}

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

x = arccos (\frac{3 - 105}{16}) + 2 πn, x = - arccos (\frac{3 - 105}{16}) + 2 πn

x = arccos (\frac{3 - 105}{16}) + 2 πn, x = - arccos (\frac{3 - 105}{16}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{3 + 105}{16}) + 2 πn, x = 2 π - arccos (\frac{3 + 105}{16}) + 2 πn, x = arccos (\frac{3 - 105}{16}) + 2 πn, x = - arccos (\frac{3 - 105}{16}) + 2 πn

Mostrar soluciones en forma decimal

x = 0.59538 \dots + 2 πn, x = 2 π - 0.59538 \dots + 2 πn, x = 2.04085 \dots + 2 πn, x = - 2.04085 \dots + 2 πn

Gráfica

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