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4/(cos(x))-6=tan(x)

Solución

\frac{4}{cos ( x )} - 6 = tan (x)

Solución

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

Grados

x = 320.57910 \dots^{\circ} + 36 0^{\circ} n, x = 58.34553 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

\frac{4}{cos ( x )} - 6 = tan (x)

Restar

tan (x)

de ambos lados

\frac{4}{cos ( x )} - 6 - tan (x) = 0

Simplificar

\frac{4}{cos ( x )} - 6 - tan (x) : \frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4}{cos ( x )} - 6 - tan (x)

Convertir a fracción:

6 = \frac{6 cos ( x )}{cos ( x )}, tan (x) = \frac{tan ( x ) cos ( x )}{cos ( x )}

= \frac{4}{cos ( x )} - \frac{6 cos ( x )}{cos ( x )} - \frac{tan ( x ) cos ( x )}{cos ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )} = 0

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

4 - 6 cos (x) - tan (x) cos (x) = 0

Expresar con seno, coseno

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) = 0

Simplificar

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) : 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x)

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= sin (x)

= 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - sin (x) = 0

Sumar

sin (x)

a ambos lados

4 - 6 cos (x) = sin (x)

Elevar al cuadrado ambos lados

(4 - 6 cos (x))^{2} = sin^{2} (x)

Restar

sin^{2} (x)

de ambos lados

(4 - 6 cos (x))^{2} - sin^{2} (x) = 0

Re-escribir usando identidades trigonométricas

(4 - 6 cos (x))^{2} - sin^{2} (x)

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

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

= (4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

Simplificar

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x)) : 37 cos^{2} (x) - 48 cos (x) + 15

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

(4 - 6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

Aplicar la formula del binomio al cuadrado:

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

a = 4, b = 6 cos (x)

= 4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

Simplificar

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

2 \cdot 4 \cdot 6 cos (x) = 48 cos (x)

2 \cdot 4 \cdot 6 cos (x)

Multiplicar los numeros:

2 \cdot 4 \cdot 6 = 48

= 48 cos (x)

(6 cos (x))^{2} = 36 cos^{2} (x)

(6 cos (x))^{2}

Aplicar las leyes de los exponentes:

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

= 6^{2} cos^{2} (x)

6^{2} = 36

= 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x) - (1 - cos^{2} (x))

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

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

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

= 16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

Simplificar

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x) : 37 cos^{2} (x) - 48 cos (x) + 15

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

Agrupar términos semejantes

= - 48 cos (x) + 36 cos^{2} (x) + cos^{2} (x) + 16 - 1

Sumar elementos similares:

36 cos^{2} (x) + cos^{2} (x) = 37 cos^{2} (x)

= - 48 cos (x) + 37 cos^{2} (x) + 16 - 1

Sumar/restar lo siguiente:

16 - 1 = 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

15 + 37 cos^{2} (x) - 48 cos (x) = 0

Usando el método de sustitución

15 + 37 cos^{2} (x) - 48 cos (x) = 0

Sea:

cos (x) = u

15 + 37 u^{2} - 48 u = 0

15 + 37 u^{2} - 48 u = 0 : u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

15 + 37 u^{2} - 48 u = 0

Escribir en la forma binómica

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

37 u^{2} - 48 u + 15 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

37 u^{2} - 48 u + 15 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 37, b = - 48, c = 15

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

(- 48)^{2} - 4 \cdot 37 \cdot 15 = 221

(- 48)^{2} - 4 \cdot 37 \cdot 15

Aplicar las leyes de los exponentes:

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

n

es par

(- 48)^{2} = 4 8^{2}

= 4 8^{2} - 4 \cdot 37 \cdot 15

Multiplicar los numeros:

4 \cdot 37 \cdot 15 = 2220

= 4 8^{2} - 2220

4 8^{2} = 2304

= 2304 - 2220

Restar:

2304 - 2220 = 84

= 84

Descomposición en factores primos de

84 : 2^{2} \cdot 3 \cdot 7

84

84

divida por

284 = 42 \cdot 2

= 2 \cdot 42

42

divida por

242 = 21 \cdot 2

= 2 \cdot 2 \cdot 21

21

divida por

321 = 7 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 7

2, 3, 7

son números primos, por lo tanto, no se pueden factorizar mas

= 2 \cdot 2 \cdot 3 \cdot 7

= 2^{2} \cdot 3 \cdot 7

= 2^{2} \cdot 3 \cdot 7

Aplicar las leyes de los exponentes:

= 2^{2} 3 \cdot 7

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2 3 \cdot 7

Simplificar

= 221

u_{1, 2} = \frac{- ( - 48 ) \pm 2 21}{2 \cdot 37}

Separar las soluciones

u_{1} = \frac{- ( - 48 ) + 2 21}{2 \cdot 37}, u_{2} = \frac{- ( - 48 ) - 2 21}{2 \cdot 37}

u = \frac{- ( - 48 ) + 2 21}{2 \cdot 37} : \frac{24 + 21}{37}

\frac{- ( - 48 ) + 2 21}{2 \cdot 37}

Aplicar la regla

- (- a) = a

= \frac{48 + 2 21}{2 \cdot 37}

Multiplicar los numeros:

2 \cdot 37 = 74

= \frac{48 + 2 21}{74}

Factorizar

48 + 221 : 2 (24 + 21)

48 + 221

Reescribir como

= 2 \cdot 24 + 221

Factorizar el termino común

2

= 2 (24 + 21)

= \frac{2 ( 24 + 21 )}{74}

Eliminar los terminos comunes:

2

= \frac{24 + 21}{37}

u = \frac{- ( - 48 ) - 2 21}{2 \cdot 37} : \frac{24 - 21}{37}

\frac{- ( - 48 ) - 2 21}{2 \cdot 37}

Aplicar la regla

- (- a) = a

= \frac{48 - 2 21}{2 \cdot 37}

Multiplicar los numeros:

2 \cdot 37 = 74

= \frac{48 - 2 21}{74}

Factorizar

48 - 221 : 2 (24 - 21)

48 - 221

Reescribir como

= 2 \cdot 24 - 221

Factorizar el termino común

2

= 2 (24 - 21)

= \frac{2 ( 24 - 21 )}{74}

Eliminar los terminos comunes:

2

= \frac{24 - 21}{37}

Las soluciones a la ecuación de segundo grado son:

u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37} : x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 + 21}{37}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{24 + 21}{37}

Soluciones generales para

cos (x) = \frac{24 + 21}{37}

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

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37} : x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{24 - 21}{37}

Soluciones generales para

cos (x) = \frac{24 - 21}{37}

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

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

\frac{4}{cos ( x )} - 6 = tan (x)

Quitar las que no concuerden con la ecuación.

Verificar la solución

arccos (\frac{24 + 21}{37}) + 2 πn :

Falso

arccos (\frac{24 + 21}{37}) + 2 πn

Sustituir

n = 1

arccos (\frac{24 + 21}{37}) + 2 π 1

Multiplicar

\frac{4}{cos ( x )} - 6 = tan (x)

por

x = arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 + 21}{37}) + 2 π 1)

Simplificar

- 0.82202 \dots = 0.82202 \dots

\Rightarrow Falso

Verificar la solución

2 π - arccos (\frac{24 + 21}{37}) + 2 πn :

Verdadero

2 π - arccos (\frac{24 + 21}{37}) + 2 πn

Sustituir

n = 1

2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

Multiplicar

\frac{4}{cos ( x )} - 6 = tan (x)

por

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 + 21}{37}) + 2 π 1)

Simplificar

- 0.82202 \dots = - 0.82202 \dots

\Rightarrow Verdadero

Verificar la solución

arccos (\frac{24 - 21}{37}) + 2 πn :

Verdadero

arccos (\frac{24 - 21}{37}) + 2 πn

Sustituir

n = 1

arccos (\frac{24 - 21}{37}) + 2 π 1

Multiplicar

\frac{4}{cos ( x )} - 6 = tan (x)

por

x = arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 - 21}{37}) + 2 π 1)

Simplificar

1.62202 \dots = 1.62202 \dots

\Rightarrow Verdadero

Verificar la solución

2 π - arccos (\frac{24 - 21}{37}) + 2 πn :

Falso

2 π - arccos (\frac{24 - 21}{37}) + 2 πn

Sustituir

n = 1

2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

Multiplicar

\frac{4}{cos ( x )} - 6 = tan (x)

por

x = 2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 - 21}{37}) + 2 π 1)

Simplificar

1.62202 \dots = - 1.62202 \dots

\Rightarrow Falso

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn

Mostrar soluciones en forma decimal

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

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