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Popular >

3/(cos^2(x))=3+2tan^2(x)

Solución

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

Solución

x = 2 πn, x = π + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Restar

3 + 2 tan^{2} (x)

de ambos lados

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

Simplificar

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

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x)

Convertir a fracción:

3 = \frac{3 cos ^{2} ( x )}{cos ^{2} ( x )}, 2 tan^{2} (x) = \frac{2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

Expresar con seno, coseno

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

Simplificar

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

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

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

2 (\frac{sin ( x )}{cos ( x )})^{2} cos^{2} (x)

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

(\frac{sin ( x )}{cos ( x )})^{2}

Aplicar las leyes de los exponentes:

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

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

= 2 \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )} cos^{2} (x)

Multiplicar fracciones:

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

= \frac{sin ^{2} ( x ) \cdot 2 cos ^{2} ( x )}{cos ^{2} ( x )}

Eliminar los terminos comunes:

cos^{2} (x)

= sin^{2} (x) \cdot 2

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

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

Sumar

2 sin^{2} (x)

a ambos lados

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

Elevar al cuadrado ambos lados

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

Restar

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

de ambos lados

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x) = 0

Factorizar

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

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

Reescribir

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

como

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

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

Reescribir

4

como

2^{2}

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

Aplicar las leyes de los exponentes:

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

Simplificar

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

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

Resolver cada parte por separado

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

3 - 3 cos^{2} (x) + 2 sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

3 + 2 sin^{2} (x) - 3 (1 - sin^{2} (x)) : 5 sin^{2} (x)

3 + 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

Expandir

- 3 (1 - sin^{2} (x)) : - 3 + 3 sin^{2} (x)

- 3 (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (x)

= - 3 \cdot 1 - (- 3) sin^{2} (x)

Aplicar las reglas de los signos

- (- a) = a

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

Multiplicar los numeros:

3 \cdot 1 = 3

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

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

Simplificar

3 + 2 sin^{2} (x) - 3 + 3 sin^{2} (x) : 5 sin^{2} (x)

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

Agrupar términos semejantes

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

Sumar elementos similares:

2 sin^{2} (x) + 3 sin^{2} (x) = 5 sin^{2} (x)

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

3 - 3 = 0

= 5 sin^{2} (x)

= 5 sin^{2} (x)

= 5 sin^{2} (x)

5 sin^{2} (x) = 0

Dividir ambos lados entre

5

5 sin^{2} (x) = 0

Dividir ambos lados entre

5

5 sin^{2} (x) = 0

Dividir ambos lados entre

5

\frac{5 sin ^{2} ( x )}{5} = \frac{0}{5}

Simplificar

sin^{2} (x) = 0

sin^{2} (x) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

Soluciones generales para

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

3 - 3 cos^{2} (x) - 2 sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

= 3 - 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

Simplificar

3 - 2 sin^{2} (x) - 3 (1 - sin^{2} (x)) : sin^{2} (x)

3 - 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

Expandir

- 3 (1 - sin^{2} (x)) : - 3 + 3 sin^{2} (x)

- 3 (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (x)

= - 3 \cdot 1 - (- 3) sin^{2} (x)

Aplicar las reglas de los signos

- (- a) = a

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

Multiplicar los numeros:

3 \cdot 1 = 3

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

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

Simplificar

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

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

Agrupar términos semejantes

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

Sumar elementos similares:

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

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

3 - 3 = 0

= sin^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

sin^{2} (x) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

Soluciones generales para

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

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

Quitar las que no concuerden con la ecuación.

Verificar la solución

2 πn :

Verdadero

2 πn

Sustituir

n = 1

2 π 1

Multiplicar

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

por

x = 2 π 1

\frac{3}{cos ^{2} ( 2 π 1 )} = 3 + 2 tan^{2} (2 π 1)

Simplificar

3 = 3

\Rightarrow Verdadero

Verificar la solución

π + 2 πn :

Verdadero

π + 2 πn

Sustituir

n = 1

π + 2 π 1

Multiplicar

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

por

x = π + 2 π 1

\frac{3}{cos ^{2} ( π + 2 π 1 )} = 3 + 2 tan^{2} (π + 2 π 1)

Simplificar

3 = 3

\Rightarrow Verdadero

x = 2 πn, x = π + 2 πn

Ejemplos populares

13*sin(x)=12

13 \cdot sin (x) = 12

11sin(θ)=11cos(θ)

11 sin (θ) = 11 cos (θ)

1-2tan(x)=0

1 - 2 tan (x) = 0

sin(a)=0.747

sin (a) = 0.747

1-tan^2(y)=2sec(y)-4

1 - tan^{2} (y) = 2 sec (y) - 4