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sec^2(x)tan^2(x)+3sec^2(x)-2tan^2(x)=3

Solución

sec^{2} (x) tan^{2} (x) + 3 sec^{2} (x) - 2 tan^{2} (x) = 3

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

sec^{2} (x) tan^{2} (x) + 3 sec^{2} (x) - 2 tan^{2} (x) = 3

Restar

3

de ambos lados

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

Re-escribir usando identidades trigonométricas

- 3 - 2 tan^{2} (x) + 3 sec^{2} (x) + sec^{2} (x) tan^{2} (x)

Utilizar la identidad pitagórica:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

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

Simplificar

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

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

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

Aplicar las reglas de los signos

- (- a) = a

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

Multiplicar los numeros:

2 \cdot 1 = 2

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

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

Expandir

sec^{2} (x) (sec^{2} (x) - 1) : sec^{4} (x) - sec^{2} (x)

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

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

Simplificar

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

sec^{2} (x) sec^{2} (x) - 1 \cdot sec^{2} (x)

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

sec^{2} (x) sec^{2} (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

sec^{2} (x) sec^{2} (x) = sec^{2 + 2} (x)

= sec^{2 + 2} (x)

Sumar:

2 + 2 = 4

= sec^{4} (x)

1 \cdot sec^{2} (x) = sec^{2} (x)

1 \cdot sec^{2} (x)

Multiplicar:

1 \cdot sec^{2} (x) = sec^{2} (x)

= sec^{2} (x)

= sec^{4} (x) - sec^{2} (x)

= sec^{4} (x) - sec^{2} (x)

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

Simplificar

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

- 3 - 2 sec^{2} (x) + 2 + 3 sec^{2} (x) + sec^{4} (x) - sec^{2} (x)

Agrupar términos semejantes

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

Sumar elementos similares:

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

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

Sumar/restar lo siguiente:

- 3 + 2 = - 1

= sec^{4} (x) - 1

= sec^{4} (x) - 1

= sec^{4} (x) - 1

- 1 + sec^{4} (x) = 0

Usando el método de sustitución

- 1 + sec^{4} (x) = 0

Sea:

sec (x) = u

- 1 + u^{4} = 0

- 1 + u^{4} = 0 : u = 1, u = - 1, u = i, u = - i

- 1 + u^{4} = 0

Mover

1

al lado derecho

- 1 + u^{4} = 0

Sumar

1

a ambos lados

- 1 + u^{4} + 1 = 0 + 1

Simplificar

u^{4} = 1

u^{4} = 1

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

v^{2} = 1

Resolver

v^{2} = 1 : v = 1, v = - 1

v^{2} = 1

Para

(g (x))^{2} = f (a)

las soluciones son

g (x) = f (a), - f (a)

v = 1, v = - 1

v = 1, v = - 1

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

Aplicar la regla

1 = 1

u^{2} = 1

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Aplicar la regla

1 = 1

= 1

- 1 = - 1

- 1

Aplicar la regla

1 = 1

= - 1

u = 1, u = - 1

Resolver

u^{2} = - 1 : u = i, u = - i

u^{2} = - 1

Aplicar la regla

1 = 1

u^{2} = - 1

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = - 1, u = - - 1

Simplificar

- 1 : i

- 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= i

Simplificar

- - 1 : - i

- - 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= - i

u = i, u = - i

Las soluciones son

u = 1, u = - 1, u = i, u = - i

Sustituir en la ecuación

u = sec (x)

sec (x) = 1, sec (x) = - 1, sec (x) = i, sec (x) = - i

sec (x) = 1, sec (x) = - 1, sec (x) = i, sec (x) = - i

sec (x) = 1 : x = 2 πn

sec (x) = 1

Soluciones generales para

sec (x) = 1

sec (x)

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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 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

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

Soluciones generales para

sec (x) = - 1

sec (x)

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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

sec (x) = i :

Sin solución

sec (x) = i

Sin soluci \overset{o}{ˊ} n

sec (x) = - i :

Sin solución

sec (x) = - i

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Ejemplos populares

2cos(x)-3=0

2 cos (x) - 3 = 0

cot(x)+6sin(x)-2cos(x)=3

cot (x) + 6 sin (x) - 2 cos (x) = 3

cos(2θ)=cos^2(θ)

cos (2 θ) = cos^{2} (θ)

cos(a)= 12/13

cos (a) = \frac{12}{13}

picos(pix)=0

π cos (π x) = 0