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Popular >

sec^2(x)-2tan^2(x)=0

Solución

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

Solución

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Grados

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Factorizar

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

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

Reescribir

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

como

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

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

Aplicar las leyes de los exponentes:

a = (a)^{2}

2 = (2)^{2}

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

Aplicar las leyes de los exponentes:

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

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

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

(sec (x) + 2 tan (x)) (sec (x) - 2 tan (x)) = 0

Resolver cada parte por separado

sec (x) + 2 tan (x) = 0 or sec (x) - 2 tan (x) = 0

sec (x) + 2 tan (x) = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) + 2 tan (x) = 0

Expresar con seno, coseno

sec (x) + 2 tan (x)

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{cos ( x )} + 2 tan (x)

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )}

Simplificar

\frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )} : \frac{1 + 2 sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )}

Multiplicar

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

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

Multiplicar fracciones:

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

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

= \frac{1}{cos ( x )} + \frac{2 sin ( x )}{cos ( x )}

Aplicar la regla

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

= \frac{1 + 2 sin ( x )}{cos ( x )}

= \frac{1 + 2 sin ( x )}{cos ( x )}

\frac{1 + sin ( x ) 2}{cos ( x )} = 0

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

1 + sin (x) 2 = 0

Mover

1

al lado derecho

1 + sin (x) 2 = 0

Restar

1

de ambos lados

1 + sin (x) 2 - 1 = 0 - 1

Simplificar

sin (x) 2 = - 1

sin (x) 2 = - 1

Dividir ambos lados entre

2

sin (x) 2 = - 1

Dividir ambos lados entre

2

\frac{sin ( x ) 2}{2} = \frac{- 1}{2}

Simplificar

\frac{sin ( x ) 2}{2} = \frac{- 1}{2}

Simplificar

\frac{sin ( x ) 2}{2} : sin (x)

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

Eliminar los terminos comunes:

2

= sin (x)

Simplificar

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

Racionalizar

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiplicar por el conjugado

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

Soluciones generales para

sin (x) = - \frac{2}{2}

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 = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) - 2 tan (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sec (x) - 2 tan (x) = 0

Expresar con seno, coseno

sec (x) - 2 tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

Aplicar la regla

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

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

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

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

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

1 - sin (x) 2 = 0

Mover

1

al lado derecho

1 - sin (x) 2 = 0

Restar

1

de ambos lados

1 - sin (x) 2 - 1 = 0 - 1

Simplificar

- sin (x) 2 = - 1

- sin (x) 2 = - 1

Dividir ambos lados entre

- 2

- sin (x) 2 = - 1

Dividir ambos lados entre

- 2

\frac{- sin ( x ) 2}{- 2} = \frac{- 1}{- 2}

Simplificar

\frac{- sin ( x ) 2}{- 2} = \frac{- 1}{- 2}

Simplificar

\frac{- sin ( x ) 2}{- 2} : sin (x)

\frac{- sin ( x ) 2}{- 2}

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

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

Eliminar los terminos comunes:

2

= sin (x)

Simplificar

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

= \frac{1}{2}

Racionalizar

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiplicar por el conjugado

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

Soluciones generales para

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

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 = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Combinar toda las soluciones

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Ejemplos populares

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