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(sin^2(a)+1)/(tan^2(a))=1

Solución

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} = 1

Solución

a = 0.90455 \dots + 2 πn, a = π - 0.90455 \dots + 2 πn, a = - 0.90455 \dots + 2 πn, a = π + 0.90455 \dots + 2 πn

Grados

a = 51.82729 \dots^{\circ} + 36 0^{\circ} n, a = 128.17270 \dots^{\circ} + 36 0^{\circ} n, a = - 51.82729 \dots^{\circ} + 36 0^{\circ} n, a = 231.82729 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} = 1

Restar

1

de ambos lados

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} - 1 = 0

Simplificar

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} - 1 : \frac{sin ^{2} ( a ) + 1 - tan ^{2} ( a )}{tan ^{2} ( a )}

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} - 1

Convertir a fracción:

1 = \frac{1 tan ^{2} ( a )}{tan ^{2} ( a )}

= \frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} - \frac{1 \cdot tan ^{2} ( a )}{tan ^{2} ( a )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{sin ^{2} ( a ) + 1 - 1 \cdot tan ^{2} ( a )}{tan ^{2} ( a )}

Multiplicar:

1 \cdot tan^{2} (a) = tan^{2} (a)

= \frac{sin ^{2} ( a ) + 1 - tan ^{2} ( a )}{tan ^{2} ( a )}

\frac{sin ^{2} ( a ) + 1 - tan ^{2} ( a )}{tan ^{2} ( a )} = 0

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

sin^{2} (a) + 1 - tan^{2} (a) = 0

Re-escribir usando identidades trigonométricas

1 + sin^{2} (a) - tan^{2} (a)

Utilizar la identidad trigonométrica básica:

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

= 1 + sin^{2} (a) - (\frac{sin ( a )}{cos ( a )})^{2}

Aplicar las leyes de los exponentes:

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

= 1 + sin^{2} (a) - \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

Utilizar la identidad pitagórica:

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

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

= 1 - \frac{sin ^{2} ( a )}{1 - sin ^{2} ( a )} + sin^{2} (a)

Combinar las fracciones usando el mínimo común denominador:

- \frac{sin ^{4} ( a )}{- sin ^{2} ( a ) + 1}

- \frac{sin ^{2} ( a )}{- sin ^{2} ( a ) + 1} + sin^{2} (a)

Convertir a fracción:

sin^{2} (a) = \frac{sin ^{2} ( a ) ( 1 - sin ^{2} ( a ) )}{1 - sin ^{2} ( a )}

= - \frac{sin ^{2} ( a )}{1 - sin ^{2} ( a )} + \frac{sin ^{2} ( a ) ( 1 - sin ^{2} ( a ) )}{1 - sin ^{2} ( a )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- sin ^{2} ( a ) + sin ^{2} ( a ) ( 1 - sin ^{2} ( a ) )}{1 - sin ^{2} ( a )}

Expandir

- sin^{2} (a) + sin^{2} (a) (1 - sin^{2} (a)) : - sin^{4} (a)

- sin^{2} (a) + sin^{2} (a) (1 - sin^{2} (a))

Expandir

sin^{2} (a) (1 - sin^{2} (a)) : sin^{2} (a) - sin^{4} (a)

sin^{2} (a) (1 - sin^{2} (a))

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

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

Simplificar

1 \cdot sin^{2} (a) - sin^{2} (a) sin^{2} (a) : sin^{2} (a) - sin^{4} (a)

1 \cdot sin^{2} (a) - sin^{2} (a) sin^{2} (a)

1 \cdot sin^{2} (a) = sin^{2} (a)

1 \cdot sin^{2} (a)

Multiplicar:

1 \cdot sin^{2} (a) = sin^{2} (a)

= sin^{2} (a)

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

sin^{2} (a) sin^{2} (a)

Aplicar las leyes de los exponentes:

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

sin^{2} (a) sin^{2} (a) = sin^{2 + 2} (a)

= sin^{2 + 2} (a)

Sumar:

2 + 2 = 4

= sin^{4} (a)

= sin^{2} (a) - sin^{4} (a)

= sin^{2} (a) - sin^{4} (a)

= - sin^{2} (a) + sin^{2} (a) - sin^{4} (a)

Sumar elementos similares:

- sin^{2} (a) + sin^{2} (a) = 0

= - sin^{4} (a)

= \frac{- sin ^{4} ( a )}{1 - sin ^{2} ( a )}

Aplicar las propiedades de las fracciones:

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

= - \frac{sin ^{4} ( a )}{1 - sin ^{2} ( a )}

1 - \frac{sin ^{4} ( a )}{1 - sin ^{2} ( a )} = 0

1 - \frac{sin ^{4} ( a )}{1 - sin ^{2} ( a )} = 0

Usando el método de sustitución

1 - \frac{sin ^{4} ( a )}{1 - sin ^{2} ( a )} = 0

Sea:

sin (a) = u

1 - \frac{u ^{4}}{1 - u ^{2}} = 0

1 - \frac{u ^{4}}{1 - u ^{2}} = 0 : u = i \frac{1 + 5}{2}, u = - i \frac{1 + 5}{2}, u = \frac{5 - 1}{2}, u = - \frac{5 - 1}{2}

1 - \frac{u ^{4}}{1 - u ^{2}} = 0

Multiplicar ambos lados por

1 - u^{2}

1 - \frac{u ^{4}}{1 - u ^{2}} = 0

Multiplicar ambos lados por

1 - u^{2}

1 \cdot (1 - u^{2}) - \frac{u ^{4}}{1 - u ^{2}} (1 - u^{2}) = 0 \cdot (1 - u^{2})

Simplificar

1 \cdot (1 - u^{2}) - \frac{u ^{4}}{1 - u ^{2}} (1 - u^{2}) = 0 \cdot (1 - u^{2})

Simplificar

1 \cdot (1 - u^{2}) : 1 - u^{2}

1 \cdot (1 - u^{2})

Multiplicar:

1 \cdot (1 - u^{2}) = (1 - u^{2})

= (1 - u^{2})

Quitar los parentesis:

(a) = a

= 1 - u^{2}

Simplificar

- \frac{u ^{4}}{1 - u ^{2}} (1 - u^{2}) : - u^{4}

- \frac{u ^{4}}{1 - u ^{2}} (1 - u^{2})

Multiplicar fracciones:

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

= - \frac{u ^{4} ( 1 - u ^{2} )}{1 - u ^{2}}

Eliminar los terminos comunes:

1 - u^{2}

= - u^{4}

Simplificar

0 \cdot (1 - u^{2}) : 0

0 \cdot (1 - u^{2})

Aplicar la regla

0 \cdot a = 0

= 0

1 - u^{2} - u^{4} = 0

1 - u^{2} - u^{4} = 0

1 - u^{2} - u^{4} = 0

Resolver

1 - u^{2} - u^{4} = 0 : u = i \frac{1 + 5}{2}, u = - i \frac{1 + 5}{2}, u = \frac{5 - 1}{2}, u = - \frac{5 - 1}{2}

1 - u^{2} - u^{4} = 0

Escribir en la forma binómica

a_{n} x^{n} + \dots + a_{1} x + a = 0

- u^{4} - u^{2} + 1 = 0

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

- v^{2} - v + 1 = 0

Resolver

- v^{2} - v + 1 = 0 : v = - \frac{1 + 5}{2}, v = \frac{5 - 1}{2}

- v^{2} - v + 1 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

- v^{2} - v + 1 = 0

Formula general para ecuaciones de segundo grado:

Para

a = - 1, b = - 1, c = 1

v_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

v_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

(- 1)^{2} - 4 (- 1) \cdot 1 = 5

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

Aplicar la regla

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

Aplicar las leyes de los exponentes:

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

n

es par

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

= 1^{2}

Aplicar la regla

1^{a} = 1

= 1

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

Multiplicar los numeros:

4 \cdot 1 \cdot 1 = 4

= 4

= 1 + 4

Sumar:

1 + 4 = 5

= 5

v_{1, 2} = \frac{- ( - 1 ) \pm 5}{2 ( - 1 )}

Separar las soluciones

v_{1} = \frac{- ( - 1 ) + 5}{2 ( - 1 )}, v_{2} = \frac{- ( - 1 ) - 5}{2 ( - 1 )}

v = \frac{- ( - 1 ) + 5}{2 ( - 1 )} : - \frac{1 + 5}{2}

\frac{- ( - 1 ) + 5}{2 ( - 1 )}

Quitar los parentesis:

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

= \frac{1 + 5}{- 2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{1 + 5}{- 2}

Aplicar las propiedades de las fracciones:

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

= - \frac{1 + 5}{2}

v = \frac{- ( - 1 ) - 5}{2 ( - 1 )} : \frac{5 - 1}{2}

\frac{- ( - 1 ) - 5}{2 ( - 1 )}

Quitar los parentesis:

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

= \frac{1 - 5}{- 2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{1 - 5}{- 2}

Aplicar las propiedades de las fracciones:

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

1 - 5 = - (5 - 1)

= \frac{5 - 1}{2}

Las soluciones a la ecuación de segundo grado son:

v = - \frac{1 + 5}{2}, v = \frac{5 - 1}{2}

v = - \frac{1 + 5}{2}, v = \frac{5 - 1}{2}

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

u^{2} = - \frac{1 + 5}{2} : u = i \frac{1 + 5}{2}, u = - i \frac{1 + 5}{2}

u^{2} = - \frac{1 + 5}{2}

Para

x^{2} = f (a)

las soluciones son

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

u = - \frac{1 + 5}{2}, u = - - \frac{1 + 5}{2}

Simplificar

- \frac{1 + 5}{2} : i \frac{1 + 5}{2}

- \frac{1 + 5}{2}

Aplicar las leyes de los exponentes:

- a = - 1 a

- \frac{1 + 5}{2} = - 1 \frac{1 + 5}{2}

= - 1 \frac{1 + 5}{2}

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= i \frac{1 + 5}{2}

Simplificar

- - \frac{1 + 5}{2} : - i \frac{1 + 5}{2}

- - \frac{1 + 5}{2}

Simplificar

- \frac{1 + 5}{2} : i \frac{1 + 5}{2}

- \frac{1 + 5}{2}

Aplicar las leyes de los exponentes:

- a = - 1 a

- \frac{1 + 5}{2} = - 1 \frac{1 + 5}{2}

= - 1 \frac{1 + 5}{2}

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= i \frac{1 + 5}{2}

= - i \frac{1 + 5}{2}

u = i \frac{1 + 5}{2}, u = - i \frac{1 + 5}{2}

Resolver

u^{2} = \frac{5 - 1}{2} : u = \frac{5 - 1}{2}, u = - \frac{5 - 1}{2}

u^{2} = \frac{5 - 1}{2}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{5 - 1}{2}, u = - \frac{5 - 1}{2}

Las soluciones son

u = i \frac{1 + 5}{2}, u = - i \frac{1 + 5}{2}, u = \frac{5 - 1}{2}, u = - \frac{5 - 1}{2}

u = i \frac{1 + 5}{2}, u = - i \frac{1 + 5}{2}, u = \frac{5 - 1}{2}, u = - \frac{5 - 1}{2}

Verificar las soluciones

Encontrar los puntos no definidos (singularidades):

u = 1, u = - 1

Tomar el(los) denominador(es) de

1 - \frac{u ^{4}}{1 - u ^{2}}

y comparar con cero

Resolver

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

Mover

1

al lado derecho

1 - u^{2} = 0

Restar

1

de ambos lados

1 - u^{2} - 1 = 0 - 1

Simplificar

- u^{2} = - 1

- u^{2} = - 1

Dividir ambos lados entre

- 1

- u^{2} = - 1

Dividir ambos lados entre

- 1

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 1}

Simplificar

u^{2} = 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 las leyes de los exponentes:

1 = 1

= 1

- 1 = - 1

- 1

Aplicar las leyes de los exponentes:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

Los siguientes puntos no están definidos

u = 1, u = - 1

Combinar los puntos no definidos con las soluciones:

u = i \frac{1 + 5}{2}, u = - i \frac{1 + 5}{2}, u = \frac{5 - 1}{2}, u = - \frac{5 - 1}{2}

Sustituir en la ecuación

u = sin (a)

sin (a) = i \frac{1 + 5}{2}, sin (a) = - i \frac{1 + 5}{2}, sin (a) = \frac{5 - 1}{2}, sin (a) = - \frac{5 - 1}{2}

sin (a) = i \frac{1 + 5}{2}, sin (a) = - i \frac{1 + 5}{2}, sin (a) = \frac{5 - 1}{2}, sin (a) = - \frac{5 - 1}{2}

sin (a) = i \frac{1 + 5}{2} :

Sin solución

sin (a) = i \frac{1 + 5}{2}

Sin soluci \overset{o}{ˊ} n

sin (a) = - i \frac{1 + 5}{2} :

Sin solución

sin (a) = - i \frac{1 + 5}{2}

Sin soluci \overset{o}{ˊ} n

sin (a) = \frac{5 - 1}{2} : a = arcsin \frac{5 - 1}{2} + 2 πn, a = π - arcsin \frac{5 - 1}{2} + 2 πn

sin (a) = \frac{5 - 1}{2}

Aplicar propiedades trigonométricas inversas

sin (a) = \frac{5 - 1}{2}

Soluciones generales para

sin (a) = \frac{5 - 1}{2}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

a = arcsin \frac{5 - 1}{2} + 2 πn, a = π - arcsin \frac{5 - 1}{2} + 2 πn

a = arcsin \frac{5 - 1}{2} + 2 πn, a = π - arcsin \frac{5 - 1}{2} + 2 πn

sin (a) = - \frac{5 - 1}{2} : a = arcsin - \frac{5 - 1}{2} + 2 πn, a = π + arcsin \frac{5 - 1}{2} + 2 πn

sin (a) = - \frac{5 - 1}{2}

Aplicar propiedades trigonométricas inversas

sin (a) = - \frac{5 - 1}{2}

Soluciones generales para

sin (a) = - \frac{5 - 1}{2}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

a = arcsin - \frac{5 - 1}{2} + 2 πn, a = π + arcsin \frac{5 - 1}{2} + 2 πn

a = arcsin - \frac{5 - 1}{2} + 2 πn, a = π + arcsin \frac{5 - 1}{2} + 2 πn

Combinar toda las soluciones

a = arcsin \frac{5 - 1}{2} + 2 πn, a = π - arcsin \frac{5 - 1}{2} + 2 πn, a = arcsin - \frac{5 - 1}{2} + 2 πn, a = π + arcsin \frac{5 - 1}{2} + 2 πn

Mostrar soluciones en forma decimal

a = 0.90455 \dots + 2 πn, a = π - 0.90455 \dots + 2 πn, a = - 0.90455 \dots + 2 πn, a = π + 0.90455 \dots + 2 πn

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