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tan^2(x)+csc^2(x)-3=0

Solución

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

Solución

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

Grados

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

Pasos de solución

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{1}{cot ( x )}

= - 3 + csc^{2} (x) + (\frac{1}{cot ( x )})^{2}

(\frac{1}{cot ( x )})^{2} = \frac{1}{cot ^{2} ( x )}

(\frac{1}{cot ( x )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{cot ^{2} ( x )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( x )}

= - 3 + csc^{2} (x) + \frac{1}{cot ^{2} ( x )}

Utilizar la identidad pitagórica:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= - 3 + csc^{2} (x) + \frac{1}{csc ^{2} ( x ) - 1}

- 3 + csc^{2} (x) + \frac{1}{- 1 + csc ^{2} ( x )} = 0

Usando el método de sustitución

- 3 + csc^{2} (x) + \frac{1}{- 1 + csc ^{2} ( x )} = 0

Sea:

csc (x) = u

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0 : u = 2, u = - 2

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0

Multiplicar ambos lados por

- 1 + u^{2}

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0

Multiplicar ambos lados por

- 1 + u^{2}

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + \frac{1}{- 1 + u ^{2}} (- 1 + u^{2}) = 0 \cdot (- 1 + u^{2})

Simplificar

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + \frac{1}{- 1 + u ^{2}} (- 1 + u^{2}) = 0 \cdot (- 1 + u^{2})

Simplificar

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

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

Multiplicar fracciones:

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

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

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}

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

Aplicar la regla

\frac{a}{a} = 1

= 1

Simplificar

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

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

Aplicar la regla

0 \cdot a = 0

= 0

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

Resolver

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0 : u = 2, u = - 2

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

Desarrollar

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

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1

Expandir

- 3 (- 1 + u^{2}) : 3 - 3 u^{2}

- 3 (- 1 + u^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = - 3, b = - 1, c = u^{2}

= - 3 (- 1) + (- 3) u^{2}

Aplicar las reglas de los signos

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

= 3 \cdot 1 - 3 u^{2}

Multiplicar los numeros:

3 \cdot 1 = 3

= 3 - 3 u^{2}

= 3 - 3 u^{2} + u^{2} (- 1 + u^{2}) + 1

Expandir

u^{2} (- 1 + u^{2}) : - u^{2} + u^{4}

u^{2} (- 1 + u^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = u^{2}, b = - 1, c = u^{2}

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

Aplicar las reglas de los signos

+ (- a) = - a

= - 1 \cdot u^{2} + u^{2} u^{2}

Simplificar

- 1 \cdot u^{2} + u^{2} u^{2} : - u^{2} + u^{4}

- 1 \cdot u^{2} + u^{2} u^{2}

1 \cdot u^{2} = u^{2}

1 \cdot u^{2}

Multiplicar:

1 \cdot u^{2} = u^{2}

= u^{2}

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Aplicar las leyes de los exponentes:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Sumar:

2 + 2 = 4

= u^{4}

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

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

= 3 - 3 u^{2} - u^{2} + u^{4} + 1

Simplificar

3 - 3 u^{2} - u^{2} + u^{4} + 1 : u^{4} - 4 u^{2} + 4

3 - 3 u^{2} - u^{2} + u^{4} + 1

Sumar elementos similares:

- 3 u^{2} - u^{2} = - 4 u^{2}

= 3 - 4 u^{2} + u^{4} + 1

Agrupar términos semejantes

= u^{4} - 4 u^{2} + 3 + 1

Sumar:

3 + 1 = 4

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

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

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

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

v^{2} - 4 v + 4 = 0

Resolver

v^{2} - 4 v + 4 = 0 : v = 2

v^{2} - 4 v + 4 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

v^{2} - 4 v + 4 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = - 4, c = 4

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

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

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

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

Aplicar las leyes de los exponentes:

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

n

es par

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

= 4^{2} - 4 \cdot 1 \cdot 4

Multiplicar los numeros:

4 \cdot 1 \cdot 4 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Restar:

16 - 16 = 0

= 0

v_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 \cdot 1}

v = \frac{- ( - 4 )}{2 \cdot 1}

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

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

Aplicar la regla

- (- a) = a

= \frac{4}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{4}{2}

Dividir:

\frac{4}{2} = 2

= 2

v = 2

La solución a la ecuación de segundo grado es:

v = 2

v = 2

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

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

u^{2} = 2

Para

x^{2} = f (a)

las soluciones son

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

u = 2, u = - 2

Las soluciones son

u = 2, u = - 2

u = 2, u = - 2

Verificar las soluciones

Encontrar los puntos no definidos (singularidades):

u = 1, u = - 1

Tomar el(los) denominador(es) de

- 3 + u^{2} + \frac{1}{- 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

Sumar

1

a ambos lados

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

Sustituir en la ecuación

u = csc (x)

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

csc (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = 2

Soluciones generales para

csc (x) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 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

csc (x) = - 2 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

csc (x) = - 2

Soluciones generales para

csc (x) = - 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 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

Combinar toda las soluciones

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

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