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2(arcsin(x))^2+(2-pi)arcsin(x)-pi=0

Solución

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

Solución

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4}), x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

Pasos de solución

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

Usando el método de sustitución

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

Sea:

arcsin (x) = u

2 u^{2} + (2 - π) u - π = 0

2 u^{2} + (2 - π) u - π = 0 : u = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, u = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

2 u^{2} + (2 - π) u - π = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

2 u^{2} + (2 - π) u - π = 0

Formula general para ecuaciones de segundo grado:

Para

a = 2, b = 2 - π, c = - π

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

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

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

(2 - π)^{2} - 4 \cdot 2 (- π)

Aplicar la regla

- (- a) = a

= (2 - π)^{2} + 4 \cdot 2 π

Multiplicar los numeros:

4 \cdot 2 = 8

= (- π + 2)^{2} + 8 π

Expandir

(2 - π)^{2} + 8 π : π^{2} + 4 π + 4

(2 - π)^{2} + 8 π

(2 - π)^{2} : 4 - 4 π + π^{2}

Aplicar la formula del binomio al cuadrado:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2, b = π

= 2^{2} - 2 \cdot 2 π + π^{2}

Simplificar

2^{2} - 2 \cdot 2 π + π^{2} : 4 - 4 π + π^{2}

2^{2} - 2 \cdot 2 π + π^{2}

2^{2} = 4

= 4 - 2 \cdot 2 π + π^{2}

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 - 4 π + π^{2}

= 4 - 4 π + π^{2}

= 4 - 4 π + π^{2} + 8 π

Simplificar

4 - 4 π + π^{2} + 8 π : π^{2} + 4 π + 4

4 - 4 π + π^{2} + 8 π

Agrupar términos semejantes

= π^{2} - 4 π + 8 π + 4

Sumar elementos similares:

- 4 π + 8 π = 4 π

= π^{2} + 4 π + 4

= π^{2} + 4 π + 4

= π^{2} + 4 π + 4

u_{1, 2} = \frac{- ( 2 - π ) \pm π ^{2} + 4 π + 4}{2 \cdot 2}

Separar las soluciones

u_{1} = \frac{- ( 2 - π ) + π ^{2} + 4 π + 4}{2 \cdot 2}, u_{2} = \frac{- ( 2 - π ) - π ^{2} + 4 π + 4}{2 \cdot 2}

u = \frac{- ( 2 - π ) + π ^{2} + 4 π + 4}{2 \cdot 2} : \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

\frac{- ( 2 - π ) + π ^{2} + 4 π + 4}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{- ( - π + 2 ) + π ^{2} + 4 π + 4}{4}

- (2 - π) : - 2 + π

- (2 - π)

Poner los parentesis

= - (2) - (- π)

Aplicar las reglas de los signos

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

= - 2 + π

= \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

u = \frac{- ( 2 - π ) - π ^{2} + 4 π + 4}{2 \cdot 2} : \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

\frac{- ( 2 - π ) - π ^{2} + 4 π + 4}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{- ( - π + 2 ) - π ^{2} + 4 π + 4}{4}

- (2 - π) : - 2 + π

- (2 - π)

Poner los parentesis

= - (2) - (- π)

Aplicar las reglas de los signos

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

= - 2 + π

= \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

Las soluciones a la ecuación de segundo grado son:

u = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, u = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

Sustituir en la ecuación

u = arcsin (x)

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4} : x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4})

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

Aplicar propiedades trigonométricas inversas

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

arcsin (x) = a \Rightarrow x = sin (a)

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4})

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4})

arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4} : x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

Aplicar propiedades trigonométricas inversas

arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

arcsin (x) = a \Rightarrow x = sin (a)

x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

Combinar toda las soluciones

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4}), x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

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