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La ecuación:
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Expresiones con funciones
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1+sinx^4-2sinx^2+a*sinx^2-2a=1 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

       4           2           2             
1 + sin (x) - 2*sin (x) + a*sin (x) - 2*a = 1

- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right) = 1

Simplificar

Solución detallada

Tenemos la ecuación

- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right) = 1

cambiamos

- a \cos^{2}{\left(x \right)} - a + \cos^{4}{\left(x \right)} - 1 = 0

\left(- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right)\right) - 1 = 0

Sustituimos

w = \sin{\left(x \right)}

Tenemos la ecuación:

a w^{2} - 2 a + w^{4} - 2 w^{2} = 0

Sustituimos

v = w^{2}

entonces la ecuación será así:

- 2 a + v^{2} + v \left(a - 2\right) = 0

Es la ecuación de la forma

a*v^2 + b*v + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:

v_{1} = \frac{\sqrt{D} - b}{2 a}

v_{2} = \frac{- \sqrt{D} - b}{2 a}

donde D = b^2 - 4*a*c es el discriminante.
Como

a = 1

b = a - 2

c = - 2 a

, entonces

D = b^2 - 4 * a * c =

(-2 + a)^2 - 4 * (1) * (-2*a) = (-2 + a)^2 + 8*a

La ecuación tiene dos raíces.

v1 = (-b + sqrt(D)) / (2*a)

v2 = (-b - sqrt(D)) / (2*a)

v_{1} = - \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1

v_{2} = - \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1

Entonces la respuesta definitiva es:
Como

v = w^{2}

entonces

w_{1} = \sqrt{v_{1}}

w_{2} = - \sqrt{v_{1}}

w_{3} = \sqrt{v_{2}}

w_{4} = - \sqrt{v_{2}}

entonces:

w_{1} =

\frac{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = \sqrt{- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}

w_{2} =

\frac{\left(-1\right) \left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}

w_{3} =

\frac{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = \sqrt{- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}

w_{4} =

\frac{\left(-1\right) \left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}

hacemos cambio inverso

\sin{\left(x \right)} = w

Tenemos la ecuación

\sin{\left(x \right)} = w

es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en

x = 2 \pi n + \operatorname{asin}{\left(w \right)}

x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi

x = 2 \pi n + \operatorname{asin}{\left(w \right)}

x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi

, donde n es cualquier número entero
sustituimos w:

x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}

x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}

x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}

x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}

x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}

x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} - 1 \right)}

x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi

x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi

x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi

x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi

x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi

x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} - 1 \right)} + \pi

Gráfica

Suma y producto de raíces [src]

suma

       /    /  ____\\       /    /  ____\\            /    /  ____\\     /    /  ____\\       /    /  ____\\       /    /  ____\\       /    /  ____\\     /    /  ____\\          /    /  ___\\       /    /  ___\\            /    /  ___\\     /    /  ___\\       /    /  ___\\       /    /  ___\\       /    /  ___\\     /    /  ___\\
pi - re\asin\\/ -a // - I*im\asin\\/ -a // + pi + I*im\asin\\/ -a // + re\asin\\/ -a // + - re\asin\\/ -a // - I*im\asin\\/ -a // + I*im\asin\\/ -a // + re\asin\\/ -a // + pi - re\asin\\/ 2 // - I*im\asin\\/ 2 // + pi + I*im\asin\\/ 2 // + re\asin\\/ 2 // + - re\asin\\/ 2 // - I*im\asin\\/ 2 // + I*im\asin\\/ 2 // + re\asin\\/ 2 //

\left(\left(\left(\left(\left(\left(\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)

4*pi

4 \pi

producto

/       /    /  ____\\       /    /  ____\\\ /         /    /  ____\\     /    /  ____\\\ /    /    /  ____\\       /    /  ____\\\ /    /    /  ____\\     /    /  ____\\\ /       /    /  ___\\       /    /  ___\\\ /         /    /  ___\\     /    /  ___\\\ /    /    /  ___\\       /    /  ___\\\ /    /    /  ___\\     /    /  ___\\\
\pi - re\asin\\/ -a // - I*im\asin\\/ -a ///*\pi + I*im\asin\\/ -a // + re\asin\\/ -a ///*\- re\asin\\/ -a // - I*im\asin\\/ -a ///*\I*im\asin\\/ -a // + re\asin\\/ -a ///*\pi - re\asin\\/ 2 // - I*im\asin\\/ 2 ///*\pi + I*im\asin\\/ 2 // + re\asin\\/ 2 ///*\- re\asin\\/ 2 // - I*im\asin\\/ 2 ///*\I*im\asin\\/ 2 // + re\asin\\/ 2 ///

\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)

                                     2                                        2                                                                                                                                                                                  
/    /    /  ___\\     /    /  ___\\\  /    /    /  ____\\     /    /  ____\\\  /         /    /  ___\\     /    /  ___\\\ /         /    /  ____\\     /    /  ____\\\ /          /    /  ___\\     /    /  ___\\\ /          /    /  ____\\     /    /  ____\\\
\I*im\asin\\/ 2 // + re\asin\\/ 2 /// *\I*im\asin\\/ -a // + re\asin\\/ -a /// *\pi + I*im\asin\\/ 2 // + re\asin\\/ 2 ///*\pi + I*im\asin\\/ -a // + re\asin\\/ -a ///*\-pi + I*im\asin\\/ 2 // + re\asin\\/ 2 ///*\-pi + I*im\asin\\/ -a // + re\asin\\/ -a ///

\left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right)^{2} \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - \pi\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right)

(i*im(asin(sqrt(2))) + re(asin(sqrt(2))))^2*(i*im(asin(sqrt(-a))) + re(asin(sqrt(-a))))^2*(pi + i*im(asin(sqrt(2))) + re(asin(sqrt(2))))*(pi + i*im(asin(sqrt(-a))) + re(asin(sqrt(-a))))*(-pi + i*im(asin(sqrt(2))) + re(asin(sqrt(2))))*(-pi + i*im(asin(sqrt(-a))) + re(asin(sqrt(-a))))

Respuesta rápida [src]

            /    /  ____\\       /    /  ____\\
x1 = pi - re\asin\\/ -a // - I*im\asin\\/ -a //

x_{1} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi

              /    /  ____\\     /    /  ____\\
x2 = pi + I*im\asin\\/ -a // + re\asin\\/ -a //

x_{2} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi

         /    /  ____\\       /    /  ____\\
x3 = - re\asin\\/ -a // - I*im\asin\\/ -a //

x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}

         /    /  ____\\     /    /  ____\\
x4 = I*im\asin\\/ -a // + re\asin\\/ -a //

x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}

            /    /  ___\\       /    /  ___\\
x5 = pi - re\asin\\/ 2 // - I*im\asin\\/ 2 //

x_{5} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}

              /    /  ___\\     /    /  ___\\
x6 = pi + I*im\asin\\/ 2 // + re\asin\\/ 2 //

x_{6} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}

         /    /  ___\\       /    /  ___\\
x7 = - re\asin\\/ 2 // - I*im\asin\\/ 2 //

x_{7} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}

         /    /  ___\\     /    /  ___\\
x8 = I*im\asin\\/ 2 // + re\asin\\/ 2 //

x_{8} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}

x8 = re(asin(sqrt(2))) + i*im(asin(sqrt(2)))

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Expresiones con funciones

1+sinx^4-2sinx^2+a*sinx^2-2a=1 la ecuación

Solución numérica:

Solución