sin(sin(x))=1 la ecuación
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Solución
Solución detallada
Tenemos la ecuación
$$\sin{\left(\sin{\left(x \right)} \right)} = 1$$
cambiamos
$$\sin{\left(\sin{\left(x \right)} \right)} - 1 = 0$$
$$\sin{\left(\sin{\left(x \right)} \right)} - 1 = 0$$
Sustituimos
$$w = \sin{\left(\sin{\left(x \right)} \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = 1$$
Obtenemos la respuesta: w = 1
hacemos cambio inverso
$$\sin{\left(\sin{\left(x \right)} \right)} = w$$
sustituimos w:
Suma y producto de raíces
[src]
/ /pi\\ / /pi\\ / /pi\\ / /pi\\
pi - re|asin|--|| - I*im|asin|--|| + I*im|asin|--|| + re|asin|--||
\ \2 // \ \2 // \ \2 // \ \2 //
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}\right)$$
$$\pi$$
/ / /pi\\ / /pi\\\ / / /pi\\ / /pi\\\
|pi - re|asin|--|| - I*im|asin|--|||*|I*im|asin|--|| + re|asin|--|||
\ \ \2 // \ \2 /// \ \ \2 // \ \2 ///
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}\right)$$
/ / /pi\\ / /pi\\\ / / /pi\\ / /pi\\\
-|I*im|asin|--|| + re|asin|--|||*|-pi + I*im|asin|--|| + re|asin|--|||
\ \ \2 // \ \2 /// \ \ \2 // \ \2 ///
$$- \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}\right)$$
-(i*im(asin(pi/2)) + re(asin(pi/2)))*(-pi + i*im(asin(pi/2)) + re(asin(pi/2)))
/ /pi\\ / /pi\\
x1 = pi - re|asin|--|| - I*im|asin|--||
\ \2 // \ \2 //
$$x_{1} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}$$
/ /pi\\ / /pi\\
x2 = I*im|asin|--|| + re|asin|--||
\ \2 // \ \2 //
$$x_{2} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\pi}{2} \right)}\right)}$$
x2 = re(asin(pi/2)) + i*im(asin(pi/2))
x1 = 1.5707963267949 + 1.02322747854755*i
x2 = 1.5707963267949 - 1.02322747854755*i
x2 = 1.5707963267949 - 1.02322747854755*i