Solución detallada
Tenemos la ecuación
$$\frac{\sin{\left(\pi x \right)}}{3} = - \frac{1}{2}$$
es la ecuación trigonométrica más simple
Dividamos ambos miembros de la ecuación en 1/3
La ecuación se convierte en
$$\sin{\left(\pi x \right)} = - \frac{3}{2}$$
Como el miembro derecho de la ecuación
en el módulo =
True
pero sin
no puede ser más de 1 o menos de -1
significa que la ecuación correspondiente no tiene solución.
Suma y producto de raíces
[src]
pi + re(asin(3/2)) I*im(asin(3/2)) re(asin(3/2)) I*im(asin(3/2))
------------------ + --------------- + - ------------- - ---------------
pi pi pi pi
$$\left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi}\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi}\right)$$
pi + re(asin(3/2)) re(asin(3/2))
------------------ - -------------
pi pi
$$- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi} + \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi}{\pi}$$
/pi + re(asin(3/2)) I*im(asin(3/2))\ / re(asin(3/2)) I*im(asin(3/2))\
|------------------ + ---------------|*|- ------------- - ---------------|
\ pi pi / \ pi pi /
$$\left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi}\right) \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi}\right)$$
-(I*im(asin(3/2)) + re(asin(3/2)))*(pi + I*im(asin(3/2)) + re(asin(3/2)))
--------------------------------------------------------------------------
2
pi
$$- \frac{\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right)}{\pi^{2}}$$
-(i*im(asin(3/2)) + re(asin(3/2)))*(pi + i*im(asin(3/2)) + re(asin(3/2)))/pi^2
pi + re(asin(3/2)) I*im(asin(3/2))
x1 = ------------------ + ---------------
pi pi
$$x_{1} = \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi}$$
re(asin(3/2)) I*im(asin(3/2))
x2 = - ------------- - ---------------
pi pi
$$x_{2} = - \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}}{\pi}$$
x2 = -re(asin(3/2))/pi - i*im(asin(3/2))/pi