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cosx-2|cosx|=y la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
cos(x) - 2*|cos(x)| = y
$$\cos{\left(x \right)} - 2 \left|{\cos{\left(x \right)}}\right| = y$$
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Gráfica
Respuesta rápida [src] 
         //-acos(-y) + 2*pi  for y <= 0\     //-acos(-y) + 2*pi  for y <= 0\
x1 = I*im|<                            | + re|<                            |
         \\      nan         otherwise /     \\      nan         otherwise /
$$x_{1} = \operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
         //      /y\                  \     //      /y\                  \
         ||- acos|-| + 2*pi  for y < 0|     ||- acos|-| + 2*pi  for y < 0|
x2 = I*im|<      \3/                  | + re|<      \3/                  |
         ||                           |     ||                           |
         \\      nan         otherwise/     \\      nan         otherwise/
$$x_{2} = \operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
         //acos(-y)  for y <= 0\     //acos(-y)  for y <= 0\
x3 = I*im|<                    | + re|<                    |
         \\  nan     otherwise /     \\  nan     otherwise /
$$x_{3} = \operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
         //    /y\           \     //    /y\           \
         ||acos|-|  for y < 0|     ||acos|-|  for y < 0|
x4 = I*im|<    \3/           | + re|<    \3/           |
         ||                  |     ||                  |
         \\  nan    otherwise/     \\  nan    otherwise/
$$x_{4} = \operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
x4 = re(Piecewise((acos(y/3, y < 0), (nan, True))) + i*im(Piecewise((acos(y/3), y < 0), (nan, True))))
Suma y producto de raíces [src] 
suma
                                                                              //      /y\                  \     //      /y\                  \                                                                 //    /y\           \     //    /y\           \
    //-acos(-y) + 2*pi  for y <= 0\     //-acos(-y) + 2*pi  for y <= 0\       ||- acos|-| + 2*pi  for y < 0|     ||- acos|-| + 2*pi  for y < 0|       //acos(-y)  for y <= 0\     //acos(-y)  for y <= 0\       ||acos|-|  for y < 0|     ||acos|-|  for y < 0|
I*im|<                            | + re|<                            | + I*im|<      \3/                  | + re|<      \3/                  | + I*im|<                    | + re|<                    | + I*im|<    \3/           | + re|<    \3/           |
    \\      nan         otherwise /     \\      nan         otherwise /       ||                           |     ||                           |       \\  nan     otherwise /     \\  nan     otherwise /       ||                  |     ||                  |
                                                                              \\      nan         otherwise/     \\      nan         otherwise/                                                                 \\  nan    otherwise/     \\  nan    otherwise/
$$\left(\left(\left(\operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$ 
=
                                          //      /y\                  \                                     //    /y\           \                                         //      /y\                  \                                 //    /y\           \
    //-acos(-y) + 2*pi  for y <= 0\       ||- acos|-| + 2*pi  for y < 0|       //acos(-y)  for y <= 0\       ||acos|-|  for y < 0|     //-acos(-y) + 2*pi  for y <= 0\     ||- acos|-| + 2*pi  for y < 0|     //acos(-y)  for y <= 0\     ||acos|-|  for y < 0|
I*im|<                            | + I*im|<      \3/                  | + I*im|<                    | + I*im|<    \3/           | + re|<                            | + re|<      \3/                  | + re|<                    | + re|<    \3/           |
    \\      nan         otherwise /       ||                           |       \\  nan     otherwise /       ||                  |     \\      nan         otherwise /     ||                           |     \\  nan     otherwise /     ||                  |
                                          \\      nan         otherwise/                                     \\  nan    otherwise/                                         \\      nan         otherwise/                                 \\  nan    otherwise/
$$\operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
producto
                                                                          /    //      /y\                  \     //      /y\                  \\                                                           /    //    /y\           \     //    /y\           \\
/    //-acos(-y) + 2*pi  for y <= 0\     //-acos(-y) + 2*pi  for y <= 0\\ |    ||- acos|-| + 2*pi  for y < 0|     ||- acos|-| + 2*pi  for y < 0|| /    //acos(-y)  for y <= 0\     //acos(-y)  for y <= 0\\ |    ||acos|-|  for y < 0|     ||acos|-|  for y < 0||
|I*im|<                            | + re|<                            ||*|I*im|<      \3/                  | + re|<      \3/                  ||*|I*im|<                    | + re|<                    ||*|I*im|<    \3/           | + re|<    \3/           ||
\    \\      nan         otherwise /     \\      nan         otherwise // |    ||                           |     ||                           || \    \\  nan     otherwise /     \\  nan     otherwise // |    ||                  |     ||                  ||
                                                                          \    \\      nan         otherwise/     \\      nan         otherwise//                                                           \    \\  nan    otherwise/     \\  nan    otherwise//
$$\left(\operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(- y \right)} + 2 \pi & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \operatorname{acos}{\left(\frac{y}{3} \right)} + 2 \pi & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(- y \right)} & \text{for}\: y \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \operatorname{acos}{\left(\frac{y}{3} \right)} & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$ 
=
/                                /    /    /y\\     /    /y\\\                                         /            /    /y\\     /    /y\\\           
|(I*im(acos(-y)) + re(acos(-y)))*|I*im|acos|-|| + re|acos|-|||*(-2*pi + I*im(acos(-y)) + re(acos(-y)))*|-2*pi + I*im|acos|-|| + re|acos|-|||  for y < 0
<                                \    \    \3//     \    \3///                                         \            \    \3//     \    \3///           
|                                                                                                                                                      
\                                                                    nan                                                                      otherwise
$$\begin{cases} \left(\operatorname{re}{\left(\operatorname{acos}{\left(- y \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- y \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{y}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{y}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- y \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- y \right)}\right)} - 2 \pi\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{y}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{y}{3} \right)}\right)} - 2 \pi\right) & \text{for}\: y < 0 \\\text{NaN} & \text{otherwise} \end{cases}$$ 
Piecewise(((i*im(acos(-y)) + re(acos(-y)))*(i*im(acos(y/3)) + re(acos(y/3)))*(-2*pi + i*im(acos(-y)) + re(acos(-y)))*(-2*pi + i*im(acos(y/3)) + re(acos(y/3))), y < 0), (nan, True))
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