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atan(y)=sqrt(4-x^2) la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
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atan(y) = \/  4 - x  
$$\operatorname{atan}{\left(y \right)} = \sqrt{4 - x^{2}}$$
Solución detallada
Tenemos la ecuación
$$\operatorname{atan}{\left(y \right)} = \sqrt{4 - x^{2}}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- \sqrt{4 - x^{2}} = - \operatorname{atan}{\left(y \right)}$$
Elevemos las dos partes de la ecuación a la potencia 2
$$4 - x^{2} = \operatorname{atan}^{2}{\left(y \right)}$$
$$4 - x^{2} = \operatorname{atan}^{2}{\left(y \right)}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- x^{2} - \operatorname{atan}^{2}{\left(y \right)} + 4 = 0$$
Es la ecuación de la forma
a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = 0$$
$$c = 4 - \operatorname{atan}^{2}{\left(y \right)}$$
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (-1) * (4 - atan(y)^2) = 16 - 4*atan(y)^2

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
$$x_{1} = - \frac{\sqrt{16 - 4 \operatorname{atan}^{2}{\left(y \right)}}}{2}$$
$$x_{2} = \frac{\sqrt{16 - 4 \operatorname{atan}^{2}{\left(y \right)}}}{2}$$
Gráfica
Suma y producto de raíces [src]
suma
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  4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|     4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|   4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|     4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|
- \/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *cos|------------------------------------------------------------------| - I*\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *sin|------------------------------------------------------------------| + \/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *cos|------------------------------------------------------------------| + I*\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *sin|------------------------------------------------------------------|
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$$\left(- i \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)}\right)$$
=
0
$$0$$
producto
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|     /                                  2                                   /     /                                  2              2         \\        /                                  2                                   /     /                                  2              2         \\| |   /                                  2                                   /     /                                  2              2         \\        /                                  2                                   /     /                                  2              2         \\|
|  4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|     4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|| |4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|     4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/||
|- \/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *cos|------------------------------------------------------------------| - I*\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *sin|------------------------------------------------------------------||*|\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *cos|------------------------------------------------------------------| + I*\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *sin|------------------------------------------------------------------||
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$$\left(- i \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)}\right)$$
=
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    /                                  2                                        /                                  2              2         \
   /  /      2              2         \        2            2            I*atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/
-\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *e                                                                    
$$- \sqrt{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}$$
-sqrt((4 + im(atan(y))^2 - re(atan(y))^2)^2 + 4*im(atan(y))^2*re(atan(y))^2)*exp(i*atan2(-2*im(atan(y))*re(atan(y)), 4 + im(atan(y))^2 - re(atan(y))^2))
Respuesta rápida [src]
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          /                                  2                                   /     /                                  2              2         \\        /                                  2                                   /     /                                  2              2         \\
       4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|     4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|
x1 = - \/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *cos|------------------------------------------------------------------| - I*\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *sin|------------------------------------------------------------------|
                                                                                 \                                2                                 /                                                                               \                                2                                 /
$$x_{1} = - i \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)}$$
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     4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|     4 /  /      2              2         \        2            2              |atan2\-2*im(atan(y))*re(atan(y)), 4 + im (atan(y)) - re (atan(y))/|
x2 = \/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *cos|------------------------------------------------------------------| + I*\/   \4 + im (atan(y)) - re (atan(y))/  + 4*im (atan(y))*re (atan(y)) *sin|------------------------------------------------------------------|
                                                                               \                                2                                 /                                                                               \                                2                                 /
$$x_{2} = i \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)} \operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)},- \left(\operatorname{re}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{atan}{\left(y \right)}\right)}\right)^{2} + 4 \right)}}{2} \right)}$$
x2 = i*((-re(atan(y))^2 + im(atan(y))^2 + 4)^2 + 4*re(atan(y))^2*im(atan(y))^2)^(1/4)*sin(atan2(-2*re(atan(y))*im(atan(y), -re(atan(y))^2 + im(atan(y))^2 + 4)/2) + ((-re(atan(y))^2 + im(atan(y))^2 + 4)^2 + 4*re(atan(y))^2*im(atan(y))^2)^(1/4)*cos(atan2(-2*re(atan(y))*im(atan(y)), -re(atan(y))^2 + im(atan(y))^2 + 4)/2))