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sin^3(x)+sinx+2^0.5=-sqrt(3)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
                              ___ 
   3                 ___   -\/ 3  
sin (x) + sin(x) + \/ 2  = -------
                              2   
$$\left(\sin^{3}{\left(x \right)} + \sin{\left(x \right)}\right) + \sqrt{2} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Gráfica
Respuesta rápida [src]
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            |    |                                                                                            /       /    108 + |27*\/ 2  + --------|          ___        ___  /          ___\||       |    |                                                                                            /       /    108 + |27*\/ 2  + --------|          ___        ___  /          ___\||
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            |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - - -------|||       |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - - -------|||
            |    |                                          1                                             \/                         2                        2          4      \  2      2   /||       |    |                                          1                                             \/                         2                        2          4      \  2      2   /||
x1 = pi - re|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------|| - I*im|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------||
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            |    |                        /       ______________________________                                                                                                               ||       |    |                        /       ______________________________                                                                                                               ||
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            |    |                     /       /         |     ___   27*\/ 3 |                                                                                                                 ||       |    |                     /       /         |     ___   27*\/ 3 |                                                                                                                 ||
            |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||       |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||
            |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||       |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||
            |    ||- - - -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||       |    ||- - - -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||
            \    \\  2      2   / \/                         2                        2          4                                                                                             //       \    \\  2      2   / \/                         2                        2          4                                                                                             //
$$x_{1} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} \right)}\right)}$$
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            |    |                                                                                           /      \/           \              2    /     27*\/ 2    27*\/ 3   |  1   I*\/ 3 |||       |    |                                                                                           /      \/           \              2    /     27*\/ 2    27*\/ 3   |  1   I*\/ 3 |||
            |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - + -------|||       |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - + -------|||
            |    |                                          1                                             \/                         2                        2          4      \  2      2   /||       |    |                                          1                                             \/                         2                        2          4      \  2      2   /||
x2 = pi - re|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------|| - I*im|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------||
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            |    |                        /       ______________________________                                                                                                               ||       |    |                        /       ______________________________                                                                                                               ||
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            |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||       |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||
            |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||       |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||
            |    ||- - + -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||       |    ||- - + -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||
            \    \\  2      2   / \/                         2                        2          4                                                                                             //       \    \\  2      2   / \/                         2                        2          4                                                                                             //
$$x_{2} = - \operatorname{re}{\left(\operatorname{asin}{\left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)}$$
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         |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - - -------|||     |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - - -------|||
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x3 = I*im|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------|| + re|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------||
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         |    |                        /       ______________________________                                                                                                               ||     |    |                        /       ______________________________                                                                                                               ||
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         |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||     |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||
         |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||     |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||
         |    ||- - - -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||     |    ||- - - -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||
         \    \\  2      2   / \/                         2                        2          4                                                                                             //     \    \\  2      2   / \/                         2                        2          4                                                                                             //
$$x_{3} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} \right)}\right)}$$
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         |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - + -------|||     |    |                                                                                        3 /       ------------------------------------ + -------- + -------- *|- - + -------|||
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x4 = I*im|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------|| + re|asin|------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------||
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         |    |                        /       ______________________________                                                                                                               ||     |    |                        /       ______________________________                                                                                                               ||
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         |    |                      /       /        /                ___\                                                                                                                 ||     |    |                      /       /        /                ___\                                                                                                                 ||
         |    |                     /       /         |     ___   27*\/ 3 |                                                                                                                 ||     |    |                     /       /         |     ___   27*\/ 3 |                                                                                                                 ||
         |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||     |    |/          ___\     /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                                         ||
         |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||     |    ||  1   I*\/ 3 |    /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                                          ||
         |    ||- - + -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||     |    ||- - + -------|*3 /       ------------------------------------ + -------- + --------                                                                                         ||
         \    \\  2      2   / \/                         2                        2          4                                                                                             //     \    \\  2      2   / \/                         2                        2          4                                                                                             //
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)}$$
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            |    |                                                                        3 /       ------------------------------------ + -------- + -------- ||       |    |                                                                        3 /       ------------------------------------ + -------- + -------- ||
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x5 = pi - re|asin|--------------------------------------------------------------------- - ---------------------------------------------------------------------|| - I*im|asin|--------------------------------------------------------------------- - ---------------------------------------------------------------------||
            |    |         ____________________________________________________________                                     3                                  ||       |    |         ____________________________________________________________                                     3                                  ||
            |    |        /       ______________________________                                                                                               ||       |    |        /       ______________________________                                                                                               ||
            |    |       /       /                            2                                                                                                ||       |    |       /       /                            2                                                                                                ||
            |    |      /       /        /                ___\                                                                                                 ||       |    |      /       /        /                ___\                                                                                                 ||
            |    |     /       /         |     ___   27*\/ 3 |                                                                                                 ||       |    |     /       /         |     ___   27*\/ 3 |                                                                                                 ||
            |    |    /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                         ||       |    |    /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                         ||
            |    |   /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                          ||       |    |   /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                          ||
            |    |3 /       ------------------------------------ + -------- + --------                                                                         ||       |    |3 /       ------------------------------------ + -------- + --------                                                                         ||
            \    \\/                         2                        2          4                                                                             //       \    \\/                         2                        2          4                                                                             //
$$x_{5} = - \operatorname{re}{\left(\operatorname{asin}{\left(- \frac{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(- \frac{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)}$$
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         |    |                                                                            /       /    108 + |27*\/ 2  + --------|          ___        ___ ||     |    |                                                                            /       /    108 + |27*\/ 2  + --------|          ___        ___ ||
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         |    |                                                                        3 /       ------------------------------------ + -------- + -------- ||     |    |                                                                        3 /       ------------------------------------ + -------- + -------- ||
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x6 = I*im|asin|--------------------------------------------------------------------- - ---------------------------------------------------------------------|| + re|asin|--------------------------------------------------------------------- - ---------------------------------------------------------------------||
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         |    |        /       ______________________________                                                                                               ||     |    |        /       ______________________________                                                                                               ||
         |    |       /       /                            2                                                                                                ||     |    |       /       /                            2                                                                                                ||
         |    |      /       /        /                ___\                                                                                                 ||     |    |      /       /        /                ___\                                                                                                 ||
         |    |     /       /         |     ___   27*\/ 3 |                                                                                                 ||     |    |     /       /         |     ___   27*\/ 3 |                                                                                                 ||
         |    |    /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                         ||     |    |    /       /    108 + |27*\/ 2  + --------|          ___        ___                                                                         ||
         |    |   /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                          ||     |    |   /      \/           \              2    /     27*\/ 2    27*\/ 3                                                                          ||
         |    |3 /       ------------------------------------ + -------- + --------                                                                         ||     |    |3 /       ------------------------------------ + -------- + --------                                                                         ||
         \    \\/                         2                        2          4                                                                             //     \    \\/                         2                        2          4                                                                             //
$$x_{6} = \operatorname{re}{\left(\operatorname{asin}{\left(- \frac{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(- \frac{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27 \sqrt{3}}{4} + \frac{27 \sqrt{2}}{2} + \frac{\sqrt{108 + \left(\frac{27 \sqrt{3}}{2} + 27 \sqrt{2}\right)^{2}}}{2}}} \right)}\right)}$$
x6 = re(asin(-(27*sqrt(3)/4 + 27*sqrt(2)/2 + sqrt(108 + (27*sqrt(3)/2 + 27*sqrt(2))^2)/2)^(1/3)/3 + (27*sqrt(3)/4 + 27*sqrt(2)/2 + sqrt(108 + (27*sqrt(3)/2 + 27*sqrt(2))^2)/2)^(-1/3))) + i*im(asin(-(27*sqrt(3)/4 + 27*sqrt(2)/2 + sqrt(108 + (27*sqrt(3)/2 + 27*sqrt(2))^2)/2)^(1/3)/3 + (27*sqrt(3)/4 + 27*sqrt(2)/2 + sqrt(108 + (27*sqrt(3)/2 + 27*sqrt(2))^2)/2)^(-1/3)))
Respuesta numérica [src]
x1 = 2.83095385521158 - 1.15517967752692*i
x2 = 2.83095385521158 + 1.15517967752692*i
x3 = 4.71238898038469 - 0.363114609498322*i
x4 = 0.310638798378211 + 1.15517967752692*i
x5 = 0.310638798378211 - 1.15517967752692*i
x6 = -1.5707963267949 + 0.363114609498322*i
x6 = -1.5707963267949 + 0.363114609498322*i