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La ecuación:
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2sin(x)cos(x)-1,1=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

                  11    
2*sin(x)*cos(x) - -- = 0
                  10

2 \sin{\left(x \right)} \cos{\left(x \right)} - \frac{11}{10} = 0

Simplificar

Gráfica

Trazar el gráfico f1(x)

Trazar el gráfico f2(x)

Respuesta rápida [src]

         /    /                             _______________\\         /    /                             _______________\\
         |    |           ____       ___   /          ____ ||         |    |           ____       ___   /          ____ ||
         |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||
x1 = 2*re|atan|- -- + -------- + --------------------------|| + 2*I*im|atan|- -- + -------- + --------------------------||
         \    \  11      11                  11            //         \    \  11      11                  11            //

x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}

           /    /                           _______________\\         /    /                           _______________\\
           |    |         ____       ___   /          ____ ||         |    |         ____       ___   /          ____ ||
           |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||
x2 = - 2*re|atan|-- - -------- + --------------------------|| - 2*I*im|atan|-- - -------- + --------------------------||
           \    \11      11                  11            //         \    \11      11                  11            //

x_{2} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)}

           /    /                _______________           \\         /    /                _______________           \\
           |    |         ___   /          ____        ____||         |    |         ___   /          ____        ____||
           |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||         |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||
x3 = - 2*re|atan|-- - -------------------------- + --------|| - 2*I*im|atan|-- - -------------------------- + --------||
           \    \11               11                  11   //         \    \11               11                  11   //

x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}

           /    /                           _______________\\         /    /                           _______________\\
           |    |         ____       ___   /          ____ ||         |    |         ____       ___   /          ____ ||
           |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||
x4 = - 2*re|atan|-- + -------- + --------------------------|| - 2*I*im|atan|-- + -------- + --------------------------||
           \    \11      11                  11            //         \    \11      11                  11            //

x_{4} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}

x4 = -2*re(atan(10/11 + 2*sqrt(5)*sqrt(10 + sqrt(21)*i)/11 + sqrt(21)*i/11)) - 2*i*im(atan(10/11 + 2*sqrt(5)*sqrt(10 + sqrt(21)*i)/11 + sqrt(21)*i/11))

Suma y producto de raíces [src]

suma

    /    /                             _______________\\         /    /                             _______________\\         /    /                           _______________\\         /    /                           _______________\\         /    /                _______________           \\         /    /                _______________           \\         /    /                           _______________\\         /    /                           _______________\\
    |    |           ____       ___   /          ____ ||         |    |           ____       ___   /          ____ ||         |    |         ____       ___   /          ____ ||         |    |         ____       ___   /          ____ ||         |    |         ___   /          ____        ____||         |    |         ___   /          ____        ____||         |    |         ____       ___   /          ____ ||         |    |         ____       ___   /          ____ ||
    |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||         |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||
2*re|atan|- -- + -------- + --------------------------|| + 2*I*im|atan|- -- + -------- + --------------------------|| + - 2*re|atan|-- - -------- + --------------------------|| - 2*I*im|atan|-- - -------- + --------------------------|| + - 2*re|atan|-- - -------------------------- + --------|| - 2*I*im|atan|-- - -------------------------- + --------|| + - 2*re|atan|-- + -------- + --------------------------|| - 2*I*im|atan|-- + -------- + --------------------------||
    \    \  11      11                  11            //         \    \  11      11                  11            //         \    \11      11                  11            //         \    \11      11                  11            //         \    \11               11                  11   //         \    \11               11                  11   //         \    \11      11                  11            //         \    \11      11                  11            //

\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right) + \left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right) + \left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right)\right)\right)

      /    /                _______________           \\       /    /                           _______________\\       /    /                           _______________\\       /    /                             _______________\\         /    /                _______________           \\         /    /                           _______________\\         /    /                           _______________\\         /    /                             _______________\\
      |    |         ___   /          ____        ____||       |    |         ____       ___   /          ____ ||       |    |         ____       ___   /          ____ ||       |    |           ____       ___   /          ____ ||         |    |         ___   /          ____        ____||         |    |         ____       ___   /          ____ ||         |    |         ____       ___   /          ____ ||         |    |           ____       ___   /          ____ ||
      |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||       |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||       |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||       |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||         |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||
- 2*re|atan|-- - -------------------------- + --------|| - 2*re|atan|-- - -------- + --------------------------|| - 2*re|atan|-- + -------- + --------------------------|| + 2*re|atan|- -- + -------- + --------------------------|| - 2*I*im|atan|-- - -------------------------- + --------|| - 2*I*im|atan|-- - -------- + --------------------------|| - 2*I*im|atan|-- + -------- + --------------------------|| + 2*I*im|atan|- -- + -------- + --------------------------||
      \    \11               11                  11   //       \    \11      11                  11            //       \    \11      11                  11            //       \    \  11      11                  11            //         \    \11               11                  11   //         \    \11      11                  11            //         \    \11      11                  11            //         \    \  11      11                  11            //

- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}

producto

/    /    /                             _______________\\         /    /                             _______________\\\ /      /    /                           _______________\\         /    /                           _______________\\\ /      /    /                _______________           \\         /    /                _______________           \\\ /      /    /                           _______________\\         /    /                           _______________\\\
|    |    |           ____       ___   /          ____ ||         |    |           ____       ___   /          ____ ||| |      |    |         ____       ___   /          ____ ||         |    |         ____       ___   /          ____ ||| |      |    |         ___   /          ____        ____||         |    |         ___   /          ____        ____||| |      |    |         ____       ___   /          ____ ||         |    |         ____       ___   /          ____ |||
|    |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||| |      |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||| |      |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||         |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||| |      |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||         |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  |||
|2*re|atan|- -- + -------- + --------------------------|| + 2*I*im|atan|- -- + -------- + --------------------------|||*|- 2*re|atan|-- - -------- + --------------------------|| - 2*I*im|atan|-- - -------- + --------------------------|||*|- 2*re|atan|-- - -------------------------- + --------|| - 2*I*im|atan|-- - -------------------------- + --------|||*|- 2*re|atan|-- + -------- + --------------------------|| - 2*I*im|atan|-- + -------- + --------------------------|||
\    \    \  11      11                  11            //         \    \  11      11                  11            /// \      \    \11      11                  11            //         \    \11      11                  11            /// \      \    \11               11                  11   //         \    \11               11                  11   /// \      \    \11      11                  11            //         \    \11      11                  11            ///

\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right)

    /    /    /                             _______________\\     /    /                             _______________\\\ /    /    /                _______________           \\     /    /                _______________           \\\ /    /    /                           _______________\\     /    /                           _______________\\\ /    /    /                           _______________\\     /    /                           _______________\\\
    |    |    |           ____       ___   /          ____ ||     |    |           ____       ___   /          ____ ||| |    |    |         ___   /          ____        ____||     |    |         ___   /          ____        ____||| |    |    |         ____       ___   /          ____ ||     |    |         ____       ___   /          ____ ||| |    |    |         ____       ___   /          ____ ||     |    |         ____       ___   /          ____ |||
    |    |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||     |    |  10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||| |    |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||     |    |10   2*\/ 5 *\/  10 + I*\/ 21     I*\/ 21 ||| |    |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||     |    |10   I*\/ 21    2*\/ 5 *\/  10 - I*\/ 21  ||| |    |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  ||     |    |10   I*\/ 21    2*\/ 5 *\/  10 + I*\/ 21  |||
-16*|I*im|atan|- -- + -------- + --------------------------|| + re|atan|- -- + -------- + --------------------------|||*|I*im|atan|-- - -------------------------- + --------|| + re|atan|-- - -------------------------- + --------|||*|I*im|atan|-- - -------- + --------------------------|| + re|atan|-- - -------- + --------------------------|||*|I*im|atan|-- + -------- + --------------------------|| + re|atan|-- + -------- + --------------------------|||
    \    \    \  11      11                  11            //     \    \  11      11                  11            /// \    \    \11               11                  11   //     \    \11               11                  11   /// \    \    \11      11                  11            //     \    \11      11                  11            /// \    \    \11      11                  11            //     \    \11      11                  11            ///

- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{\sqrt{21} i}{11} + \frac{2 \sqrt{5} \sqrt{10 - \sqrt{21} i}}{11} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} - \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{10}{11} + \frac{2 \sqrt{5} \sqrt{10 + \sqrt{21} i}}{11} + \frac{\sqrt{21} i}{11} \right)}\right)}\right)

-16*(i*im(atan(-10/11 + i*sqrt(21)/11 + 2*sqrt(5)*sqrt(10 - i*sqrt(21))/11)) + re(atan(-10/11 + i*sqrt(21)/11 + 2*sqrt(5)*sqrt(10 - i*sqrt(21))/11)))*(i*im(atan(10/11 - 2*sqrt(5)*sqrt(10 + i*sqrt(21))/11 + i*sqrt(21)/11)) + re(atan(10/11 - 2*sqrt(5)*sqrt(10 + i*sqrt(21))/11 + i*sqrt(21)/11)))*(i*im(atan(10/11 - i*sqrt(21)/11 + 2*sqrt(5)*sqrt(10 - i*sqrt(21))/11)) + re(atan(10/11 - i*sqrt(21)/11 + 2*sqrt(5)*sqrt(10 - i*sqrt(21))/11)))*(i*im(atan(10/11 + i*sqrt(21)/11 + 2*sqrt(5)*sqrt(10 + i*sqrt(21))/11)) + re(atan(10/11 + i*sqrt(21)/11 + 2*sqrt(5)*sqrt(10 + i*sqrt(21))/11)))

Respuesta numérica [src]

x1 = 0.785398163397448 + 0.221784127192558*i

x2 = -2.35619449019234 + 0.221784127192558*i

x3 = 0.785398163397448 - 0.221784127192558*i

x4 = -2.35619449019234 - 0.221784127192558*i

x4 = -2.35619449019234 - 0.221784127192558*i

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2*sin(x)*cos(x)-1,1=0 la ecuación

Solución numérica:

Solución

2sin(x)cos(x)-1,1=0 la ecuación