sin(z)+cos(z)=3 (уравнение)

Найду корень уравнения: sin(z)+cos(z)=3

Вы ввели

sin(z) + cos(z) = 3

$$\sin{\left(z \right)} + \cos{\left(z \right)} = 3$$

Быстрый ответ

         /    /        ___\\         /    /        ___\\
         |    |1   I*\/ 7 ||         |    |1   I*\/ 7 ||
z1 = 2*re|atan|- - -------|| + 2*I*im|atan|- - -------||
         \    \4      4   //         \    \4      4   //

$$z_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}$$

         /    /        ___\\         /    /        ___\\
         |    |1   I*\/ 7 ||         |    |1   I*\/ 7 ||
z2 = 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
         \    \4      4   //         \    \4      4   //

$$z_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}$$

Сумма и произведение корней

сумма

        /    /        ___\\         /    /        ___\\       /    /        ___\\         /    /        ___\\
        |    |1   I*\/ 7 ||         |    |1   I*\/ 7 ||       |    |1   I*\/ 7 ||         |    |1   I*\/ 7 ||
0 + 2*re|atan|- - -------|| + 2*I*im|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
        \    \4      4   //         \    \4      4   //       \    \4      4   //         \    \4      4   //

$$\left(0 + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right)$$

=

    /    /        ___\\       /    /        ___\\         /    /        ___\\         /    /        ___\\
    |    |1   I*\/ 7 ||       |    |1   I*\/ 7 ||         |    |1   I*\/ 7 ||         |    |1   I*\/ 7 ||
2*re|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- - -------|| + 2*I*im|atan|- + -------||
    \    \4      4   //       \    \4      4   //         \    \4      4   //         \    \4      4   //

$$2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}$$

произведение

  /    /    /        ___\\         /    /        ___\\\ /    /    /        ___\\         /    /        ___\\\
  |    |    |1   I*\/ 7 ||         |    |1   I*\/ 7 ||| |    |    |1   I*\/ 7 ||         |    |1   I*\/ 7 |||
1*|2*re|atan|- - -------|| + 2*I*im|atan|- - -------|||*|2*re|atan|- + -------|| + 2*I*im|atan|- + -------|||
  \    \    \4      4   //         \    \4      4   /// \    \    \4      4   //         \    \4      4   ///

$$1 \cdot \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right)$$

=

  /    /    /        ___\\     /    /        ___\\\ /    /    /        ___\\     /    /        ___\\\
  |    |    |1   I*\/ 7 ||     |    |1   I*\/ 7 ||| |    |    |1   I*\/ 7 ||     |    |1   I*\/ 7 |||
4*|I*im|atan|- - -------|| + re|atan|- - -------|||*|I*im|atan|- + -------|| + re|atan|- + -------|||
  \    \    \4      4   //     \    \4      4   /// \    \    \4      4   //     \    \4      4   ///

$$4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right)$$

Численный ответ

z1 = 0.785398163397448 - 1.38432969165679*i

z2 = 0.785398163397448 + 1.38432969165679*i

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