sin(z)+cos(z)=i (уравнение)

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

Вы ввели

sin(z) + cos(z) = I

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

Быстрый ответ

           /    /            ___   ____\\         /    /            ___   ____\\
           |    |  1   I   \/ 6 *\/ -I ||         |    |  1   I   \/ 6 *\/ -I ||
z1 = - 2*re|atan|- - + - + ------------|| - 2*I*im|atan|- - + - + ------------||
           \    \  2   2        2      //         \    \  2   2        2      //

$$z_{1} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)}$$

         /    /          ___   ____\\         /    /          ___   ____\\
         |    |1   I   \/ 6 *\/ -I ||         |    |1   I   \/ 6 *\/ -I ||
z2 = 2*re|atan|- - - + ------------|| + 2*I*im|atan|- - - + ------------||
         \    \2   2        2      //         \    \2   2        2      //

$$z_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)}$$

Сумма и произведение корней

сумма

          /    /            ___   ____\\         /    /            ___   ____\\       /    /          ___   ____\\         /    /          ___   ____\\
          |    |  1   I   \/ 6 *\/ -I ||         |    |  1   I   \/ 6 *\/ -I ||       |    |1   I   \/ 6 *\/ -I ||         |    |1   I   \/ 6 *\/ -I ||
0 + - 2*re|atan|- - + - + ------------|| - 2*I*im|atan|- - + - + ------------|| + 2*re|atan|- - - + ------------|| + 2*I*im|atan|- - - + ------------||
          \    \  2   2        2      //         \    \  2   2        2      //       \    \2   2        2      //         \    \2   2        2      //

$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)}\right) - \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)}\right)$$

=

      /    /            ___   ____\\       /    /          ___   ____\\         /    /            ___   ____\\         /    /          ___   ____\\
      |    |  1   I   \/ 6 *\/ -I ||       |    |1   I   \/ 6 *\/ -I ||         |    |  1   I   \/ 6 *\/ -I ||         |    |1   I   \/ 6 *\/ -I ||
- 2*re|atan|- - + - + ------------|| + 2*re|atan|- - - + ------------|| - 2*I*im|atan|- - + - + ------------|| + 2*I*im|atan|- - - + ------------||
      \    \  2   2        2      //       \    \2   2        2      //         \    \  2   2        2      //         \    \2   2        2      //

$$- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)}$$

произведение

  /      /    /            ___   ____\\         /    /            ___   ____\\\ /    /    /          ___   ____\\         /    /          ___   ____\\\
  |      |    |  1   I   \/ 6 *\/ -I ||         |    |  1   I   \/ 6 *\/ -I ||| |    |    |1   I   \/ 6 *\/ -I ||         |    |1   I   \/ 6 *\/ -I |||
1*|- 2*re|atan|- - + - + ------------|| - 2*I*im|atan|- - + - + ------------|||*|2*re|atan|- - - + ------------|| + 2*I*im|atan|- - - + ------------|||
  \      \    \  2   2        2      //         \    \  2   2        2      /// \    \    \2   2        2      //         \    \2   2        2      ///

$$1 \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)}\right)$$

=

   /    /    /          ___   ____\\     /    /          ___   ____\\\ /    /    /            ___   ____\\     /    /            ___   ____\\\
   |    |    |1   I   \/ 6 *\/ -I ||     |    |1   I   \/ 6 *\/ -I ||| |    |    |  1   I   \/ 6 *\/ -I ||     |    |  1   I   \/ 6 *\/ -I |||
-4*|I*im|atan|- - - + ------------|| + re|atan|- - - + ------------|||*|I*im|atan|- - + - + ------------|| + re|atan|- - + - + ------------|||
   \    \    \2   2        2      //     \    \2   2        2      /// \    \    \  2   2        2      //     \    \  2   2        2      ///

$$- 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} + \frac{i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{6} \sqrt{- i}}{2} - \frac{i}{2} \right)}\right)}\right)$$

Численный ответ

z1 = 14.9225651045515 - 0.658478948462408*i

z2 = 2.35619449019234 - 0.658478948462408*i

z3 = -13.3517687777566 + 0.658478948462408*i

z4 = 36.9137136796801 + 0.658478948462408*i

z5 = 99.7455667514759 + 0.658478948462408*i

z6 = -85.6083998103219 - 0.658478948462408*i

z7 = 68.329640215578 + 0.658478948462408*i

z8 = -25.9181393921158 + 0.658478948462408*i

z9 = 80.8960108299372 + 0.658478948462408*i

z10 = -47.9092879672443 - 0.658478948462408*i

z11 = -57.3340659280137 + 0.658478948462408*i

z12 = -69.9004365423729 + 0.658478948462408*i

z13 = -38.484510006475 + 0.658478948462408*i

z14 = -44.7676953136546 + 0.658478948462408*i

z15 = -91.8915851175014 - 0.658478948462408*i

z16 = 49.4800842940392 + 0.658478948462408*i

z17 = 30.6305283725005 + 0.658478948462408*i

z18 = -19.6349540849362 + 0.658478948462408*i

z19 = -3.92699081698724 - 0.658478948462408*i

z20 = 8.63937979737193 - 0.658478948462408*i

z21 = 96.6039740978861 - 0.658478948462408*i

z22 = 40.0553063332699 - 0.658478948462408*i

z23 = 62.0464549083984 + 0.658478948462408*i

z24 = 5.49778714378214 + 0.658478948462408*i

z25 = 18.0641577581413 + 0.658478948462408*i

z26 = -79.3252145031423 - 0.658478948462408*i

z27 = 21.2057504117311 - 0.658478948462408*i

z28 = 33.7721210260903 - 0.658478948462408*i

z29 = -88.7499924639117 + 0.658478948462408*i

z30 = 90.3207887907066 - 0.658478948462408*i

z31 = 24.3473430653209 + 0.658478948462408*i

z32 = -98.174770424681 - 0.658478948462408*i

z33 = 11.7809724509617 + 0.658478948462408*i

z34 = 46.3384916404494 - 0.658478948462408*i

z35 = -51.0508806208341 + 0.658478948462408*i

z36 = 77.7544181763474 - 0.658478948462408*i

z37 = -63.6172512351933 + 0.658478948462408*i

z38 = -60.4756585816035 - 0.658478948462408*i

z39 = 71.4712328691678 - 0.658478948462408*i

z40 = -54.1924732744239 - 0.658478948462408*i

z41 = 43.1968989868597 + 0.658478948462408*i

z42 = -66.7588438887831 - 0.658478948462408*i

z43 = -73.0420291959627 - 0.658478948462408*i

z44 = 52.621676947629 - 0.658478948462408*i

z45 = -41.6261026600648 - 0.658478948462408*i

z46 = -29.0597320457056 - 0.658478948462408*i

z47 = 93.4623814442964 + 0.658478948462408*i

z48 = -7.06858347057703 + 0.658478948462408*i

z49 = 74.6128255227576 + 0.658478948462408*i

z50 = -76.1836218495525 + 0.658478948462408*i

z51 = 58.9048622548086 - 0.658478948462408*i

z52 = -10.2101761241668 - 0.658478948462408*i

z53 = -32.2013246992954 + 0.658478948462408*i

z54 = -35.3429173528852 - 0.658478948462408*i

z55 = -82.4668071567321 + 0.658478948462408*i

z56 = -16.4933614313464 - 0.658478948462408*i

z57 = -0.785398163397448 + 0.658478948462408*i

z58 = 87.1791961371168 + 0.658478948462408*i

z59 = 65.1880475619882 - 0.658478948462408*i

z60 = 84.037603483527 - 0.658478948462408*i

z61 = 27.4889357189107 - 0.658478948462408*i

z62 = -22.776546738526 - 0.658478948462408*i

z63 = -95.0331777710912 + 0.658478948462408*i

z64 = 55.7632696012188 + 0.658478948462408*i