Sr Examen

sin(z)+cos(z)=i la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src]
sin(z) + cos(z) = I
$$\sin{\left(z \right)} + \cos{\left(z \right)} = i$$
Gráfica
Respuesta rápida [src]
           /    /            ___        \\         /    /            ___        \\
           |    |  1   I   \/ 3 *(1 - I)||         |    |  1   I   \/ 3 *(1 - I)||
z1 = - 2*re|atan|- - + - + -------------|| - 2*I*im|atan|- - + - + -------------||
           \    \  2   2         2      //         \    \  2   2         2      //
$$z_{1} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)}$$
         /    /          ___        \\         /    /          ___        \\
         |    |1   I   \/ 3 *(1 - I)||         |    |1   I   \/ 3 *(1 - I)||
z2 = 2*re|atan|- - - + -------------|| + 2*I*im|atan|- - - + -------------||
         \    \2   2         2      //         \    \2   2         2      //
$$z_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)}$$
z2 = 2*re(atan(1/2 + sqrt(3)*(1 - i)/2 - i/2)) + 2*i*im(atan(1/2 + sqrt(3)*(1 - i)/2 - i/2))
Suma y producto de raíces [src]
suma
      /    /            ___        \\         /    /            ___        \\       /    /          ___        \\         /    /          ___        \\
      |    |  1   I   \/ 3 *(1 - I)||         |    |  1   I   \/ 3 *(1 - I)||       |    |1   I   \/ 3 *(1 - I)||         |    |1   I   \/ 3 *(1 - I)||
- 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      //
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)}\right) + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)}\right)$$
=
      /    /            ___        \\       /    /          ___        \\         /    /            ___        \\         /    /          ___        \\
      |    |  1   I   \/ 3 *(1 - I)||       |    |1   I   \/ 3 *(1 - I)||         |    |  1   I   \/ 3 *(1 - I)||         |    |1   I   \/ 3 *(1 - 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      //
$$- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)}$$
producto
/      /    /            ___        \\         /    /            ___        \\\ /    /    /          ___        \\         /    /          ___        \\\
|      |    |  1   I   \/ 3 *(1 - I)||         |    |  1   I   \/ 3 *(1 - I)||| |    |    |1   I   \/ 3 *(1 - I)||         |    |1   I   \/ 3 *(1 - I)|||
|- 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      ///
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)}\right)$$
=
   /    /    /          ___        \\     /    /          ___        \\\ /    /    /            ___        \\     /    /            ___        \\\
   |    |    |1   I   \/ 3 *(1 - I)||     |    |1   I   \/ 3 *(1 - I)||| |    |    |  1   I   \/ 3 *(1 - I)||     |    |  1   I   \/ 3 *(1 - I)|||
-4*|I*im|atan|- - - + -------------|| + re|atan|- - - + -------------|||*|I*im|atan|- - + - + -------------|| + re|atan|- - + - + -------------|||
   \    \    \2   2         2      //     \    \2   2         2      /// \    \    \  2   2         2      //     \    \  2   2         2      ///
$$- 4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} + \frac{i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} \left(1 - i\right)}{2} - \frac{i}{2} \right)}\right)}\right)$$
-4*(i*im(atan(1/2 - i/2 + sqrt(3)*(1 - i)/2)) + re(atan(1/2 - i/2 + sqrt(3)*(1 - i)/2)))*(i*im(atan(-1/2 + i/2 + sqrt(3)*(1 - i)/2)) + re(atan(-1/2 + i/2 + sqrt(3)*(1 - i)/2)))
Respuesta numérica [src]
z1 = -0.785398163397448 + 0.658478948462408*i
z2 = -44.7676953136546 + 0.658478948462408*i
z3 = 93.4623814442964 + 0.658478948462408*i
z4 = 80.8960108299372 + 0.658478948462408*i
z5 = -38.484510006475 + 0.658478948462408*i
z6 = 68.329640215578 + 0.658478948462408*i
z7 = 14.9225651045515 - 0.658478948462408*i
z8 = 2.35619449019234 - 0.658478948462408*i
z9 = -66.7588438887831 - 0.658478948462408*i
z10 = 58.9048622548086 - 0.658478948462408*i
z11 = 40.0553063332699 - 0.658478948462408*i
z12 = 55.7632696012188 + 0.658478948462408*i
z13 = -73.0420291959627 - 0.658478948462408*i
z14 = 99.7455667514759 + 0.658478948462408*i
z15 = 52.621676947629 - 0.658478948462408*i
z16 = 71.4712328691678 - 0.658478948462408*i
z17 = -60.4756585816035 - 0.658478948462408*i
z18 = 46.3384916404494 - 0.658478948462408*i
z19 = 43.1968989868597 + 0.658478948462408*i
z20 = 77.7544181763474 - 0.658478948462408*i
z21 = -13.3517687777566 + 0.658478948462408*i
z22 = -19.6349540849362 + 0.658478948462408*i
z23 = -82.4668071567321 + 0.658478948462408*i
z24 = -54.1924732744239 - 0.658478948462408*i
z25 = -98.174770424681 - 0.658478948462408*i
z26 = 33.7721210260903 - 0.658478948462408*i
z27 = 21.2057504117311 - 0.658478948462408*i
z28 = 8.63937979737193 - 0.658478948462408*i
z29 = 24.3473430653209 + 0.658478948462408*i
z30 = -22.776546738526 - 0.658478948462408*i
z31 = 27.4889357189107 - 0.658478948462408*i
z32 = -63.6172512351933 + 0.658478948462408*i
z33 = 62.0464549083984 + 0.658478948462408*i
z34 = -57.3340659280137 + 0.658478948462408*i
z35 = -16.4933614313464 - 0.658478948462408*i
z36 = -79.3252145031423 - 0.658478948462408*i
z37 = -85.6083998103219 - 0.658478948462408*i
z38 = 36.9137136796801 + 0.658478948462408*i
z39 = -32.2013246992954 + 0.658478948462408*i
z40 = -10.2101761241668 - 0.658478948462408*i
z41 = 18.0641577581413 + 0.658478948462408*i
z42 = -91.8915851175014 - 0.658478948462408*i
z43 = 90.3207887907066 - 0.658478948462408*i
z44 = 30.6305283725005 + 0.658478948462408*i
z45 = 96.6039740978861 - 0.658478948462408*i
z46 = -29.0597320457056 - 0.658478948462408*i
z47 = -95.0331777710912 + 0.658478948462408*i
z48 = 65.1880475619882 - 0.658478948462408*i
z49 = -47.9092879672443 - 0.658478948462408*i
z50 = 74.6128255227576 + 0.658478948462408*i
z51 = -41.6261026600648 - 0.658478948462408*i
z52 = -69.9004365423729 + 0.658478948462408*i
z53 = 84.037603483527 - 0.658478948462408*i
z54 = -88.7499924639117 + 0.658478948462408*i
z55 = -76.1836218495525 + 0.658478948462408*i
z56 = -35.3429173528852 - 0.658478948462408*i
z57 = -51.0508806208341 + 0.658478948462408*i
z58 = -3.92699081698724 - 0.658478948462408*i
z59 = 11.7809724509617 + 0.658478948462408*i
z60 = 49.4800842940392 + 0.658478948462408*i
z61 = 87.1791961371168 + 0.658478948462408*i
z62 = -25.9181393921158 + 0.658478948462408*i
z63 = -7.06858347057703 + 0.658478948462408*i
z64 = 5.49778714378214 + 0.658478948462408*i
z64 = 5.49778714378214 + 0.658478948462408*i