linearalgebra_practice107108109.ipynb

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   "execution_count": 8,
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    "%%html\n",
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    "@import url('https://fonts.googleapis.com/css?family=Roboto&effect=3d|brick-sign|');\n",
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  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math, sympy, numpy\n",
    "sympy.init_printing(use_unicode=True)\n",
    "from sympy import *\n",
    "from mpmath import *\n",
    "\n",
    "import pylab as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Computer Algebra I: Mathematica, SymPy, Sage, Maxima](http://hyperpolyglot.org/computer-algebra2)\n",
    "\n",
    "[SymPy. Numerical evaluation](https://docs.sympy.org/latest/modules/evalf.html)\n",
    "\n",
    "[LaTeX/Mathematics](https://en.wikibooks.org/wiki/LaTeX/Mathematics)\n",
    "\n",
    "[Wolfram Mathworld. Trigonometric Identities](http://mathworld.wolfram.com/topics/TrigonometricIdentities.html)\n",
    "\n",
    "# Типовой расчет 7 / Standard Calculation 7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "\n",
    "Выполнить указанные действия с комплексными числами, \n",
    "\n",
    "представив ответ в алгебраической форме:\n",
    "\n",
    "$\\large{Z_1 + Z_2, Z_2 - Z_3, Z_1 * Z_4, Z_2 / Z_5}$\n",
    "\n",
    "$\\large{Z_1 = 4 + 3 i; Z_2 = 1 + 2 i; Z_3 = 1 + 3 i; Z_4 = – 10 + i; Z_5 = – 7 + 6 i}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((4+3j), (1+2j), (1+3j), (-10+1j), (-7+6j))"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z1 = complex(4 + 3 * I)\n",
    "z2 = complex(1 + 2 * I)\n",
    "z3 = complex(1 + 3 * I)\n",
    "z4 = complex(-10 + 1 * I)\n",
    "z5 = complex(-7 + 6 * I)\n",
    "\n",
    "z1, z2, z3, z4, z5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decision\n",
    "\n",
    "Сложение комплексных чисел.\n",
    "\n",
    "$(a + bi) + (c + di) = (a + c) + (b + d)i$\n",
    "\n",
    "$Z_1 + Z_2 =  (4 + 3 i) + (1 + 2 i) = 5 + 5i$\n",
    "\n",
    "Вычитание комплексных чисел.\n",
    "\n",
    "$(a + bi) - (c + di) = (a - c) + (b - d)i$\n",
    "\n",
    "$Z_2 - Z_3 =  (1 + 2 i) - (1 + 3i)= - i$\n",
    "\n",
    "Умножение комплексных чисел.\n",
    "\n",
    "$(a + bi) * (c + di) = (ac - bd) + (ad + bc)i$\n",
    "\n",
    "$Z_1 + Z_4 =  (4 + 3 i) + (-10 + i) = (4 * (-10) - 3 * 1) + (4 * 1 + 3 * (-10))i = -43 -26i$\n",
    "\n",
    "Деление комплексных чисел.\n",
    "\n",
    "$\\large{\\frac{a + bi}{c + di} = \\frac{(a + bi) * (c - di)}{(c + di) * (c - di)} = \\frac{ac+ bd}{c^2 + d^2} + \\frac{bc - ad}{c^2 + d^2}i}$\n",
    "\n",
    "$\\large{Z_2 / Z_5 = \\frac{1 + 2i}{-7 + 6i} = \\frac{1 * (-7) + 2 * 6}{(-7)^2 + 6^2} + \\frac{2 * (-7) - 1 * 6}{(-7)^2 + 6^2}i} = \\frac{1}{17} - \\frac{4}{17}i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((5+5j), -1j, (-43-26j))"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z1 + z2, z2 - z3, z1 * z4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((0.0588235294117647-0.23529411764705885j),\n",
       " (0.058823529411764705, -0.23529411764705882))"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z2 / z5, (1/17, -4/17)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "C = numpy.array([4,1,1,-10,-7,5,0]) + 1j * numpy.array([3,2,3,1,6,5,-1])\n",
    "x = [c.real for c in C]; y = [c.imag for c in C]; Z=['Z1','Z2','Z3','Z4','Z5','Z1+Z2','Z2-Z3']\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10,10)); ax.grid(True)\n",
    "\n",
    "for i in range(7):\n",
    "    ax.plot([0,x[i]], [0,y[i]], '-o', lw=4, ls='--')\n",
    "    ax.annotate(Z[i], (x[i]-0.5, y[i]+0.1), fontsize=12)\n",
    "    \n",
    "plt.text(6.5,-0.3,r'$x$', fontsize=20)\n",
    "plt.text(-0.5,6.5,r'$y$', fontsize=20)\n",
    "plt.text(0.3,-0.3,r'$0$', fontsize=20)\n",
    "\n",
    "plt.scatter(0,6.5, marker='^', s=100, c='black')\n",
    "plt.scatter(6.5,0, marker='>', s=100, c='black') \n",
    "ax.axvline(0,0.02,0.96,c='black'); ax.axhline(0,0.02,0.96,c='black'); "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет  8 / Standard Calculation 8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "\n",
    "По заданным комплексным числам $Z_1, Z_2, Z_3$ и показателям $K, L, M, N$ вычислить\n",
    "\n",
    "$\\large{Z_1^K * Z_2^L, \\ Z_2^M \\ / \\ Z_3^N}$.\n",
    "\n",
    "Ответ представить в алгебраической форме.\n",
    "\n",
    "$Z_1 = -\\sqrt{2 - \\sqrt{3}} + \\sqrt{2 + \\sqrt{3}} i; \\ Z_2 = \\sqrt{3} + i; \\ Z_3 = -1 + i$\n",
    "\n",
    "$K = 10; \\ L = 6; \\ M = 15; \\ N = 27$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decision\n",
    "\n",
    "### I.\n",
    "\n",
    "Для получения тригонометрической формы заданных комплексных чисел найдем их модули и аргументы.\n",
    "\n",
    "---\n",
    "\n",
    "$Z_1 = -\\sqrt{2 - \\sqrt{3}} + \\sqrt{2 + \\sqrt{3}} i$\n",
    "\n",
    "$r_1 = |Z_1| = \\sqrt{\\Big(-\\sqrt{2 - \\sqrt{3}}\\Big)^2 + \\Big(\\sqrt{2 + \\sqrt{3}}\\Big)^2} = \\sqrt{2 - \\sqrt{3} + 2 + \\sqrt{3}} = \\sqrt{4} = 2$\n",
    "\n",
    "$-\\sqrt{2 - \\sqrt{3}} < 0; \\sqrt{2 + \\sqrt{3}} > 0 \\implies \\varphi_1 \\in II $ четверти.\n",
    "\n",
    "$\\large{\\varphi_1 = \\pi - arctg\\frac{\\sqrt{2 + \\sqrt{3}}}{\\sqrt{2 - \\sqrt{3}}} = \\pi - arctg\\frac{\\sqrt{2 + \\sqrt{3}} * \\sqrt{2 + \\sqrt{3}}}{\\sqrt{2 - \\sqrt{3}} * \\sqrt{2 + \\sqrt{3}}} = }$\n",
    "\n",
    "$\\large{= \\pi - arctg \\frac{2 + \\sqrt{3}}{\\sqrt{2^2 - \\sqrt{3}^2}}  = \\pi - arctg (2 + \\sqrt{3}) = \\frac{\\pi}{2} + arcctg(2 + \\sqrt{3}) }$ ,\n",
    "\n",
    "т.к. $\\large{arctg \\ \\alpha + arcctg \\ \\alpha = \\frac{\\pi}{2}, \\ \\forall \\alpha}$.\n",
    "\n",
    "Применим формулу котангенса половинного аргумента:\n",
    "\n",
    "$\\large{ctg \\frac{\\alpha}{2} = \\frac{1 + cos \\ \\alpha}{sin \\ \\alpha} }$\n",
    "\n",
    "$\\large{arcctg(2 + \\sqrt{3}) = arcctg \\frac{1 + \\frac{\\sqrt{3}}{2}} {\\frac{1}{2}} = arcctg \\frac{1 + cos \\frac{\\pi}{6}}{sin \\frac{\\pi}{6}} = arcctg(ctg \\frac{\\pi}{12}) = \\frac{\\pi}{12} }$\n",
    "\n",
    "$\\large{\\varphi_1 = \\frac{\\pi}{2} + \\frac{\\pi}{12} = \\frac{7\\pi}{12} }$\n",
    "\n",
    "\n",
    "---\n",
    "\n",
    "$Z_2 = \\sqrt{3} + i$\n",
    "\n",
    "$r_2 = |Z_2| = \\sqrt{\\sqrt{3}^2 + 1^2} = \\sqrt{4} = 2$\n",
    "\n",
    "$\\sqrt{3} > 0; 1 > 0 \\implies \\varphi_2 \\in I $ четверти.\n",
    "\n",
    "$\\large{\\varphi_2 = arctg \\frac{1}{\\sqrt{3}} = \\frac{\\pi}{6}}$\n",
    "\n",
    "---\n",
    "\n",
    "$Z_3 = -1 + i$\n",
    "\n",
    "$r_3 = |Z_3| = \\sqrt{(-1)^2 + 1^2} = \\sqrt{2}$\n",
    "\n",
    "$-1 < 0; 1 > 0 \\implies \\varphi_3 \\in II $ четверти.\n",
    "\n",
    "$\\large{\\varphi_3 = \\pi - arctg \\frac{1}{1} = \\pi - \\frac{\\pi}{4} = \\frac{3\\pi}{4}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### II.\n",
    "\n",
    "Возведение комплексного числа в степень в тригонометрической форме:\n",
    "    \n",
    "$z^k = [r * (\\cos \\varphi + i \\sin \\varphi)]^n = r^n * [\\cos(n\\varphi) + i \\sin(n\\varphi) ]$\n",
    "\n",
    "Умножение комплексных чисел в тригонометрической форме:\n",
    "    \n",
    "$z_1 * z_2 = r_1 * (\\cos \\varphi_1 + i \\sin \\varphi_1) *  r_2 * (\\cos \\varphi_2 + i \\sin \\varphi_2) = r_1r_2 * [\\cos(\\varphi_1 + \\varphi_2) + i \\sin(\\varphi_1 + \\varphi_2) ]$\n",
    "\n",
    "Деление комплексных чисел в тригонометрической форме:\n",
    "\n",
    "$\\large{\\frac{z_1}{z_2} = \\frac{r_1 * (\\cos \\varphi_1 + i \\sin \\varphi_1)} { r_2 * (\\cos \\varphi_2 + i \\sin \\varphi_2)} = \\frac {r_1}{r_2} }$$* [\\cos(\\varphi_1 - \\varphi_2) + i \\sin(\\varphi_1 - \\varphi_2) ]$\n",
    "\n",
    "Вычислим $Z_1^K * Z_2^L, \\ Z_2^M \\ / \\ Z_3^N$  в тригонометрической форме, используя приведенные формулы."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{Z_1 ^ K = 2 ^ {10} * (\\cos (10 * \\frac{7\\pi}{12}) + i \\sin(10 * \\frac{7\\pi}{12})) = 2 ^ {10} * (\\cos \\frac{35\\pi}{6} + i \\sin \\frac{35\\pi}{6}) =}$\n",
    "\n",
    "$\\large{= 2 ^ {10} * (\\cos (6\\pi - \\frac{\\pi}{6}) + i \\sin (6\\pi - \\frac{\\pi}{6})) =  2 ^ {10} * (\\cos \\frac{\\pi}{6} + i \\sin(- \\frac{\\pi}{6}))}$\n",
    "\n",
    "---\n",
    "\n",
    "$\\large{Z_2 ^ L = 2 ^ 6 * (\\cos (6 * \\frac{\\pi}{6}) + i \\sin(6 * \\frac{\\pi}{6})) = 2 ^ 6 * (\\cos \\pi + i \\sin \\pi) }$\n",
    "\n",
    "---\n",
    "\n",
    "$\\large{Z_1 ^ K * Z_2 ^ L = 2 ^ {10} * 2 ^ 6 * (\\cos (\\pi + \\frac{\\pi}{6}) + i \\sin(\\pi - \\frac{\\pi}{6})) = 2 ^ {16} * (\\cos (-\\frac{\\pi}{6}) + i \\sin\\frac{\\pi}{6}) = }$\n",
    "\n",
    "$\\large{= - 2 ^ {15} \\sqrt{3} + 2 ^ {15} i = -32768 \\sqrt{3} + 32768 i }$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "$\\large{Z_2 ^ M = 2 ^ {15} * (\\cos (15 * \\frac{\\pi}{6}) + i \\sin(15 * \\frac{\\pi}{6})) = }$\n",
    "\n",
    "$\\large{= 2 ^ {15} * (\\cos (2\\pi + \\frac{\\pi}{2}) + i \\sin (2\\pi + \\frac{\\pi}{2})) =  2 ^ {15} * (\\cos \\frac{\\pi}{2} + i \\sin \\frac{\\pi}{2})}$\n",
    "\n",
    "---\n",
    "\n",
    "$\\large{Z_3 ^ N = (\\sqrt 2 )^ {27} * (\\cos (27 * \\frac{3\\pi}{4}) + i \\sin(27 * \\frac{3\\pi}{4})) = (\\sqrt 2) ^ {27} * (\\cos \\frac{81\\pi}{4} + i \\sin \\frac{81\\pi}{4}) = }$\n",
    "\n",
    "$\\large{= (\\sqrt 2) ^ {27} *  (\\cos (20\\pi + \\frac{\\pi}{4}) + i \\sin (20\\pi + \\frac{\\pi}{4})) = (\\sqrt 2) ^ {27} *  (\\cos \\frac{\\pi}{4} + i \\sin \\frac{\\pi}{4}) }$\n",
    "\n",
    "---\n",
    "\n",
    "$\\large{Z_2 ^ M / Z_3 ^ N = \\frac{2 ^ {15}} {(\\sqrt{2}) ^ {27}} * (\\cos (\\frac{\\pi}{2} - \\frac{\\pi}{4}) + i \\sin(\\frac{\\pi}{2} - \\frac{\\pi}{4})) = }$\n",
    "\n",
    "$\\large{=  \\big(\\sqrt{2}\\big) ^ {30 - 27} * (\\cos \\frac{\\pi}{4} + i \\sin \\frac{\\pi}{4}) = \\big(\\sqrt{2}\\big) ^ 3 * (\\frac{\\sqrt{2} } {2} + i \\frac{\\sqrt{2} } {2}) = 2 + 2 i}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### III."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((-0.5176380902050416+1.9318516525781366j),\n",
       " (mpf('2.0'), mpf('1.8325957145940461')))"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z1 = complex(-sqrt(2 - sqrt(3)), sqrt(2 + sqrt(3)))\n",
    "z1, polar(z1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$1.832595714594046$$"
      ],
      "text/plain": [
       "1.832595714594046"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "7 * math.pi / 12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((1.7320508075688772+1j), (mpf('2.0'), mpf('0.52359877559829893')))"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z2 = complex(sqrt(3), 1)\n",
    "z2, polar(z2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$0.5235987755982988$$"
      ],
      "text/plain": [
       "0.5235987755982988"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "math.pi/6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((-1+1j), (mpf('1.4142135623730951'), mpf('2.3561944901923448')))"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z3 = complex(-1, 1)\n",
    "z3, polar(z3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2.356194490192345$$"
      ],
      "text/plain": [
       "2.356194490192345"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 * math.pi / 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### IV."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-56755.840862417+32767.999999999975j)"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(z1 ** 10) * (z2 ** 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-56755.84086241697+32768j)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-32768 * (3 ** 0.5) + 32768 * 1j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.9999999999999973+1.9999999999999991j)"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(z2 ** 15) / (z3 ** 27)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2+2j)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 + 2 * 1j"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет  9 / Standard Calculation 9"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "\n",
    "Найти все значения корней k-ой степени из заданного комплексного числа Z.\n",
    "\n",
    "Ответ представить в алгебраической форме.\n",
    "\n",
    "На чертеже комплексной плоскости изобразить полученные значения корней.\n",
    "\n",
    "$Z = -\\sqrt{2 - \\sqrt{3}} + \\sqrt{2 + \\sqrt{3}} i; \\ k = 4$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decision\n",
    "\n",
    "### I.\n",
    "\n",
    "Для получения тригонометрической формы заданного комплексного числа найдем его модуль и аргумент.\n",
    "\n",
    "$r = |Z| = \\sqrt{\\Big(-\\sqrt{2 - \\sqrt{3}}\\Big)^2 + \\Big(\\sqrt{2 + \\sqrt{3}}\\Big)^2} = \\sqrt{2 - \\sqrt{3} + 2 + \\sqrt{3}} = \\sqrt{4} = 2$\n",
    "\n",
    "$-\\sqrt{2 - \\sqrt{3}} < 0; \\sqrt{2 + \\sqrt{3}} > 0 \\implies \\varphi \\in II $ четверти.\n",
    "\n",
    "$\\large{\\varphi = \\pi - arctg\\frac{\\sqrt{2 + \\sqrt{3}}}{\\sqrt{2 - \\sqrt{3}}} = \\pi - arctg\\frac{\\sqrt{2 + \\sqrt{3}} * \\sqrt{2 + \\sqrt{3}}}{\\sqrt{2 - \\sqrt{3}} * \\sqrt{2 + \\sqrt{3}}} = }$\n",
    "\n",
    "$\\large{= \\pi - arctg \\frac{2 + \\sqrt{3}}{\\sqrt{2^2 - \\sqrt{3}^2}}  = \\pi - arctg (2 + \\sqrt{3}) = \\frac{\\pi}{2} + arcctg(2 + \\sqrt{3}) }$ ,\n",
    "\n",
    "т.к. $\\large{arctg \\ \\alpha + arcctg \\ \\alpha = \\frac{\\pi}{2}, \\ \\forall \\alpha}$.\n",
    "\n",
    "Применим формулу котангенса половинного аргумента $\\large{ctg \\frac{\\alpha}{2} = \\frac{1 + cos \\ \\alpha}{sin \\ \\alpha} }$:\n",
    "\n",
    "$\\large{arcctg(2 + \\sqrt{3}) = arcctg \\frac{1 + \\frac{\\sqrt{3}}{2}} {\\frac{1}{2}} = arcctg \\frac{1 + cos \\frac{\\pi}{6}}{sin \\frac{\\pi}{6}} = arcctg(ctg \\frac{\\pi}{12}) = \\frac{\\pi}{12} }$\n",
    "\n",
    "$\\large{\\varphi = \\frac{\\pi}{2} + \\frac{\\pi}{12} = \\frac{7\\pi}{12} }$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### II.\n",
    "\n",
    "Извлечение корня из комплексного числа в тригонометрической форме:\n",
    "\n",
    "$\\sqrt[n] {z} = \\sqrt[n] {r * (\\cos \\varphi + i \\sin \\varphi)} = \\sqrt[n]{r} * \\bigg(\\cos \\frac{\\varphi + 2\\pi m}{n} + i \\sin \\frac{\\varphi + 2\\pi m}{n} \\bigg), \\ m = 0, 1, ..., n-1$\n",
    "\n",
    "$\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(\\cos \\frac{\\frac{7\\pi}{12} + 2\\pi m}{4} + i \\sin \\frac{\\frac{7\\pi}{12} + 2\\pi m}{4} \\bigg), \\ m = 0, 1, 2, 3}$\n",
    "\n",
    "1) $\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(\\cos \\frac{7\\pi}{12 * 4} + i \\sin \\frac{7\\pi}{12 * 4} \\bigg) = \\sqrt[4] {2} * \\bigg(\\cos \\frac{7\\pi}{48} + i \\sin \\frac{7\\pi}{48} \\bigg) }$\n",
    "\n",
    "2) $\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(\\cos \\frac{(7 + 24)\\pi}{12 * 4} + i \\sin \\frac{(7 + 24)\\pi}{12 * 4} \\bigg) = } $\n",
    "\n",
    "$\\large{= \\sqrt[4] {2} * \\bigg(\\cos (\\frac{\\pi}{2} + \\frac{7\\pi}{48}) + i \\sin (\\frac{\\pi}{2} + \\frac{7\\pi}{48}) \\bigg)  = \\sqrt[4] {2} * \\bigg(-\\sin \\frac{7\\pi}{48} + i \\cos \\frac{7\\pi}{48} \\bigg) }$\n",
    "\n",
    "3) $\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(\\cos \\frac{(7 + 48)\\pi}{12 * 4} + i \\sin \\frac{(7 + 48)\\pi}{12 * 4} \\bigg) = }$\n",
    "\n",
    "$\\large{= \\sqrt[4] {2} * \\bigg(\\cos (\\pi + \\frac{7\\pi}{48}) + i \\sin (\\pi + \\frac{7\\pi}{48}) \\bigg) = -\\sqrt[4] {2} * \\bigg(\\cos \\frac{7\\pi}{48} + i \\sin \\frac{7\\pi}{48} \\bigg) }$\n",
    "\n",
    "4) $\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(\\cos \\frac{(7 + 72)\\pi}{12 * 4} + i \\sin \\frac{(7 + 72)\\pi}{12 * 4} \\bigg) = }$\n",
    "\n",
    "$\\large{=\\sqrt[4] {2} * \\bigg(\\cos (\\frac{3\\pi}{2} + \\frac{7\\pi}{48}) + i \\sin (\\frac{3\\pi}{2} + \\frac{7\\pi}{48}) \\bigg) = \\sqrt[4] {2} * \\bigg(\\sin \\frac{7\\pi}{48} - i \\cos \\frac{7\\pi}{48} \\bigg) }$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### III.\n",
    "\n",
    "$\\large{cos \\frac{7\\pi}{12} = \\cos (\\frac{\\pi}{2} + \\frac{\\pi}{12}) = -\\sin \\frac{\\pi}{12} =  -\\sin (\\frac{\\pi}{3} -  \\frac{\\pi}{4}) =  }$\n",
    "\n",
    "$\\large{= -sin\\frac{\\pi}{3} cos \\frac{\\pi}{4} + sin\\frac{\\pi}{4} cos \\frac{\\pi}{3} = \\frac{\\sqrt{2} - \\sqrt{6}}{4} }$\n",
    "\n",
    "$\\large{cos \\frac{7\\pi}{24} = \\sqrt \\frac{1 + cos \\frac{7\\pi}{12}}{2}  = \\sqrt \\frac{1 + \\frac{\\sqrt{2} - \\sqrt{6}}{4}}{2} = \\sqrt \\frac{4 + \\sqrt{2} - \\sqrt{6}}{8} }$\n",
    "\n",
    "$\\large{cos \\frac{7\\pi}{48} = \\sqrt \\frac{1 + cos \\frac{7\\pi}{24}}{2}  = \\sqrt \\frac{1 + \\sqrt \\frac{4 + \\sqrt{2} - \\sqrt{6}}{8}}{2} = \\sqrt \\frac{\\sqrt{8} + \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}}{4\\sqrt{2}} = }$\n",
    "\n",
    "$\\large{ = \\frac{ \\sqrt{\\sqrt{8} + \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2\\sqrt[4]{2}} }$\n",
    "\n",
    "$\\large{sin \\frac{7\\pi}{48} = \\sqrt \\frac{1 - cos \\frac{7\\pi}{24}}{2}  = \\sqrt \\frac{1 - \\sqrt \\frac{4 + \\sqrt{2} - \\sqrt{6}}{8}}{2} = \\sqrt \\frac{\\sqrt{8} - \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}}{4\\sqrt{2}} = }$\n",
    "\n",
    "$\\large{ = \\frac{ \\sqrt{\\sqrt{8} - \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2\\sqrt[4]{2}} }$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### IV.\n",
    "1) $\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(\\cos \\frac{7\\pi}{48} + i \\sin \\frac{7\\pi}{48} \\bigg)  = \\frac{ \\sqrt{\\sqrt{8} + \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2} + i \\frac{ \\sqrt{\\sqrt{8} - \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2}}$\n",
    "\n",
    "2) $\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(-\\sin \\frac{7\\pi}{48} + i \\cos \\frac{7\\pi}{48} \\bigg)  = -\\frac{ \\sqrt{\\sqrt{8} - \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2} + i \\frac{ \\sqrt{\\sqrt{8} + \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2}}$\n",
    "\n",
    "3) $\\large{\\sqrt[k] {Z} = -\\sqrt[4] {2} * \\bigg(\\cos \\frac{7\\pi}{48} + i \\sin \\frac{7\\pi}{48} \\bigg)  = -\\frac{ \\sqrt{\\sqrt{8} + \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2} - i \\frac{ \\sqrt{\\sqrt{8} - \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2}}$\n",
    "\n",
    "4) $\\large{\\sqrt[k] {Z} = \\sqrt[4] {2} * \\bigg(\\sin \\frac{7\\pi}{48} - i \\cos \\frac{7\\pi}{48} \\bigg)  = \\frac{ \\sqrt{\\sqrt{8} - \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2} - i \\frac{ \\sqrt{\\sqrt{8} + \\sqrt{4 + \\sqrt{2} - \\sqrt{6}}} } {2}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### V."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((-0.5176380902050416+1.9318516525781366j),\n",
       " (mpf('2.0'), mpf('1.8325957145940461')))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = complex(-sqrt(2 - sqrt(3)), sqrt(2 + sqrt(3)))\n",
    "Z, polar(Z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.0665674454826692+0.5259728572936707j)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z ** 0.25"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$1.0665674454826695$$"
      ],
      "text/plain": [
       "1.0665674454826695"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# cos 7π/48\n",
    "(8 ** 0.5  + (4 + 2 ** 0.5 - 6 ** 0.5) ** 0.5) ** 0.5 / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$0.5259728572936708$$"
      ],
      "text/plain": [
       "0.5259728572936708"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sin 7π/48\n",
    "(8 ** 0.5  - (4 + 2 ** 0.5 - 6 ** 0.5) ** 0.5) ** 0.5 / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ \\left ( -1.06656744548267, \\quad -0.525972857293671\\right ), \\quad \\left ( -0.525972857293671, \\quad 1.06656744548267\\right ), \\quad \\left ( 0.525972857293671, \\quad -1.06656744548267\\right ), \\quad \\left ( 1.06656744548267, \\quad 0.525972857293671\\right )\\right ]$$"
      ],
      "text/plain": [
       "[(-1.06656744548267, -0.525972857293671), (-0.525972857293671, 1.0665674454826\n",
       "7), (0.525972857293671, -1.06656744548267), (1.06656744548267, 0.5259728572936\n",
       "71)]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y = symbols('x y', real=True)\n",
    "roots = solve((x + I * y) ** 4 - Z , (x, y))\n",
    "roots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(10,10)) \n",
    "ax.grid(True); n = [3,2,4,1]; i = 0\n",
    "for (x,y) in roots:\n",
    "    ax.plot([0,x],[0,y],'-o')\n",
    "    ax.annotate('$\\sqrt[4]{Z}_{%i}$ ' %n[i], (x,y), fontsize=15);\n",
    "    i +=1\n",
    "    \n",
    "plt.suptitle('$\\sqrt[4]{-\\sqrt{2 - \\sqrt{3}} + \\sqrt{2 + \\sqrt{3}} \\ i } $', fontsize=20)\n",
    "plt.text(1.5,-0.1,r'$x$', fontsize=20)\n",
    "plt.text(-0.1,1.5,r'$y$', fontsize=20)\n",
    "plt.text(0.1,-0.1,r'$0$', fontsize=20)\n",
    "\n",
    "plt.scatter(0,1.5, marker='^', s=100, c='black')\n",
    "plt.scatter(1.5,0, marker='>', s=100, c='black') \n",
    "ax.axvline(0,0.02,0.96,c='black'); ax.axhline(0,0.02,0.96,c='black'); "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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