MathWithSumpy_TRIGONOMETERY_082918.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Just about any TRIG equation !!!\n",
    "\n",
    "### For TRIGONOMETRY: sympy library in Jupyter Notebook allows you to:\n",
    " \n",
    " - Expand  [Careful !]\n",
    " - Expand_Trig\n",
    " - Trig_Simp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x, y, z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a, b,c,d = symbols ('a b c d')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAL0AAAA2BAMAAABgjoUYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niFOub7qi5+Gp/4WoXMEJS2yfAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{2} + x}{x \\sin^{2}{\\left (y \\right )} + x \\cos^{2}{\\left (y \\right )}}$$"
      ],
      "text/plain": [
       "         2           \n",
       "        x  + x       \n",
       "─────────────────────\n",
       "     2           2   \n",
       "x⋅sin (y) + x⋅cos (y)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "leeTrig = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)\n",
    "leeTrig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAWBAMAAADeGsQjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$x^{2} + 2 x + 1$$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 2⋅x + 1"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expandAlgebra = expand((x+1)**2)\n",
    "expandAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAAUBAMAAABc0JOuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHPb/NO9O8cQhoUfLeigfDNybbQsX31+4JZtaGyDpQAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 \\sin{\\left (x \\right )} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "2⋅sin(x)⋅cos(x)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expandTrig = expand_trig((sin(2*x)))\n",
    "expandTrig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATsAAAAZBAMAAAC84ZR9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niy/fAWJs9y9dWFNjf74XOgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- 8 \\sin^{3}{\\left (x \\right )} \\cos{\\left (x \\right )} + 4 \\sin{\\left (x \\right )} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "       3                            \n",
       "- 8⋅sin (x)⋅cos(x) + 4⋅sin(x)⋅cos(x)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expandTrig2 = expand_trig((sin(4*x)))\n",
    "expandTrig2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAUBAMAAAB2RTUoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDzFYPDHiWIKOSPGrfDKhSdnIKvMKW08lWnQBpCsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (x \\right )} \\cos{\\left (y \\right )} + \\sin{\\left (y \\right )} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "sin(x)⋅cos(y) + sin(y)⋅cos(x)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expandTrig3 = expand_trig((sin(x+y)))\n",
    "expandTrig3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplifyTrig4 = trigsimp(sin(x)**2 + cos(x)**2)\n",
    "simplifyTrig4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAL0AAAA2BAMAAABgjoUYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niFOub7qi5+Gp/4WoXMEJS2yfAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{2} + x}{x \\sin^{2}{\\left (y \\right )} + x \\cos^{2}{\\left (y \\right )}}$$"
      ],
      "text/plain": [
       "         2           \n",
       "        x  + x       \n",
       "─────────────────────\n",
       "     2           2   \n",
       "x⋅sin (y) + x⋅cos (y)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mixedTrig = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)\n",
    "mixedTrig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAAqBAMAAAAjeEqJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{1}{x} \\left(x^{2} + x\\right)$$"
      ],
      "text/plain": [
       " 2    \n",
       "x  + x\n",
       "──────\n",
       "  x   "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigSimp = trigsimp((x + x**2)/(x*sin(y)**2 + x*cos(y)**2))\n",
    "trigSimp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAZBAMAAACIvQJIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRpI+mvw/APY90pgxKQH8ARuYvXIGpgj2AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$4 \\cos^{3}{\\left (x \\right )} - 3 \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "     3              \n",
       "4⋅cos (x) - 3⋅cos(x)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#  Simplify Trig\n",
    "expandTrig5 = expand_trig(cos(3*x))\n",
    "expandTrig5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAUBAMAAAAkb50PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nKfh7QSZaF/MXyGpVuSOLIkUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos{\\left (3 x \\right )}$$"
      ],
      "text/plain": [
       "cos(3⋅x)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigSimp2 = trigsimp(4 * cos(x)**3 - 3*cos(x))\n",
    "trigSimp2"
   ]
  }
 ],
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