Trigonometric Identities the Euler Way

1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "2c9e7ca0",
   "metadata": {},
   "source": [
    "**Euler's formula** states that for any real number $\\theta$,\n",
    "\n",
    "$$\\label{eq:1}\n",
    "e^{i \\theta} = \\cos\\theta + i \\sin\\theta, \\tag{1}\n",
    "$$\n",
    "\n",
    "where $i$ is the imaginary unit, defined as $i^2 = -1$. The formula has a geometric interpretation in the complex plane that provides insight into trigonometry and complex numbers, **Figure 1**.\n",
    "\n",
    "In **Figure 1**, the horizontal axis represents the real part, the vertical axis represents the imaginary part, and any complex number $z = x + i y$ is plotted as the point $(x, y)$. Euler's formula represents a point on the unit circle at angle $\\theta$ from the positive real axis, where:\n",
    "\n",
    "- **Real part**: $\\cos\\theta$ (horizontal coordinate)\n",
    "- **Imaginary part**: $\\sin\\theta$ (vertical coordinate)\n",
    "- **Magnitude**: $|e^{i\\theta}| = \\sqrt{\\cos^2\\theta + \\sin^2\\theta} = 1$\n",
    "\n",
    "As $\\theta$ varies from $0$ to $2\\pi$, the point $e^{i\\theta}$ traces out the entire unit circle. The famous **Euler's identity** emerges naturally:\n",
    "\n",
    "$$\n",
    "e^{i \\pi} = \\cos(\\pi) + i \\sin(\\pi) = -1 + 0 i = -1,\n",
    "$$\n",
    "\n",
    "or\n",
    "\n",
    "$$\n",
    "e^{i \\pi} + 1 = 0.\n",
    "$$\n",
    "\n",
    "Now, let's go back to $\\eqref{eq:1}$. We know that $\\cos(x)$ is an even function, meaning $\\cos(-x) = \\cos(x)$, and $\\sin(x)$ is an odd function, meaning $\\sin(-x) = -\\sin(x)$. Using that, we can derive the following identity:\n",
    "\n",
    "$$\\label{eq:2}\n",
    "e^{-i \\theta} = e^{i (-\\theta)} = \\cos(-\\theta) + i \\sin(-\\theta) = \\cos\\theta - i \\sin\\theta. \\tag{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56fc4761",
   "metadata": {},
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "blog",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "name": "python",
   "version": "3.11.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}

2.ipynb

3.ipynb