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Created a notebook describing a new, easy memorization technique I le…
…arned for remembering the values of sin and cos for standard angles on the unit circle. Also added a Trigonometry directory.
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Tom Malone
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Oct 31, 2015
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Trigonometry/.ipynb_checkpoints/Easy_Way_to_Memorize_Unit_Circle-checkpoint.ipynb
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Memorizing the Unit Circle\n", | ||
| "\n", | ||
| "<p class='lead'>This is a great, very easy technique for remembering the values of the sine and cosine of any angle on the unit circle.</p>\n", | ||
| "\n", | ||
| "<p class='lead'>All you have to remember is: $0, 1, 2, 3, \\text{ and } 4$ and $\\frac{\\sqrt{x}}{2}$.</p>\n", | ||
| "\n", | ||
| "---" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Sine \n", | ||
| "\n", | ||
| "Beginning in __Quadrant I__ with angle $\\angle \\theta = 0$, _all_ you have to do is remember to replace $x$ in $\\frac{\\sqrt{x}}{2}$ with $0, 1, 2, 3, \\text{ or } 4$:\n", | ||
| " \n", | ||
| "<table width='100%' align='left'>\n", | ||
| "<tr>\n", | ||
| " <th>Number</th>\n", | ||
| " <th>$\\text{Value of } \\theta$</th>\n", | ||
| " <th>$\\sin(\\theta)$</th>\n", | ||
| " <th>Actual Value of $\\sin$</th>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>0:</td>\n", | ||
| " <td>$0$</td>\n", | ||
| " <td>$\\sin(0) = \\frac{\\sqrt{\\color{red}0}}{2}$</td>\n", | ||
| " <td>$0$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>1:</td>\n", | ||
| " <td>$\\frac{\\pi}{6}$ / $30\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{6}) = \\frac{\\sqrt{\\color{red}1}}{2} = \\frac{1}{2}$</td>\n", | ||
| " <td>$\\frac{1}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>2:</td>\n", | ||
| " <td>$\\frac{\\pi}{4}$ / $45\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{4}) = \\frac{\\sqrt{\\color{red}2}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{2}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>3:</td>\n", | ||
| " <td>$\\frac{\\pi}{3}$ / $60\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{3}) = \\frac{\\sqrt{\\color{red}3}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{3}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>4:</td>\n", | ||
| " <td>$\\frac{\\pi}{2}$ / $90\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{2}) = \\frac{\\sqrt{\\color{red}4}}{2} = \\frac{2}{2} = 1$</td>\n", | ||
| " <td>$1$</td>\n", | ||
| "</tr>\n", | ||
| "</table>\n", | ||
| "\n", | ||
| "<br/><br/>\n", | ||
| "For __Quadrant II__, the values count back down from $\\frac{\\sqrt{4}}{2}$ to $\\frac{\\sqrt{0}}{2}:$\n", | ||
| "\n", | ||
| "<table width='100%' align='left'>\n", | ||
| "<tr>\n", | ||
| " <th>Number</th>\n", | ||
| " <th>$\\text{Value of } \\theta$</th>\n", | ||
| " <th>$\\sin(\\theta)$</th>\n", | ||
| " <th>Actual Value of $\\sin$</th>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>4:</td>\n", | ||
| " <td>$\\frac{\\pi}{2}$ / $90\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{2}) = \\frac{\\sqrt{\\color{red}4}}{2} = \\frac{2}{2} = 1$</td>\n", | ||
| " <td>$1$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>3:</td>\n", | ||
| " <td>$\\frac{2\\pi}{3} (= \\frac{4\\pi}{6})$ / $120\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{3}) = \\frac{\\sqrt{\\color{red}3}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{3}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>2:</td>\n", | ||
| " <td>$\\frac{\\pi}{4}$ / $45\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{4}) = \\frac{\\sqrt{\\color{red}2}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{2}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>1:</td>\n", | ||
| " <td>$\\frac{\\pi}{6}$ / $30\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{6}) = \\frac{\\sqrt{\\color{red}1}}{2} = \\frac{1}{2}$</td>\n", | ||
| " <td>$\\frac{1}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>0:</td>\n", | ||
| " <td>$\\frac{2\\pi}{2} (= \\pi)$ / $180\\circ$</td>\n", | ||
| " <td>$\\sin(\\pi) = \\frac{\\sqrt{\\color{red}0}}{2} = 0$</td>\n", | ||
| " <td>$1$</td>\n", | ||
| "</tr>\n", | ||
| "</table>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": { | ||
| "collapsed": true | ||
| }, | ||
| "outputs": [], | ||
| "source": [] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 2", | ||
| "language": "python", | ||
| "name": "python2" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 2 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython2", | ||
| "version": "2.7.10" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 0 | ||
| } |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,137 @@ | ||
| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Memorizing the Unit Circle\n", | ||
| "\n", | ||
| "<p class='lead'>This is a great, very easy technique for remembering the values of the sine and cosine of any angle on the unit circle.</p>\n", | ||
| "\n", | ||
| "<p class='lead'>All you have to remember is: $0, 1, 2, 3, \\text{ and } 4$ and $\\frac{\\sqrt{x}}{2}$.</p>\n", | ||
| "\n", | ||
| "---" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Sine \n", | ||
| "\n", | ||
| "Beginning in __Quadrant I__ with angle $\\angle \\theta = 0$, _all_ you have to do is remember to replace $x$ in $\\frac{\\sqrt{x}}{2}$ with $0, 1, 2, 3, \\text{ or } 4$:\n", | ||
| " \n", | ||
| "<table width='100%' align='left'>\n", | ||
| "<tr>\n", | ||
| " <th>Number</th>\n", | ||
| " <th>$\\text{Value of } \\theta$</th>\n", | ||
| " <th>$\\sin(\\theta)$</th>\n", | ||
| " <th>Actual Value of $\\sin$</th>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>0:</td>\n", | ||
| " <td>$0$</td>\n", | ||
| " <td>$\\sin(0) = \\frac{\\sqrt{\\color{red}0}}{2}$</td>\n", | ||
| " <td>$0$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>1:</td>\n", | ||
| " <td>$\\frac{\\pi}{6}$ / $30\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{6}) = \\frac{\\sqrt{\\color{red}1}}{2} = \\frac{1}{2}$</td>\n", | ||
| " <td>$\\frac{1}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>2:</td>\n", | ||
| " <td>$\\frac{\\pi}{4}$ / $45\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{4}) = \\frac{\\sqrt{\\color{red}2}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{2}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>3:</td>\n", | ||
| " <td>$\\frac{\\pi}{3}$ / $60\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{3}) = \\frac{\\sqrt{\\color{red}3}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{3}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>4:</td>\n", | ||
| " <td>$\\frac{\\pi}{2}$ / $90\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{2}) = \\frac{\\sqrt{\\color{red}4}}{2} = \\frac{2}{2} = 1$</td>\n", | ||
| " <td>$1$</td>\n", | ||
| "</tr>\n", | ||
| "</table>\n", | ||
| "\n", | ||
| "<br/><br/>\n", | ||
| "For __Quadrant II__, the values count back down from $\\frac{\\sqrt{4}}{2}$ to $\\frac{\\sqrt{0}}{2}:$\n", | ||
| "\n", | ||
| "<table width='100%' align='left'>\n", | ||
| "<tr>\n", | ||
| " <th>Number</th>\n", | ||
| " <th>$\\text{Value of } \\theta$</th>\n", | ||
| " <th>$\\sin(\\theta)$</th>\n", | ||
| " <th>Actual Value of $\\sin$</th>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>4:</td>\n", | ||
| " <td>$\\frac{\\pi}{2}$ / $90\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{2}) = \\frac{\\sqrt{\\color{red}4}}{2} = \\frac{2}{2} = 1$</td>\n", | ||
| " <td>$1$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>3:</td>\n", | ||
| " <td>$\\frac{2\\pi}{3} (= \\frac{4\\pi}{6})$ / $120\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{3}) = \\frac{\\sqrt{\\color{red}3}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{3}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>2:</td>\n", | ||
| " <td>$\\frac{\\pi}{4}$ / $45\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{4}) = \\frac{\\sqrt{\\color{red}2}}{2}$</td>\n", | ||
| " <td>$\\frac{\\sqrt{2}}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>1:</td>\n", | ||
| " <td>$\\frac{\\pi}{6}$ / $30\\circ$</td>\n", | ||
| " <td>$\\sin(\\frac{\\pi}{6}) = \\frac{\\sqrt{\\color{red}1}}{2} = \\frac{1}{2}$</td>\n", | ||
| " <td>$\\frac{1}{2}$</td>\n", | ||
| "</tr>\n", | ||
| "<tr>\n", | ||
| " <td>0:</td>\n", | ||
| " <td>$\\frac{2\\pi}{2} (= \\pi)$ / $180\\circ$</td>\n", | ||
| " <td>$\\sin(\\pi) = \\frac{\\sqrt{\\color{red}0}}{2} = 0$</td>\n", | ||
| " <td>$1$</td>\n", | ||
| "</tr>\n", | ||
| "</table>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": { | ||
| "collapsed": true | ||
| }, | ||
| "outputs": [], | ||
| "source": [] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 2", | ||
| "language": "python", | ||
| "name": "python2" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 2 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython2", | ||
| "version": "2.7.10" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 0 | ||
| } |