学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第4章(正弦と余弦)、3(加法公式)、練習問題4、5、6.を取り組んでみる。
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sin2x=sin(x+x)=sinxcosx+cosxsinx=2sinxcosx
cos2x=cos(x+x)=cosxcosx−sinxsinx=cos2x−sin2x
cos2x=cos2x−(1−cos2x)cos2x=2cos2x−1cos2x=1+cos2x2
cos2x=1−sin2x−sin2xcos2x=1−2sin2xsin2x=1−cos2x2
sin3x=sin(2x+x)=sin2xcosx+cos2xsinx=2sinxcos2x+(cos2x−sin2x)sinx=3sinxcos2x−sin3x=3sinx(1−sin2x)−sin3x=3sinx−4sin3xsin4x=sin2(2x)=2sin2xcos2x=2(2sinxcosx)(cos2x−sin2x)=2(2sinxcosx)(1−2sin2x)=4sinxcosx−8sinx3cosxsin5x=sin(4x+x)=sin4xcosx+cos4xsinx=4sinxcos2x−8sinx3cos2x+cos2(2x)sinx=4sinxcos2x−8sin3xcos2x+(cos22x−sin22x)sinx=4sincos2x−8sin3xcos2x+((cos2x−sin2x)2−4sin2xcos2x)sinx
sinnx=∑aij(sinx)i(cosx)jsin(n+1)x=sinnxcosx+cosnxsinxcos2kx=cos2kx+sin2kxcos(2k+1)x=cos2kxcosx+sin2kxsinxsin(n+1)x=∑aij(sinx)i(cosx)j
コード(Emacs)
Python 3
#!/usr/bin/env python3 # -*- coding: utf-8 -*- from sympy import pprint, symbols, sin, cos, plot, pi x = symbols('x') fs = [sin(x), cos(x), sin(2 * x), cos(2 * x), cos(x) ** 2, sin(x) ** 2] p = plot(*fs, (x, -pi, pi), show=False, legend=True) for i, c in enumerate(['red', 'green', 'blue', 'brown', 'purple', 'yellow']): p[i].line_color = c p.save('sample4.svg')
入出力結果(Terminal, IPython)
$ ./sample4.py $
HTML5
<div id="graph0"></div> <pre id="output0"></pre> <label for="r0">r = </label> <input id="r0" type="number" min="0" value="0.5"> <label for="dx">dx = </label> <input id="dx" type="number" min="0" step="0.0001" value="0.001"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-5"> <label for="x2">x2 = </label> <input id="x2" type="number" value="5"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-5"> <label for="y2">y2 = </label> <input id="y2" type="number" value="5"> <br> <label for="n0">n = </label> <input id="n0" type="number" min="0" step="1" value="2"> <button id="draw0">draw</button> <button id="clear0">clear</button> <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script> <script src="sample4.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'), pre0 = document.querySelector('#output0'), width = 600, height = 600, padding = 50, btn0 = document.querySelector('#draw0'), btn1 = document.querySelector('#clear0'), input_r = document.querySelector('#r0'), input_dx = document.querySelector('#dx'), input_x1 = document.querySelector('#x1'), input_x2 = document.querySelector('#x2'), input_y1 = document.querySelector('#y1'), input_y2 = document.querySelector('#y2'), input_n0 = document.querySelector('#n0'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_n0], p = (x) => pre0.textContent += x + '\n', range = (start, end, step=1) => { let res = []; for (let i = start; i < end; i += step) { res.push(i); } return res; }; let draw = () => { pre0.textContent = ''; let r = parseFloat(input_r.value), dx = parseFloat(input_dx.value), x1 = parseFloat(input_x1.value), x2 = parseFloat(input_x2.value), y1 = parseFloat(input_y1.value), y2 = parseFloat(input_y2.value), n0 = parseFloat(input_n0.value); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return;n } let points = [], lines = [], f = (x) => Math.sin(n0 * x); for (let x = x1; x <= x2; x += dx) { let y = Math.sin(x); if (Math.abs(y) < Infinity) { points.push([x, y]); } } let t1 = points.length; for (let x = x1; x <= x2; x += dx) { let y = Math.cos(x); if (Math.abs(y) < Infinity) { points.push([x, y]); } } let t2 = points.length; for (let x = x1; x <= x2; x += dx) { let y = f(x); if (Math.abs(y) < Infinity) { points.push([x, y]); } } let xscale = d3.scaleLinear() .domain([x1, x2]) .range([padding, width - padding]); let yscale = d3.scaleLinear() .domain([y1, y2]) .range([height - padding, padding]); let xaxis = d3.axisBottom().scale(xscale); let yaxis = d3.axisLeft().scale(yscale); div0.innerHTML = ''; let svg = d3.select('#graph0') .append('svg') .attr('width', width) .attr('height', height); svg.selectAll('line') .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines)) .enter() .append('line') .attr('x1', (d) => xscale(d[0])) .attr('y1', (d) => yscale(d[1])) .attr('x2', (d) => xscale(d[2])) .attr('y2', (d) => yscale(d[3])) .attr('stroke', 'black'); svg.selectAll('circle') .data(points) .enter() .append('circle') .attr('cx', (d) => xscale(d[0])) .attr('cy', (d) => yscale(d[1])) .attr('r', r) .attr('fill', (d, i) => i < t1 ? 'red' : i < t2 ? 'green' : 'blue'); svg.append('g') .attr('transform', `translate(0, ${height - padding})`) .call(xaxis); svg.append('g') .attr('transform', `translate(${padding}, 0)`) .call(yaxis); }; inputs.forEach((input) => input.onchange = draw); btn0.onclick = draw; btn1.onclick = () => pre0.textContent = ''; draw();
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