单位圆与对称性：三角函数可视化探索

几何画板及绘图内容加载中，请稍等……

2017-05-08 22:34

圆角三角函数数列变换

该范例通过单位圆上的点及其关于坐标轴、原点的对称点，直观展示三角函数值的对称关系。核心数学概念包括单位圆定义、正弦/余弦的符号变化、对称变换（关于x轴、y轴、原点）以及角度与弧度的对应。作图要点：使用GeoGebra的Sequence命令生成单位圆上的系列点，通过Mirror命令创建对称点，并用Segment、Polygon、Angle等对象构造三角形和角度标注，同时涉及对数函数、参数曲线等辅助元素。适合高中或大学预科阶段学习三角函数对称性、单位圆性质以及几何变换的课堂演示或自主探究，也可用于理解周期性。

GeoGebra 命令及参数

Sequence((cos(k°), sin(k°)), k, 0, 360) → l1, {(1, 0), (0.9998476951563913, 0.01745240643728351), (0.9993908270190958, 0.03489949670250097), (0.9986295347545738, 0.052335956242943835), (0.9975640502598242, 0.0697564737441253), (0.9961946980917455, 0.08715574274765817), (0.9945218953682733, 0.10452846326765347), (0.992546151641322, 0.12186934340514748), (0.9902680687415704, 0.13917310096006544), (0.9876883405951378, 0.15643446504023087), (0.984807753012208, 0.17364817766693033), (0.981627183447664, 0.1908089953765448), (0.9781476007338057, 0.20791169081775934), (0.9743700647852352, 0.224951054343865), (0.9702957262759965, 0.24192189559966773), (0.9659258262890683, 0.25881904510252074), (0.9612616959383189, 0.27563735581699916), (0.9563047559630354, 0.29237170472273677), (0.9510565162951535, 0.3090169943749474), (0.9455185755993168, 0.3255681544571567), (0.9396926207859084, 0.3420201433256687), (0.9335804264972017, 0.35836794954530027), (0.9271838545667874, 0.374606593415912), (0.9205048534524404, 0.39073112848927377), (0.9135454576426009, 0.4067366430758002), (0.9063077870366499, 0.42261826174069944), (0.898794046299167, 0.4383711467890774), (0.8910065241883679, 0.45399049973954675), (0.882947592858927, 0.4694715627858908), (0.8746197071393957, 0.48480962024633706), (0.8660254037844387, 0.49999999999999994), (0.8571673007021123, 0.5150380749100542), (0.848048096156426, 0.5299192642332049), (0.838670567945424, 0.5446390350150271), (0.8290375725550416, 0.5591929034707469), (0.8191520442889918, 0.573576436351046), (0.8090169943749475, 0.5877852522924731), (0.7986355100472928, 0.6018150231520483), (0.7880107536067219, 0.6156614753256583), (0.7771459614569709, 0.6293203910498374), (0.7660444431189781, 0.6427876096865393), (0.754709580222772, 0.6560590289905073), (0.7431448254773942, 0.6691306063588582), (0.7313537016191705, 0.6819983600624985), (0.7193398003386512, 0.6946583704589973), (0.7071067811865476, 0.7071067811865475), (0.6946583704589974, 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(0.3420201433256688, 0.9396926207859083), (0.32556815445715676, 0.9455185755993167), (0.30901699437494745, 0.9510565162951535), (0.29237170472273677, 0.9563047559630354), (0.27563735581699916, 0.9612616959383189), (0.25881904510252074, 0.9659258262890683), (0.24192189559966767, 0.9702957262759965), (0.22495105434386492, 0.9743700647852352), (0.20791169081775945, 0.9781476007338056), (0.19080899537654492, 0.981627183447664), (0.17364817766693041, 0.984807753012208), (0.15643446504023092, 0.9876883405951378), (0.13917310096006547, 0.9902680687415704), (0.12186934340514748, 0.992546151641322), (0.10452846326765344, 0.9945218953682733), (0.08715574274765812, 0.9961946980917455), (0.06975647374412523, 0.9975640502598242), (0.052335956242943966, 0.9986295347545738), (0.03489949670250108, 0.9993908270190958), (0.0174524064372836, 0.9998476951563913), (0, 1), (-0.017452406437283473, 0.9998476951563913), (-0.034899496702500955, 0.9993908270190958), (-0.05233595624294384, 0.9986295347545738), 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(0.9702957262759965, -0.24192189559966787), (0.9743700647852351, -0.22495105434386534), (0.9781476007338056, -0.20791169081775987), (0.981627183447664, -0.19080899537654467), (0.984807753012208, -0.1736481776669304), (0.9876883405951377, -0.15643446504023112), (0.9902680687415703, -0.13917310096006588), (0.9925461516413221, -0.12186934340514724), (0.9945218953682733, -0.10452846326765342), (0.9961946980917455, -0.08715574274765832), (0.9975640502598242, -0.06975647374412565), (0.9986295347545738, -0.05233595624294437), (0.9993908270190958, -0.03489949670250082), (0.9998476951563913, -0.01745240643728356), (1, 0)}
Point(l1) → A
Mirror(A, yAxis) → A'
Mirror(A, xAxis) → A'_1
Mirror(A', xAxis) → A''
Mirror(B, yAxis) → B'
Mirror(B, xAxis) → B'_1
Mirror(B', xAxis) → B''
Segment(A, C) → f
Segment(B, C) → g
Segment(B', C) → h
Segment(A', C) → i
Segment(A'', C) → j
Segment(B'', C) → k
Segment(B'_1, C) → l
Segment(A'_1, C) → m
Segment(A, A') → n
Segment(A', A'') → p
Segment(A'', A'_1) → q
Segment(A'_1, A) → r
Segment(B, B') → s
Segment(B', B'') → t
Segment(B'', B'_1) → a
Segment(B'_1, B) → b
Polygon(A, C, A'_1) → poly1, a'_1, a_1, c_1
Polygon(A', C, A'') → poly2, a'', a', c_2
Polygon(B, B'_1, C) → poly3, c_3, b_1, b'_1
Polygon(B', C, B'') → poly4, b'', b', c_4
Polygon(A, B, A'', B'') → poly5, a_2, b_2, a''_1, b''_1
Polygon(B', A', B'_1, A'_1) → poly6, b'_2, a'_2, b'_3, a'_3
Segment(B', B'') → f_1
Segment(B, B'_1) → g_1
Segment(B', B'_1) → h_1
Segment(B, B'') → i_1
Polygon(A, B', A'', B'_1) → poly7, a_3, b'_4, a''_2, b'_5
Polygon(B, A', B'', A'_1) → poly8, b_3, a'_4, b''_2, a'_5
Angle(D, C, A) → α