目 录
一、初中代数 ················································································································································· 2
二、高中代数 ················································································································································· 5 2.1、函数 ············································································································································ 5 2.1.1不等式 ··································································································································· 9 2.1.1数列 ····································································································································· 11 2.1.1三角函数 ····························································································································· 12 2.1.1复数 ····································································································································· 15 2.2排列、组合 ································································································································· 16 2.3平面几何 ····································································································································· 18 2.3.1直线与角 ····························································································································· 18 2.3.2三角形 ································································································································· 19 2.4立体几何 ····································································································································· 20 2.4.1直线与平面 ························································································································· 20 2.4.2多面体、棱柱、棱锥 ········································································································· 23 2.5解析几何 ····································································································································· 24 2.5.1方程与曲线 ························································································································· 24 2.5.2直线 ····································································································································· 25 2.5.3圆 ········································································································································· 27 2.5.4椭圆 ····································································································································· 28 2.5.5双曲线 ································································································································· 28 2.5抛物线 ····································································································································· 30 2.6向量部分 ····································································································································· 31 2.6.1空间向量 ····························································································································· 31 2.6.2平面向量 ····························································································································· 32
三、常用公式 ··············································································································································· 33 3.1常用公式 ····································································································································· 33 3.2几何图形及计算公式 ················································································································· 35
四、坐标几何和二维、三维图形 ············································································································· 38 4.1坐标几何 ····································································································································· 38 4.2二维图形 ····································································································································· 40 4.3三维图形 ····································································································································· 41
一、初中代数
二、高中代数
2.1、函数
2.1.1不等式
2.1.1数列
名
定 义 称 通 项 公 式 前n 项的和公式
其它
如果一个数列{an}的第n 项an 与n 之间的关系
数 按照一定次序排成一
可以用一个公式
列列的数叫做数列，记为来表示，这个公式就叫这个数列的通项公式 等差数列
等比数列
数列前n 项和与通项的关系：
无穷等比数列所有项的和：
适 用 范 围 数学
归只适用于证明与自然数n 有纳关的数学命题 法 证 明 步 骤
注 意 事 项
设P(n)是关于自然n 的一个命题，如果(1)
（1）第一步是递推的基础，第二步的推理
当n 取第一个值n0(例如：n=1或n=2)根据，两步缺一不可
命题成立(2)假设n=k时，命题成立，由此
（2）第二步的证明过程中必须使用归纳假
推出n=k+1时成立。那么P(n)对于一切自
设。
然数n 都成立。
2.1.1三角函数
2.1.1复数
2.2排列、组合
2.3平面几何
2.3.1直线与角
2.3.2三角形
2.4立体几何
2.4.1直线与平面
2.4.2多面体、棱柱、棱锥
2.5解析几何
2.5.1方程与曲线
2.5.2直线
2.5.3圆
2.5.4椭圆
2.5.5双曲线
2.5抛物线
2.6向量部分
2.6.1空间向量
2.6.2平面向量
三、常用公式
3.1常用公式
3.2几何图形及计算公式
四、坐标几何和二维、三维图形
4.1坐标几何
一对垂直相交于平面的轴线，可以让平面上的任意一点用一组实数来表示。轴线的交点是 (0, 0)，称为原点。水平与垂直方向的位置，分别用x 与y 代表。
一条直线可以用方程式y ＝mx ＋c 来表示，m 是直线的斜率（gradient ）。这条直线与y 轴相交于 (0, c)，与x 轴则相交于(–c/m, 0)。垂直线的方程式则是x ＝k ，x 为定值。 通过(x0, y0) 这一点，且斜率为n 的直线是 y–y 0＝n(x–x 0)
一条直线若垂直于斜率为n 的直线，则其斜率为–1/n。通过(x1, y1) 与(x2, y2) 两点的直线是y ＝(y2–y 1／x 2–x 1)(x–x 2) ＋y 2 （ x1≠x2 ）
若两直线的斜率分别为m 与n ，则它们的夹角θ满足于 tanθ＝m –n ／1＋mn
半径为r 、圆心在(a, b)的圆，以(x–a) 2＋(y–b) 2＝r 2表示。
三维空间里的坐标与二维空间类似，只是多加一个z 轴而已，例如半径为r 、中心位置在(a, b, c)的球，以(x–a) 2＋(y–b) 2＋(z–c) 2＝r 2表示。
三维空间平面的一般式为ax ＋by ＋cz ＝d 。
三角学 边长为a 、b 、c 的直角三角形，其中一个夹角为θ。它的六个三角函数分别为：正弦（sine ）、余弦（cosine ）、正切（tangent ）、余割（cosecant ）、正割（secant ）和余切（cotangent ）。 sinθ＝b/c cosθ＝a/c tanθ＝b/a cscθ＝c/b secθ＝c/a cotθ＝a/b
若圆的半径是1，则其正弦与余弦分别为直角三角形的高与底。 a ＝cosθ b ＝sinθ
依照勾股定理, 我们知道a 2＋b 2＝c 2。因此对于圆上的任何角度θ，我们都可得出下列的全等式： cos2θ＋sin 2θ＝1
三角恒等式
根据前几页所述的定义，可得到下列恒等式（identity ）： tanθ＝sinθ/cosθ，cotθ＝cosθ/sinθ secθ＝1/cosθ，cscθ＝1/sinθ
分别用cos 2θ与sin 2θ来除cos 2θ＋sin 2θ＝1，可得： sec 2θ–tan 2θ＝1 及 csc 2θ–cot 2θ＝1 对于负角度，六个三角函数分别为： sin(–θ)＝ –sinθ csc(–θ)＝ –cscθ cos(–θ)＝ cosθ sec(–θ)＝ secθ tan(–θ)＝ –tanθ cot(–θ)＝ –cotθ
当两角度相加时，运用和角公式： sin(α＋β)＝ sinαcosβ＋cosαsinβ
cos(α＋β)＝ cosαcosβ–sinαsinβ
tan(α＋β)＝ tanα＋tanβ／1–tanαtanβ
若遇到两倍角或三倍角，运用倍角公式： sin2α＝ 2sinαcosα sin3α＝ 3sinαcos2α–sin3α
cos2α＝ cos 2α–sin 2α cos3α＝ cos 3α–3sin 2αcosα tan 2α＝ 2tanα／1–tan 2α
tan3α＝ 3tanα–tan 3α／1–3tan 2α
4.2二维图形
下面是一些二维图形的周长与面积公式。 圆：
半径＝ r 直径d ＝2r 圆周长＝ 2πr ＝πd
面积＝πr2 (π＝3.1415926…….)
椭圆： 面积＝πab
a 与b 分别代表短轴与长轴的一半。
矩形： 面积＝ ab 周长＝ 2a＋2b
平行四边形（parallelogram ）： 面积＝ bh ＝ ab sinα 周长＝ 2a＋2b
梯形：
面积＝ 1/2h (a＋b)
周长＝ a＋b ＋h (secα＋secβ)
正n 边形：
面积＝ 1/2nb2 cot (180°/n) 周长＝ nb
四边形（i ）： 面积＝ 1/2ab sinα 四边形（ii ）：
面积＝ 1/2 (h1＋h 2) b＋ah 1＋ch 2
4.3三维图形
以下是三维立体的体积与表面积（包含底部）公式。
球体：
体积＝ 4/3πr3
表面积＝ 4πr2
方体：
体积＝ abc
表面积＝ 2(ab＋ac ＋bc)
圆柱体：
体积＝ πr 2h
表面积＝ 2πrh＋2πr2
圆锥体：
体积＝ 1/3πr2h
表面积＝πr√r2＋h 2 ＋πr2
三角锥体：
若底面积为A ，
体积＝ 1/3Ah
平截头体（frustum ）：
体积＝ 1/3πh (a2＋ab ＋b 2)
表面积＝π(a＋b)c ＋πa2＋πb2
椭球：
体积＝ 4/3πabc
环面（torus ）：
体积＝ 1/4π2 (a＋b) (b–a) 2
表面积＝π2 (b2–a 2)
目 录
一、初中代数 ················································································································································· 2
二、高中代数 ················································································································································· 5 2.1、函数 ············································································································································ 5 2.1.1不等式 ··································································································································· 9 2.1.1数列 ····································································································································· 11 2.1.1三角函数 ····························································································································· 12 2.1.1复数 ····································································································································· 15 2.2排列、组合 ································································································································· 16 2.3平面几何 ····································································································································· 18 2.3.1直线与角 ····························································································································· 18 2.3.2三角形 ································································································································· 19 2.4立体几何 ····································································································································· 20 2.4.1直线与平面 ························································································································· 20 2.4.2多面体、棱柱、棱锥 ········································································································· 23 2.5解析几何 ····································································································································· 24 2.5.1方程与曲线 ························································································································· 24 2.5.2直线 ····································································································································· 25 2.5.3圆 ········································································································································· 27 2.5.4椭圆 ····································································································································· 28 2.5.5双曲线 ································································································································· 28 2.5抛物线 ····································································································································· 30 2.6向量部分 ····································································································································· 31 2.6.1空间向量 ····························································································································· 31 2.6.2平面向量 ····························································································································· 32
三、常用公式 ··············································································································································· 33 3.1常用公式 ····································································································································· 33 3.2几何图形及计算公式 ················································································································· 35
四、坐标几何和二维、三维图形 ············································································································· 38 4.1坐标几何 ····································································································································· 38 4.2二维图形 ····································································································································· 40 4.3三维图形 ····································································································································· 41
一、初中代数
二、高中代数
2.1、函数
2.1.1不等式
2.1.1数列
名
定 义 称 通 项 公 式 前n 项的和公式
其它
如果一个数列{an}的第n 项an 与n 之间的关系
数 按照一定次序排成一
可以用一个公式
列列的数叫做数列，记为来表示，这个公式就叫这个数列的通项公式 等差数列
等比数列
数列前n 项和与通项的关系：
无穷等比数列所有项的和：
适 用 范 围 数学
归只适用于证明与自然数n 有纳关的数学命题 法 证 明 步 骤
注 意 事 项
设P(n)是关于自然n 的一个命题，如果(1)
（1）第一步是递推的基础，第二步的推理
当n 取第一个值n0(例如：n=1或n=2)根据，两步缺一不可
命题成立(2)假设n=k时，命题成立，由此
（2）第二步的证明过程中必须使用归纳假
推出n=k+1时成立。那么P(n)对于一切自
设。
然数n 都成立。
2.1.1三角函数
2.1.1复数
2.2排列、组合
2.3平面几何
2.3.1直线与角
2.3.2三角形
2.4立体几何
2.4.1直线与平面
2.4.2多面体、棱柱、棱锥
2.5解析几何
2.5.1方程与曲线
2.5.2直线
2.5.3圆
2.5.4椭圆
2.5.5双曲线
2.5抛物线
2.6向量部分
2.6.1空间向量
2.6.2平面向量
三、常用公式
3.1常用公式
3.2几何图形及计算公式
四、坐标几何和二维、三维图形
4.1坐标几何
一对垂直相交于平面的轴线，可以让平面上的任意一点用一组实数来表示。轴线的交点是 (0, 0)，称为原点。水平与垂直方向的位置，分别用x 与y 代表。
一条直线可以用方程式y ＝mx ＋c 来表示，m 是直线的斜率（gradient ）。这条直线与y 轴相交于 (0, c)，与x 轴则相交于(–c/m, 0)。垂直线的方程式则是x ＝k ，x 为定值。 通过(x0, y0) 这一点，且斜率为n 的直线是 y–y 0＝n(x–x 0)
一条直线若垂直于斜率为n 的直线，则其斜率为–1/n。通过(x1, y1) 与(x2, y2) 两点的直线是y ＝(y2–y 1／x 2–x 1)(x–x 2) ＋y 2 （ x1≠x2 ）
若两直线的斜率分别为m 与n ，则它们的夹角θ满足于 tanθ＝m –n ／1＋mn
半径为r 、圆心在(a, b)的圆，以(x–a) 2＋(y–b) 2＝r 2表示。
三维空间里的坐标与二维空间类似，只是多加一个z 轴而已，例如半径为r 、中心位置在(a, b, c)的球，以(x–a) 2＋(y–b) 2＋(z–c) 2＝r 2表示。
三维空间平面的一般式为ax ＋by ＋cz ＝d 。
三角学 边长为a 、b 、c 的直角三角形，其中一个夹角为θ。它的六个三角函数分别为：正弦（sine ）、余弦（cosine ）、正切（tangent ）、余割（cosecant ）、正割（secant ）和余切（cotangent ）。 sinθ＝b/c cosθ＝a/c tanθ＝b/a cscθ＝c/b secθ＝c/a cotθ＝a/b
若圆的半径是1，则其正弦与余弦分别为直角三角形的高与底。 a ＝cosθ b ＝sinθ
依照勾股定理, 我们知道a 2＋b 2＝c 2。因此对于圆上的任何角度θ，我们都可得出下列的全等式： cos2θ＋sin 2θ＝1
三角恒等式
根据前几页所述的定义，可得到下列恒等式（identity ）： tanθ＝sinθ/cosθ，cotθ＝cosθ/sinθ secθ＝1/cosθ，cscθ＝1/sinθ
分别用cos 2θ与sin 2θ来除cos 2θ＋sin 2θ＝1，可得： sec 2θ–tan 2θ＝1 及 csc 2θ–cot 2θ＝1 对于负角度，六个三角函数分别为： sin(–θ)＝ –sinθ csc(–θ)＝ –cscθ cos(–θ)＝ cosθ sec(–θ)＝ secθ tan(–θ)＝ –tanθ cot(–θ)＝ –cotθ
当两角度相加时，运用和角公式： sin(α＋β)＝ sinαcosβ＋cosαsinβ
cos(α＋β)＝ cosαcosβ–sinαsinβ
tan(α＋β)＝ tanα＋tanβ／1–tanαtanβ
若遇到两倍角或三倍角，运用倍角公式： sin2α＝ 2sinαcosα sin3α＝ 3sinαcos2α–sin3α
cos2α＝ cos 2α–sin 2α cos3α＝ cos 3α–3sin 2αcosα tan 2α＝ 2tanα／1–tan 2α
tan3α＝ 3tanα–tan 3α／1–3tan 2α
4.2二维图形
下面是一些二维图形的周长与面积公式。 圆：
半径＝ r 直径d ＝2r 圆周长＝ 2πr ＝πd
面积＝πr2 (π＝3.1415926…….)
椭圆： 面积＝πab
a 与b 分别代表短轴与长轴的一半。
矩形： 面积＝ ab 周长＝ 2a＋2b
平行四边形（parallelogram ）： 面积＝ bh ＝ ab sinα 周长＝ 2a＋2b
梯形：
面积＝ 1/2h (a＋b)
周长＝ a＋b ＋h (secα＋secβ)
正n 边形：
面积＝ 1/2nb2 cot (180°/n) 周长＝ nb
四边形（i ）： 面积＝ 1/2ab sinα 四边形（ii ）：
面积＝ 1/2 (h1＋h 2) b＋ah 1＋ch 2
4.3三维图形
以下是三维立体的体积与表面积（包含底部）公式。
球体：
体积＝ 4/3πr3
表面积＝ 4πr2
方体：
体积＝ abc
表面积＝ 2(ab＋ac ＋bc)
圆柱体：
体积＝ πr 2h
表面积＝ 2πrh＋2πr2
圆锥体：
体积＝ 1/3πr2h
表面积＝πr√r2＋h 2 ＋πr2
三角锥体：
若底面积为A ，
体积＝ 1/3Ah
平截头体（frustum ）：
体积＝ 1/3πh (a2＋ab ＋b 2)
表面积＝π(a＋b)c ＋πa2＋πb2
椭球：
体积＝ 4/3πabc
环面（torus ）：
体积＝ 1/4π2 (a＋b) (b–a) 2
表面积＝π2 (b2–a 2)