📚 Advanced Mathematics Formulae Summary | 数学常用公式汇总（进阶篇）
This comprehensive summary brings together essential formulae from key advanced topics in mathematics, including algebra, trigonometry, calculus, vectors, matrices, and statistics. It is designed for A‑Level, IB, and equivalent courses, serving as a rapid revision tool and a reliable reference for problem solving.
本篇汇总了数学进阶阶段核心课题的重要公式，涵盖代数、三角、微积分、向量、矩阵与统计等内容，适合 A‑Level、IB 及同等程度的学生使用，可作为快速复习资料与解题参考。
1. Algebraic Identities and Expansions | 代数恒等式与展开式
Expanding binomials using standard identities simplifies both algebraic manipulation and equation solving.
运用标准恒等式展开二项式可以简化代数运算与方程求解。
(a ± b)² = a² ± 2ab + b²
The square of a sum or difference returns a perfect square trinomial.
和或差的平方展开后是完全平方三项式。
a² − b² = (a − b)(a + b)
Difference of two squares factors into the product of a difference and a sum.
平方差可分解为差与和的乘积。
a³ ± b³ = (a ± b)(a² ∓ ab + b²)
Sum and difference of cubes follow a predictable pattern with a binomial and a trinomial factor.
立方和与立方差按既定规律分解为一个二项式与一个三项式的乘积。
2. Quadratic and Polynomial Equations | 二次与多项式方程
Quadratic equations form the backbone of many advanced problems. Their solutions rely on discriminant analysis and the quadratic formula.
二次方程是众多进阶问题的基础，其求解依赖判别式分析与求根公式。
x = (−b ± √(b² − 4ac)) / (2a) , a ≠ 0
The quadratic formula gives the roots of ax² + bx + c = 0 directly.
二次方程 ax² + bx + c = 0 的根可直接由求根公式得出。
Discriminant Δ = b² − 4ac determines the nature of the roots.
判别式 Δ = b² − 4ac 决定根的性质。
- Δ > 0 → two distinct real roots | 两个不等实根
- Δ = 0 → one repeated real root | 一个重实根
- Δ < 0 → two complex conjugate roots | 一对共轭复根
For polynomial equations, the Factor Theorem connects roots and linear factors.
对于多项式方程，因式定理将根与一次因式联系起来。
(x − r) is a factor of P(x) ⇔ P(r) = 0
3. Logarithms and Exponentials | 对数与指数公式
Working with exponential growth, decay, and multi‑step equations demands fluency in logarithm laws.
处理指数增长、衰减及多步方程需要熟练运用对数运算律。
logₐ(mn) = logₐm + logₐn
The log of a product is the sum of logs (valid for any admissible base).
乘积的对数等于对数之和（对任何允许的底数均成立）。
logₐ(m/n) = logₐm − logₐn
logₐ(mᵏ) = k logₐm
Power rule moves the exponent to the front as a multiplier.
幂法则将指数提到前面作为乘数。
logₐb = logₓb / logₓa
Change of base is essential when calculator functions are limited to base 10 or e.
当计算器仅支持常用对数或自然对数时，换底公式至关重要。
aˣ = eˣ ˡⁿ ᵃ
Converting an exponential to base e is common in calculus.
在微积分中常将指数函数改写为以 e 为底的形式。
4. Sequences and Series | 数列与级数
Arithmetic and geometric progressions appear frequently in finance, population modelling, and pure mathematics.
等差数列与等比数列在金融、人口模型及纯数学中频频出现。
Arithmetic progression: n‑th term uₙ = a + (n − 1)d; sum Sₙ = n/2 [2a + (n − 1)d].
等差数列：第 n 项 uₙ = a + (n − 1)d；前 n 项和 Sₙ = n/2 [2a + (n − 1)d]。
Geometric progression: n‑th term uₙ = arⁿ⁻¹; sum Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1.
等比数列：第 n 项 uₙ = arⁿ⁻¹；前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1。
Sum to infinity for |r| < 1: S∞ = a/(1 − r).
无穷等比级数和（|r| < 1）：S∞ = a/(1 − r)。
The binomial expansion (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + … is valid for |x| < 1 when n is not a positive integer.
二项展开式 (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + …，当 n 非正整数时要求 |x| < 1。
5. Trigonometry Fundamentals | 三角学基础公式
A strong command of exact values and fundamental relationships between sine, cosine, and tangent is indispensable.
熟练掌握正弦、余弦、正切的精确值及其基本关系式是必不可少的。
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | ½ | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | ½ | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | — |
Exact trigonometric values for standard angles simplify many expressions immediately.
标准角的精确三角值可直接化简许多表达式。
tan θ = sin θ / cos θ
sin²θ + cos²θ ≡ 1
The Pythagorean identity is the most important tool for proving other identities.
勾股恒等式是证明其他恒等式的最重要工具。
6. Trigonometric Equations and Identities | 三角方程与恒等式
Solving equations in a given interval relies on the symmetry and periodicity of the trigonometric functions.
在给定区间内解三角方程需要利用三角函数的对称性与周期性。
General solutions: sin θ = k → θ = nπ + (−1)ⁿα, cos θ = k → θ = 2nπ ± α, tan θ = k → θ = nπ + α (α = principal value).
通解公式：sin θ = k → θ = nπ + (−1)ⁿα，cos θ = k → θ = 2nπ ± α，tan θ = k → θ = nπ + α（α 为主值）。
Double angle formulae: sin 2A ≡ 2 sin A cos A, cos 2A ≡ cos²A − sin²A ≡ 2cos²A − 1 ≡ 1 − 2sin²A, tan 2A ≡ 2 tan A / (1 − tan²A).
倍角公式在积分及解方程中极为常用。
Addition formulae: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B.
和差公式是推导众多恒等式的基础。
Product‑to‑sum identities help integrate products of sine and cosine.
积化和差恒等式用于正弦与余弦乘积的积分。
7. Differentiation Rules | 微分法则
Derivatives describe instantaneous rates of change. The following table collects standard derivatives and rules.
导数描述瞬时变化率。下表汇集了标准导数和求导法则。
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| aˣ | aˣ ln a |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
Chain rule: dy/dx = dy/du × du/dx.
链式法则：y 对 x 的导数等于 y 对 u 的导数乘以 u 对 x 的导数。
Product rule: d(uv)/dx = u’v + uv’.
积法则：两项乘积的导数为两导数分别乘另一函数之和。
Quotient rule: d(u/v)/dx = (u’v − uv’)/v².
商法则：分子导数乘分母减分子乘分母导数，除分母的平方。
8. Integration Techniques | 积分技巧
Integration is the reverse of differentiation. Standard integrals and methods are summarised below.
积分是微分的逆运算。标准积分与方法如下。
| Function f(x) | Integral ∫ f(x) dx |
|---|---|
| xⁿ, n ≠ −1 | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x).
换元积分法：令 u = g(x)，将原积分化为关于 u 的积分。
Integration by parts: ∫ u dv = uv − ∫ v du.
分部积分法：来源于乘积法则的逆用。
For definite integrals, ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x).
定积分计算依赖于原函数的差值。
Area between curves: ∫ₐᵇ [f(x) − g(x)] dx for f(x) ≥ g(x) on [a, b].
曲线间面积公式适用于上方函数减下方函数的积分。
9. Vectors and Coordinate Geometry | 向量与坐标几何
Vectors provide a powerful language for describing directions and magnitudes in 2D and 3D.
向量为描述平面及空间中的方向与大小提供了有力工具。
If a = a₁i + a₂j + a₃k, then magnitude |a| = √(a₁² + a₂² + a₃²).
向量的模长公式如上。
Dot product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.
点积可用于计算夹角，且两向量垂直时点积为零。
Vector equation of a line: r = a + t b, where a is a point on the line and b is a direction vector.
直线向量方程：a 为线上一点，b 为方向向量。
In coordinate geometry, gradient of a line: m = (y₂ − y₁)/(x₂ − x₁); equation: y − y₁ = m(x − x₁).
坐标几何中的斜率与点斜式方程。
Distance between points (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²].
两点间距离公式。
10. Matrices and Determinants | 矩阵与行列式
Matrices are used to solve linear systems, represent transformations, and compute areas.
矩阵用于求解线性方程组、表示变换及计算面积。
For 2×2 matrix A = [[a, b], [c, d]], det(A) = ad − bc.
2×2 矩阵的行列式值为 ad − bc。
Inverse: A⁻¹ = 1/det(A) [[d, −b], [−c, a]], valid when det(A) ≠ 0.
仅当行列式非零时矩阵可逆，逆矩阵公式如上。
Solving AX = B using X = A⁻¹B (unique solution if det(A) ≠ 0).
利用逆矩阵求解线性方程组，要求系数矩阵行列式不为零。
For transformation matrices, the image of a unit square has area |det(A)|.
变换矩阵对应单位正方形的像面积等于行列式的绝对值。
Rotation matrix through angle θ: [[cos θ, −sin θ], [sin θ, cos θ]].
旋转矩阵如上。
11. Probability and Statistics | 概率与统计
Data handling and probability models appear in almost every applied mathematics module.
数据处理与概率模型几乎出现在每个应用数学模块中。
Mean of a set: x̄ = Σxᵢ / n; variance: σ² = Σ(xᵢ − x̄)² / n (population) or s² = Σ(xᵢ − x̄)² / (n − 1) (sample).
均值与方差公式，区分总体与样本。
Probability addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
概率加法法则。
Conditional probability: P(A|B) = P(A ∩ B) / P(B).
条件概率公式。
For independent events, P(A ∩ B) = P(A) × P(B).
独立事件的交事件概率为各自概率的乘积。
Binomial distribution: P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ, mean = np, variance = np(1−p).
二项分布的概率、期望与方差。
Standard normal variable: z = (x − μ)/σ.
正态分布的标准化公式。
12. Complex Numbers | 复数
Complex numbers extend the real number system and are fundamental in advanced mathematics and engineering.
复数扩展了实数系，是进阶数学与工程学的基础。
A complex number is written as z = a + bi, where a, b ∈ ℝ and i² = −1.
复数的一般形式为 a + bi，其中 i² = −1。
Modulus: |z| = √(a² + b²); argument: θ = arg(z), measured from the positive real axis.
模长与辐角如上定义。
Polar form: z = r(cos θ + i sin θ) = r eⁱθ.
极形式与欧拉公式。
De Moivre’s theorem: (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ).
棣莫弗定理用于计算复数幂次与求 n 次方根。
Complex conjugate: z̄ = a − bi; z z̄ = |z|².
共轭复数的乘积等于模长的平方。
The n distinct n‑th roots of z are given by zₖ = r^(1/n) [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)], k = 0,1,…,n−1.
求复数 n 次方根的通项公式。
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