📚 Math Formula Compendium | 数学常用公式集锦
This comprehensive guide brings together the most essential formulas across algebra, geometry, trigonometry, calculus, and statistics. Designed for exam revision and quick reference, each section pairs clear English explanations with Chinese translations, ensuring students can master the language of mathematics wherever they study.
本综合指南汇集了代数、几何、三角学、微积分和统计学中最基本的公式。专为考前复习和快速查阅设计，每个部分都清晰地将英文解释与中文翻译配对，确保学生无论在哪里学习都能掌握数学的语言。
1. Algebraic Identities | 代数恒等式
Algebraic identities form the bedrock of equation manipulation. The square of a binomial (a + b)² expands to a² + 2ab + b², while (a – b)² gives a² – 2ab + b². The difference of two squares factorises neatly: a² – b² = (a + b)(a – b).
代数恒等式是方程式变形的基础。二项式的平方 (a + b)² 展开为 a² + 2ab + b²，而 (a – b)² 给出 a² – 2ab + b²。平方差公式分解得很整齐：a² – b² = (a + b)(a – b)。
The cube expansions are frequently tested: (a + b)³ = a³ + 3a²b + 3ab² + b³, and (a – b)³ = a³ – 3a²b + 3ab² – b³. The sum and difference of cubes can be rewritten as a³ + b³ = (a + b)(a² – ab + b²) and a³ – b³ = (a – b)(a² + ab + b²).
立方展开式经常被考查：(a + b)³ = a³ + 3a²b + 3ab² + b³, 以及 (a – b)³ = a³ – 3a²b + 3ab² – b³。立方和与立方差可以改写为 a³ + b³ = (a + b)(a² – ab + b²) 和 a³ – b³ = (a – b)(a² + ab + b²)。
2. Quadratic Equations | 二次方程
A quadratic equation in standard form is ax² + bx + c = 0 (a ≠ 0). Its solutions are given by the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a. The discriminant Δ = b² – 4ac determines the nature of roots: two distinct real roots when Δ > 0, one repeated real root when Δ = 0, and two complex conjugate roots when Δ < 0.
标准形式的二次方程为 ax² + bx + c = 0 (a ≠ 0)。其解由求根公式给出：x = [-b ± √(b² – 4ac)] / 2a。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 时有两个不等实根，Δ = 0 时有一个重实根，Δ < 0 时有一对共轭复根。
Sum and product of roots (α + β = -b/a, αβ = c/a) enable quick checks and symmetrical equation construction. Completing the square rewrites ax² + bx + c as a(x + b/(2a))² + (c – b²/(4a)), providing an alternative route to the vertex form.
根的和与积 (α + β = -b/a, αβ = c/a) 可以快速检验和构造对称方程。配方法将 ax² + bx + c 改写为 a(x + b/(2a))² + (c – b²/(4a))，提供了转化为顶点式的另一种途径。
3. Geometry – Areas and Volumes | 几何 – 面积与体积
Key plane figures include the circle (area = πr², circumference = 2πr), triangle (area = ½ × base × height, or Heron’s formula for sides a,b,c: √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2), and parallelogram (area = base × height). The area of a trapezium is ½(a+b)h where a,b are parallel sides and h is the perpendicular height.
关键平面图形包括圆（面积 = πr², 周长 = 2πr），三角形（面积 = ½ × 底 × 高，或已知三边 a,b,c 的海伦公式：√[s(s-a)(s-b)(s-c)]，其中 s = (a+b+c)/2），和平行四边形（面积 = 底 × 高）。梯形的面积是 ½(a+b)h，其中 a,b 是平行边，h 是垂直高度。
For solids, the volume of a cylinder is πr²h, a sphere is (4/3)πr³, a cone is (1/3)πr²h, and a pyramid is (1/3) × base area × height. Surface areas: sphere = 4πr², open cylinder = 2πrh, cone lateral surface = πrl (where l = slant height).
对于立体，圆柱体积为 πr²h，球体积为 (4/3)πr³，圆锥体积为 (1/3)πr²h，棱锥体积为 (1/3) × 底面积 × 高。表面积：球 = 4πr²，开口圆柱 = 2πrh，圆锥侧面积 = πrl（其中 l 为斜高）。
4. Trigonometry – Right-Angled Triangles | 三角学 – 直角三角形
In a right triangle, sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. The Pythagorean identity sin²θ + cos²θ = 1 is fundamental, along with the complementary relations sin(90° – θ) = cos θ, tan(90° – θ) = cot θ.
在直角三角形中，正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。勾股恒等式 sin²θ + cos²θ = 1 是基础，余角关系同样重要：sin(90° – θ) = cos θ, tan(90° – θ) = cot θ。
Exact values for 30°, 45°, 60° should be memorised: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
应牢记 30°、45°、60° 的精确值：sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3；sin 45° = cos 45° = 1/√2, tan 45° = 1；sin 60° = √3/2, cos 60° = ½, tan 60° = √3。
5. Sine and Cosine Rules | 正弦定理与余弦定理
For any triangle with sides a, b, c and opposite angles A, B, C, the sine rule states: a/sin A = b/sin B = c/sin C = 2R (R being the circumradius). It is used when two angles and a side are known (AAS, ASA) or two sides and a non-included angle (SSA, the ambiguous case).
对于任意三角形，边长为 a, b, c，对角为 A, B, C，正弦定理指出：a/sin A = b/sin B = c/sin C = 2R（R 为外接圆半径）。当已知两角一边（AAS, ASA）或两边及一个非夹角（SSA，不确定情况）时使用。
The cosine rule extends Pythagoras: a² = b² + c² – 2bc cos A. It is applicable for SAS (two sides and included angle) and SSS (three sides). The area formula involving sine is Area = ½ ab sin C, meaning any two sides and their included angle give the area.
余弦定理扩展了勾股定理：a² = b² + c² – 2bc cos A。它适用于 SAS（两边及夹角）和 SSS（三边）情形。包含正弦的面积公式为 Area = ½ ab sin C，即任意两边及其夹角可求面积。
6. Trigonometric Identities | 三角恒等式
The double-angle formulas are critical: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ; tan 2θ = (2 tan θ) / (1 – tan²θ). The sum-to-product and product-to-sum identities help in integration and equation solving.
二倍角公式至关重要：sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ；tan 2θ = (2 tan θ) / (1 – tan²θ)。和差化积与积化和差恒等式有助于积分和方程求解。
Reciprocal functions: sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ. The Pythagorean identities extend to 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
倒数函数：sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ。勾股恒等式扩展为 1 + tan²θ = sec²θ 和 1 + cot²θ = cosec²θ。
7. Exponents and Logarithms | 指数与对数
The exponent laws: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1 (a ≠ 0). For logarithms, logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x – logₐ y, logₐ(xⁿ) = n logₐ x. The change-of-base rule: logₐ b = log_c b / log_c a.
指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1 (a ≠ 0)。对数运算：logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x – logₐ y, logₐ(xⁿ) = n logₐ x。换底公式：logₐ b = log_c b / log_c a。
The natural logarithm ln x = logₑ x, with e ≈ 2.71828. The relationship between exponentials and logs is given by aˣ = b ⇔ x = logₐ b. In calculus, d/dx(eˣ) = eˣ and d/dx(ln x) = 1/x.
自然对数 ln x = logₑ x，其中 e ≈ 2.71828。指数与对数的关系为 aˣ = b ⇔ x = logₐ b。在微积分中，d/dx(eˣ) = eˣ，d/dx(ln x) = 1/x。
8. Sequences and Series | 数列与级数
Arithmetic progression: nth term uₙ = a + (n-1)d, sum of first n terms Sₙ = n/2[2a + (n-1)d] = n/2(a + l) where l is the last term. Geometric progression: uₙ = arⁿ⁻¹, sum Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1, and sum to infinity S∞ = a/(1 – r) valid for |r| < 1.
等差数列：第 n 项 uₙ = a + (n-1)d，前 n 项和 Sₙ = n/2[2a + (n-1)d] = n/2(a + l)，其中 l 为末项。等比数列：uₙ = arⁿ⁻¹，和 Sₙ = a(1 – rⁿ)/(1 – r)（r ≠ 1），无穷项和 S∞ = a/(1 – r) 适用于 |r| < 1。
The binomial expansion for (1 + x)ⁿ where n is rational and |x| < 1 is 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + ... In particular, when n is a positive integer, the expansion terminates and coefficients are given by Pascal's triangle.
二项式展开 (1 + x)ⁿ，其中 n 为有理数且 |x| < 1，为 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + ... 特别地，当 n 为正整数时，展开式终止，系数由帕斯卡三角形给出。
9. Differentiation | 微分学
The derivative of a function f(x) is defined as f'(x) = lim_{h→0} [f(x+h) – f(x)]/h. Basic derivatives: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec² x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.
函数 f(x) 的导数定义为 f'(x) = lim_{h→0} [f(x+h) – f(x)]/h。基本导数：d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec² x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x。
Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’)/v². Chain rule: dy/dx = dy/du × du/dx. These three rules handle most composite functions encountered in A-level mathematics.
乘法法则：(uv)’ = u’v + uv’。除法法则：(u/v)’ = (u’v – uv’)/v²。链式法则：dy/dx = dy/du × du/dx。这三条法则可以处理 A-level 数学中遇到的大多数复合函数。
10. Integration | 积分学
Integration reverses differentiation. The indefinite integral ∫ f(x) dx gives the antiderivative plus constant C. Basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C.
积分是微分的逆运算。不定积分 ∫ f(x) dx 给出原函数加常数 C。基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C。
The definite integral ∫ₐᵇ f(x) dx calculates the area under the curve between x = a and x = b. Integration by substitution and integration by parts (∫ u dv = uv – ∫ v du) are the two most powerful techniques for more complex integrands.
定积分 ∫ₐᵇ f(x) dx 计算曲线在 x = a 到 x = b 之间的面积。换元积分法和分部积分法 (∫ u dv = uv – ∫ v du) 是处理更复杂被积函数的两个最有力技巧。
11. Probability and Statistics | 概率与统计
For mutually exclusive events, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A) × P(B). The addition rule for any two events is P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Conditional probability: P(A|B) = P(A ∩ B)/P(B).
对于互斥事件，P(A ∪ B) = P(A) + P(B)。对于独立事件，P(A ∩ B) = P(A) × P(B)。任意两个事件的加法法则是 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。条件概率：P(A|B) = P(A ∩ B)/P(B)。
The mean of a discrete random variable X is E(X) = Σ x·P(X=x); variance Var(X) = E(X²) – [E(X)]². For the binomial distribution X ~ B(n, p): E(X) = np, Var(X) = np(1-p). For the normal distribution X ~ N(μ, σ²), the standardised variable Z = (X – μ)/σ follows N(0,1).
离散随机变量 X 的期望为 E(X) = Σ x·P(X=x)；方差 Var(X) = E(X²) – [E(X)]²。二项分布 X ~ B(n, p)：E(X) = np, Var(X) = np(1-p)。正态分布 X ~ N(μ, σ²)，标准化变量 Z = (X – μ)/σ 服从 N(0,1)。
12. Vectors | 向量
A vector in 2D or 3D has magnitude |a| = √(x² + y² + z²). The dot product a · b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂ is used to find angles and test perpendicularity (a·b = 0). The cross product a × b (in 3D) yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ.
二维或三维向量的模为 |a| = √(x² + y² + z²)。点积 a · b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂ 用于求角度和检验垂直（a·b = 0）。叉积 a × b（在三维中）给出一个垂直于 a 和 b 的向量，大小为 |a||b| sin θ。
The position vector from origin O to point P is commonly written as OP = xi + yj + zk. The vector equation of a line is r = a + λb, where a is a point on the line and b is the direction vector. The equation of a plane can be expressed as r·n = d or ax + by + cz = d.
从原点 O 到点 P 的位置向量通常写作 OP = xi + yj + zk。直线的向量方程为 r = a + λb，其中 a 是线上一点，b 是方向向量。平面的方程可表示为 r·n = d 或 ax + by + cz = d。
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