📚 GCSE Cambridge Additional Mathematics: Formula & Theorem Quick Reference Handbook | GCSE 剑桥进阶数学：公式定理速查手册
This is a concise, exam-focused quick reference handbook for the Cambridge IGCSE Additional Mathematics (0606/4037) syllabus. All key formulae, theorems, and relationships are summarised in a logically grouped format, enabling rapid revision and easy memorisation. No detailed derivations are included — only the essential results you need to apply in examination questions. Use this as your go-to checklist before the exam.
这是一份针对剑桥 IGCSE 进阶数学（0606/4037）考纲的精简速查手册。所有核心公式、定理与关系式均已按逻辑分组汇总，便于快速复习和记忆。手册不含详细推导，只保留解题所必需的关键结果。考前用它作为你的终极检查清单。
1. Basic Algebra and Surds | 基础代数与根式
Simplifying surds: √(ab) = √a × √b, √(a/b) = √a / √b, (√a)² = a. Rationalising the denominator: multiply both numerator and denominator by the conjugate if needed.
根式化简：√(ab) = √a × √b，√(a/b) = √a / √b，(√a)² = a。分母有理化：如有需要，分子分母同乘共轭根式。
Quadratic formula: for ax² + bx + c = 0, x = (-b ± √(b² – 4ac)) / (2a). Discriminant Δ = b² – 4ac. Two distinct real roots when Δ > 0, one repeated root when Δ = 0, no real roots when Δ < 0.
一元二次方程求根公式：对 ax² + bx + c = 0，x = (-b ± √(b² – 4ac)) / (2a)。判别式 Δ = b² – 4ac。Δ > 0 有两不等实根，Δ = 0 有重根，Δ < 0 无实根。
Completing the square: for x² + bx + c, write as (x + b/2)² – (b/2)² + c. The vertex of y = a(x – h)² + k is (h, k).
配方法：对 x² + bx + c，可写为 (x + b/2)² – (b/2)² + c。y = a(x – h)² + k 的顶点为 (h, k)。
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = √[n]a.
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a^(1/n) = ⁿ√a。
2. Polynomials and Partial Fractions | 多项式与部分分式
Remainder theorem: when a polynomial P(x) is divided by (x – a), the remainder is P(a). Factor theorem: (x – a) is a factor of P(x) if and only if P(a) = 0.
余式定理：多项式 P(x) 除以 (x – a) 所得余数为 P(a)。因式定理：(x – a) 是 P(x) 的因式当且仅当 P(a) = 0。
For a cubic with roots α, β, γ: α + β + γ = –b/a, αβ + βγ + γα = c/a, αβγ = –d/a for ax³ + bx² + cx + d = 0. For quadratics: sum of roots = –b/a, product = c/a.
对三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ 时：α + β + γ = –b/a，αβ + βγ + γα = c/a，αβγ = –d/a。二次方程：根的和 = –b/a，积 = c/a。
Partial fractions: (px + q)/((x – a)(x – b)) = A/(x – a) + B/(x – b). For repeated linear factor (x – a)², use A/(x – a) + B/(x – a)². For improper fractions, perform polynomial long division first.
部分分式：(px + q)/((x – a)(x – b)) = A/(x – a) + B/(x – b)。含重因式 (x – a)² 时，用 A/(x – a) + B/(x – a)²。假分式需先进行多项式长除法。
3. Binomial Expansion | 二项式展开
For a positive integer n: (1 + x)ⁿ = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … + xⁿ. The general term: ⁿCᵣ xʳ or (ⁿᵣ) xʳ.
对于正整数 n：(1 + x)ⁿ = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … + xⁿ。通项为 ⁿCᵣ xʳ 或 (ⁿᵣ) xʳ。
For (a + b)ⁿ, the (r+1)th term is ⁿCᵣ aⁿ⁻ʳ bʳ. When |x| < 1 and n is rational, the expansion (1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + ... is infinite. Condition: |x| < 1.
(a + b)ⁿ 的第 (r+1) 项为 ⁿCᵣ aⁿ⁻ʳ bʳ。当 |x| < 1 且 n 为有理数时，展开式 (1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + ... 为无穷级数，收敛条件为 |x| < 1。
Combination formula: ⁿCᵣ = n! / (r!(n – r)!). Use symmetry: ⁿCᵣ = ⁿCₙ₋ᵣ.
组合公式：ⁿCᵣ = n! / (r!(n – r)!)。对称性：ⁿCᵣ = ⁿCₙ₋ᵣ。
4. Exponential and Logarithmic Functions | 指数与对数函数
Definition: if aˣ = N then x = logₐ N. Natural log: ln x = logₑ x, where e ≈ 2.71828. Common log: log₁₀ x is often written as lg x.
定义：若 aˣ = N，则 x = logₐ N。自然对数：ln x = logₑ x，其中 e ≈ 2.71828。常用对数：log₁₀ x 常写作 lg x。
Log laws: logₐ (MN) = logₐ M + logₐ N; logₐ (M/N) = logₐ M – logₐ N; logₐ Mⁿ = n logₐ M. Change of base: logₐ b = logₓ b / logₓ a. ln and e are inverses: e^(ln x) = x, ln(eˣ) = x.
对数运算法则：logₐ (MN) = logₐ M + logₐ N；logₐ (M/N) = logₐ M – logₐ N；logₐ Mⁿ = n logₐ M。换底公式：logₐ b = logₓ b / logₓ a。ln 与 e 互为反函数：e^(ln x) = x，ln(eˣ) = x。
Solving exponential equations: take logs on both sides and apply log laws. For growth/decay models, A = P e^(kt) or A = P (1 ± r)^t.
解指数方程：两边取对数并运用对数律。增长/衰退模型常用 A = P e^(kt) 或 A = P (1 ± r)^t。
5. Coordinate Geometry and Straight Lines | 坐标几何与直线
Distance between two points (x₁, y₁) and (x₂, y₂): d = √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 与 (x₂, y₂) 间距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Gradient of line through two points: m = (y₂ – y₁) / (x₂ – x₁). Equation of a line: y – y₁ = m(x – x₁), or y = mx + c. Parallel lines have same gradient; perpendicular lines satisfy m₁ × m₂ = –1.
过两点的斜率：m = (y₂ – y₁) / (x₂ – x₁)。直线方程：y – y₁ = m(x – x₁)，或 y = mx + c。平行线斜率相等；垂直线满足 m₁ × m₂ = –1。
Distance from a point (x₁, y₁) to line Ax + By + C = 0: |Ax₁ + By₁ + C| / √(A² + B²). Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): ½| x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂) |.
点 (x₁, y₁) 到直线 Ax + By + C = 0 的距离：|Ax₁ + By₁ + C| / √(A² + B²)。以 (x₁,y₁), (x₂,y₂), (x₃,y₃) 为顶点的三角形面积：½| x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂) |。
6. Functions and Graph Transformations | 函数与图像变换
Domain (input set) and range (output set). Composite function: (g∘f)(x) = g(f(x)). Inverse function f⁻¹(x): swap x and y and solve for y. The graph of f⁻¹ is the reflection of f in y = x. A function has an inverse only if it is one-to-one.
定义域（输入集合）和值域（输出集合）。复合函数：(g∘f)(x) = g(f(x))。反函数 f⁻¹(x)：交换 x 和 y 后解出 y。f⁻¹ 的图像是 f 关于 y = x 的反射。仅当函数是一一映射时才有反函数。
Graph transformations: y = f(x) + a → vertical shift up by a; y = f(x + a) → horizontal shift left by a; y = a f(x) → vertical stretch by factor a; y = f(ax) → horizontal stretch by factor 1/a; y = –f(x) → reflection in x-axis; y = f(–x) → reflection in y-axis.
图像变换：y = f(x) + a 表示上移 a 个单位；y = f(x + a) 表示左移 a 个单位；y = a f(x) 表示纵轴拉伸 a 倍；y = f(ax) 表示横轴拉伸 1/a 倍；y = –f(x) 关于 x 轴反射；y = f(–x) 关于 y 轴反射。
7. Trigonometry: Ratios and Graphs | 三角学：比值与图像
In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. SOHCAHTOA. For any angle θ, use CAST diagram to determine sign.
直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。（SOHCAHTOA 口诀）对任意角 θ，用 CAST 象限图判断符号。
Exact values for special angles:
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Radian measure: π radians = 180°. Arc length s = rθ, sector area A = ½ r²θ (θ in radians).
弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ（θ 以弧度为单位）。
8. Trigonometric Identities and Equations | 三角恒等式与方程
Fundamental identity: sin²θ + cos²θ = 1. Derived: tan²θ + 1 = sec²θ, 1 + cot²θ = cosec²θ. (sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ.)
基本恒等式：sin²θ + cos²θ = 1。导出：tan²θ + 1 = sec²θ，1 + cot²θ = cosec²θ。（sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ = cos θ/sin θ。）
Compound angle formulae: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).
和角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。
Double angle: 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：sin 2A = 2 sin A cos A；cos 2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A；tan 2A = 2 tan A / (1 – tan²A)。
Expressing a sin θ ± b cos θ as R sin(θ ± α) or R cos(θ ∓ α), where R = √(a² + b²) and tan α = b/a.
将 a sin θ ± b cos θ 表达为 R sin(θ ± α) 或 R cos(θ ∓ α) 的形式，其中 R = √(a² + b²)，tan α = b/a。
9. Differentiation | 微分
Derivative as rate of change: dy/dx = lim(h→0) [f(x+h) – f(x)] / h. For f(x) = xⁿ, f ‘(x) = n xⁿ⁻¹. Constant rule: derivative of constant is zero. Sum rule: (u ± v)’ = u’ ± v’.
导数表示变化率：dy/dx = lim(h→0) [f(x+h) – f(x)] / h。对 f(x) = xⁿ，f ‘(x) = n xⁿ⁻¹。常数法则：常数的导数为零。加法法则：(u ± v)’ = u’ ± v’。
Product rule: if y = u v, then dy/dx = u’v + u v’. Quotient rule: if y = u/v, then dy/dx = (u’v – u v’) / v².
乘法法则：若 y = u v，则 dy/dx = u’v + u v’。除法法则：若 y = u/v，则 dy/dx = (u’v – u v’) / v²。
Chain rule: dy/dx = dy/du × du/dx. For composite functions, differentiate outer function times derivative of inner function.
链式法则：dy/dx = dy/du × du/dx。对复合函数，先对外层函数求导再乘以内层函数的导数。
Differentiation of standard functions: d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x; d/dx (sin x) = cos x; d/dx (cos x) = –sin x; d/dx (tan x) = sec² x.
标准函数求导：d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x；d/dx (sin x) = cos x；d/dx (cos x) = –sin x；d/dx (tan x) = sec² x。
Second derivative: f ”(x) or d²y/dx². Nature of stationary points: f ‘(x)=0. If f ”(x) > 0, minimum point; if f ”(x) < 0, maximum point; if f ''(x)=0, use first derivative test.
二阶导数：f ”(x) 或 d²y/dx²。驻点性质：f ‘(x)=0。若 f ”(x) > 0 为极小点；f ”(x) < 0 为极大点；若 f ''(x)=0，用一阶导数检验法判断。
10. Integration | 积分
Integration as reverse of differentiation: ∫ f ‘(x) dx = f(x) + c. For xⁿ (n ≠ –1): ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c. For 1/x: ∫ (1/x) dx = ln|x| + c.
积分是微分的逆运算：∫ f ‘(x) dx = f(x) + c。对 xⁿ (n ≠ –1)：∫ xⁿ dx = xⁿ⁺¹/(n+1) + c。对 1/x：∫ (1/x) dx = ln|x| + c。
Standard integrals: ∫ eˣ dx = eˣ + c; ∫ sin x dx = –cos x + c; ∫ cos x dx = sin x + c; ∫ sec² x dx = tan x + c.
标准积分：∫ eˣ dx = eˣ + c；∫ sin x dx = –cos x + c；∫ cos x dx = sin x + c；∫ sec² x dx = tan x + c。
Definite integral: ∫ₐᵇ f(x) dx = F(b) – F(a). Area between curve and x-axis: A = ∫ₐᵇ |f(x)| dx if the curve crosses the axis. Volume of revolution about x-axis: V = π ∫ₐᵇ y² dx.
定积分：∫ₐᵇ f(x) dx = F(b) – F(a)。曲线与 x 轴围成的面积：若曲线穿过 x 轴，则 A = ∫ₐᵇ |f(x)| dx。绕 x 轴旋转的体积：V = π ∫ₐᵇ y² dx。
Trapezium rule (approximate integration): ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)], where h = (b – a)/n.
梯形法则（近似积分）：∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]，其中 h = (b – a)/n。
11. Kinematics and Applications of Calculus | 运动学与微积分应用
For a particle moving in a straight line: displacement s, velocity v = ds/dt, acceleration a = dv/dt = d²s/dt². Reverse: v = ∫ a dt, s = ∫ v dt. Use initial conditions to determine constants.
对于直线运动的质点：位移 s，速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。反之：v = ∫ a dt，s = ∫ v dt。利用初始条件确定积分常数。
Stationary in kinematics: v = 0. Maximum/minimum displacement occurs when v = 0. Distance travelled is total area under |v|-t graph, distinct from displacement.
运动学中的静止：v = 0。位移的极大/极小值出现在 v = 0 时。路程为 |v|-t 图像下的总面积，与位移不同。
Connected rates: if three variables are related, use chain rule e.g. dy/dt = dy/dx × dx/dt. Useful for problems where you differentiate or integrate with respect to time.
相关变化率：若三个变量相关，用链式法则，如 dy/dt = dy/dx × dx/dt。常用于对时间求导或积分的问题。
Optimisation: find max/min of a function by solving f ‘(x)=0 and testing second derivative. Common applications: maximum volume, minimum surface area.
最优化：通过解 f ‘(x)=0 并验证二阶导数求函数的极大/极小值。常见应用：最大体积、最小表面积。
12. Matrices and Transformations (if applicable) | 矩阵与变换（如有涉及）
Some syllabi include matrix operations: 2×2 matrix multiplication, determinant det A = ad – bc, inverse A⁻¹ = (1/det A) × [[d, –b], [–c, a]]. Check if transformations are in your specific syllabus.
部分考纲包含矩阵运算：2×2 矩阵乘法，行列式 det A = ad – bc，逆矩阵 A⁻¹ = (1/det A) × [[d, –b], [–c, a]]。请确认你的具体考纲是否包含变换内容。
Common transformations: rotation, reflection, enlargement. Matrix representation can be used to find image of points. This section serves as a quick reminder for matrix-based questions.
常见变换：旋转、反射、放大。矩阵可用来求点的像。本节为基于矩阵的题目提供快速提示。
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