Edexcel Mathematics: Formula Summary Handbook | Edexcel 数学：公式汇总手册

📚 Edexcel Mathematics: Formula Summary Handbook | Edexcel 数学：公式汇总手册

This handbook brings together the essential formulas needed for Edexcel IGCSE (9-1) and Edexcel AS/A Level Mathematics. It is structured by topic to help you locate key results quickly, from arithmetic and algebra through to calculus, statistics and mechanics. Each entry pairs an English statement with its Chinese equivalent to support bilingual learners.

本手册汇总了 Edexcel IGCSE (9-1) 以及 Edexcel AS/A Level 数学所需的核心公式。手册按主题编排，方便你快速查找从算术代数到微积分、统计和力学的关键结论。每个条目都配以中英双语说明，助力双语学习者。

1. Arithmetic and Number | 算术与数

Percentage change: percentage change = (change / original value) × 100. This works for both increases and decreases.

百分比变化：百分比变化 = (变化量 / 原值) × 100。该公式同时适用于增加和减少。

Simple interest: I = P × r × t, where P is the principal, r is the annual interest rate (as a decimal), and t is the time in years.

单利：I = P × r × t，其中 P 为本金，r 为年利率（用小数表示），t 为时间（年）。

Compound interest: A = P(1 + r/n)^nt for n compounding periods per year. For annual compounding, A = P(1 + r)^t.

复利：每年复利 n 次时，A = P(1 + r/n)^nt。按年复利时，A = P(1 + r)^t。

A = P(1 + r/100)ⁿ

Upper and lower bounds: if a number is given to the nearest unit, the error interval is ±0.5 of that unit. For a rounded value x, the lower bound is x – 0.5 and the upper bound is x + 0.5 (depending on the precision).

上下界：若一个数精确到个位，则误差区间为 ±0.5 个单位。对于四舍五入值 x，下界为 x – 0.5，上界为 x + 0.5（取决于精度）。

2. Algebra and Functions | 代数与函数

The quadratic formula gives the solutions of ax² + bx + c = 0:

二次公式给出了 ax² + bx + c = 0 的解：

x = (-b ± √(b² – 4ac)) / (2a)

The discriminant Δ = b² – 4ac tells us the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.

判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。

Completing the square: x² + bx + c = (x + b/2)² – (b/2)² + c. This form reveals the vertex of a parabola.

配方：x² + bx + c = (x + b/2)² – (b/2)² + c。这一形式揭示了抛物线的顶点。

Laws of indices:

指数律：

a^m × aⁿ = a^m+n (English) / 同底数幂相乘，指数相加。
a^m ÷ aⁿ = a^m-n (English) / 同底数幂相除，指数相减。
(a^m)ⁿ = a^mn (English) / 幂的乘方，指数相乘。
a⁰ = 1, a^-n = 1/aⁿ (English) / 任何非零数的 0 次幂为 1，负指数表示倒数。

Functions: f(x) = mx + c is linear; f(x) = ax² + bx + c is quadratic. For an inverse function, swap x and y and solve for y. The composite function fg(x) means apply g first, then f.

函数：f(x) = mx + c 为一次函数；f(x) = ax² + bx + c 为二次函数。反函数：交换 x 和 y 并解出 y。复合函数 fg(x) 表示先应用 g 再应用 f。

3. Coordinate Geometry | 坐标几何

The gradient (slope) of a line joining (x₁, y₁) and (x₂, y₂) is:

连接 (x₁, y₁) 和 (x₂, y₂) 的直线斜率（坡度）为：

m = (y₂ – y₁) / (x₂ – x₁)

The equation of a straight line can be written as y – y₁ = m(x – x₁) or y = mx + c, where c is the y-intercept.

直线方程可写作 y – y₁ = m(x – x₁) 或 y = mx + c，其中 c 为 y 轴截距。

For parallel lines, m₁ = m₂. For perpendicular lines, m₁ × m₂ = –1.

平行线：m₁ = m₂。垂直线：m₁ × m₂ = –1。

The midpoint of two points is ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between them is √((x₂ – x₁)² + (y₂ – y₁)²).

两点中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为 √((x₂ – x₁)² + (y₂ – y₁)²)。

The equation of a circle with centre (a, b) and radius r is:

圆心为 (a, b)、半径为 r 的圆方程为：

(x – a)² + (y – b)² = r²

4. Sequences and Series | 数列与级数

Arithmetic sequence (AP): nth term uₙ = a + (n – 1)d, where a is the first term and d is the common difference.

等差数列：第 n 项 uₙ = a + (n – 1)d，其中 a 为首项，d 为公差。

Sum of the first n terms of an AP: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l is the last term.

等差数列前 n 项和：Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。

Geometric sequence (GP): nth term uₙ = ar^n-1, where r is the common ratio.

等比数列：第 n 项 uₙ = ar^n-1，其中 r 为公比。

Sum of the first n terms of a GP: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. For an infinite convergent geometric series (|r| < 1), the sum to infinity is S_∞ = a/(1 – r).

等比数列前 n 项和：当 r ≠ 1 时，Sₙ = a(1 – rⁿ)/(1 – r)。对于无穷收敛等比级数 (|r| < 1)，无穷和为 S_∞ = a/(1 – r)。

Binomial expansion: for small n and rational index, (1 + x)ⁿ ≈ 1 + n x + [n(n – 1)/2!] x² + … . For a positive integer n, (a + b)ⁿ = Σ (nCr) a^n-r b^r.

二项式展开：对于小 n 和有理指数，(1 + x)ⁿ ≈ 1 + n x + [n(n – 1)/2!] x² + … 。对于正整数 n，(a + b)ⁿ = Σ (nCr) a^n-r b^r。

5. Trigonometry | 三角学

Basic ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

基本比值：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

Exact values for common angles:

常见角的精确值：

θ (degrees)	sin θ	cos θ	tan θ
0°, 0	0	1	0
30°, π/6	1/2	√3/2	1/√3
45°, π/4	1/√2	1/√2	1
60°, π/3	√3/2	1/2	√3
90°, π/2	1	0	undefined

Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² – 2bc cos A. Area of triangle: ½ ab sin C.

正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² – 2bc cos A。三角形面积：½ ab sin C。

Pythagorean identity: sin² θ + cos² θ = 1. Derived identities: 1 + tan² θ = sec² θ, 1 + cot² θ = cosec² θ.

勾股恒等式：sin² θ + cos² θ = 1。衍生恒等式：1 + tan² θ = sec² θ，1 + cot² θ = cosec² θ。

Radian measure: 180° = π radians. Arc length s = rθ, sector area = ½ r² θ, where θ is in radians.

弧度制：180° = π 弧度。弧长 s = rθ，扇形面积 = ½ r² θ，其中 θ 以弧度为单位。

6. Exponentials and Logarithms | 指数与对数

If a^x = b, then x = logₐ b. The natural logarithm ln x uses base e ≈ 2.718.

若 a^x = b，则 x = logₐ b。自然对数 ln x 以 e ≈ 2.718 为底。

Laws of logs:

对数律：

logₐ (m n) = logₐ m + logₐ n (product rule) / 积的对数等于对数之和。
logₐ (m/n) = logₐ m – logₐ n (quotient rule) / 商的对数等于对数之差。
logₐ (m^k) = k logₐ m (power rule) / 幂的对数等于指数乘以对数。
logₐ a = 1, logₐ 1 = 0.

Change of base: logₐ b = log_c b / log_c a. Commonly used to evaluate logs on a calculator.

换底公式：logₐ b = log_c b / log_c a。常用此式在计算器上求对数值。

Exponential growth/decay: N = N₀ e^kt, where k > 0 for growth and k < 0 for decay.

指数增长/衰减：N = N₀ e^kt，其中 k > 0 表示增长，k < 0 表示衰减。

7. Differentiation | 微分

The derivative of f(x) is f ‘(x) or dy/dx. For y = xⁿ, dy/dx = n x^n-1.

f(x) 的导数为 f ‘(x) 或 dy/dx。对于 y = xⁿ，dy/dx = n x^n-1。

Constant multiple rule: d/dx [k f(x)] = k f ‘(x). Sum rule: d/dx [f(x) ± g(x)] = f ‘(x) ± g ‘(x).

常数倍法则：d/dx [k f(x)] = k f ‘(x)。和差法则：d/dx [f(x) ± g(x)] = f ‘(x) ± g ‘(x)。

Product rule: d/dx (u v) = u’ v + u v’. Quotient rule: d/dx (u/v) = (u’ v – u v’) / v².

乘积法则：d/dx (u v) = u’ v + u v’。商法则：d/dx (u/v) = (u’ v – u v’) / v²。

Chain rule: dy/dx = dy/du × du/dx. For y = (f(x))ⁿ, dy/dx = n (f(x))^n-1 f ‘(x).

链式法则：dy/dx = dy/du × du/dx。对于 y = (f(x))ⁿ，dy/dx = n (f(x))^n-1 f ‘(x)。

Differentiating standard functions:

标准函数的导数：

d/dx (sin x) = cos x (English) / sin x 的导数为 cos x。
d/dx (cos x) = –sin x (English) / cos x 的导数为 –sin x。
d/dx (tan x) = sec² x (English) / tan x 的导数为 sec² x。
d/dx (e^x) = e^x, d/dx (a^x) = a^x ln a.
d/dx (ln x) = 1/x, d/dx (logₐ x) = 1/(x ln a).

Stationary points: solve f ‘(x) = 0. Use second derivative f ”(x) to classify: if f ”(x) > 0, local minimum; if f ”(x) < 0, local maximum.

驻点：解 f ‘(x) = 0。用二阶导数 f ”(x) 判断：若 f ”(x) > 0，为局部极小值；若 f ”(x) < 0，为局部极大值。

8. Integration | 积分

Integration is the reverse of differentiation. The indefinite integral of xⁿ (n ≠ –1) is:

积分是微分的逆运算。xⁿ（n ≠ –1）的不定积分为：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

Standard integrals:

标准积分：

∫ 1/x dx = ln |x| + C (English) / 1/x 的积分为 ln|x| + C。
∫ e^x dx = e^x + C, ∫ a^x dx = a^x/ln a + C.
∫ sin x dx = –cos x + C, ∫ cos x dx = sin x + C.
∫ sec² x dx = tan x + C.

For definite integrals, the area between a curve and the x-axis from x = a to x = b is ∫ₐᵇ f(x) dx. For area between two curves y = f(x) and y = g(x), use ∫ₐᵇ (upper – lower) dx.

对于定积分，曲线与 x 轴之间从 x = a 到 x = b 的面积为 ∫ₐᵇ f(x) dx。对于两曲线 y = f(x) 和 y = g(x) 之间的面积，使用 ∫ₐᵇ (上曲线 – 下曲线) dx。

The trapezium rule approximates the area under a curve using strips of width h: Area ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ].

梯形法则用宽度为 h 的条形近似曲线下的面积：面积 ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]。

9. Vectors | 向量

A vector has magnitude and direction. In 2D, v = ai + bj or as a column vector (a, b)^T. Magnitude: |v| = √(a² + b²).

向量具有大小和方向。二维中，v = ai + bj 或列向量 (a, b)^T。模：|v| = √(a² + b²)。

Unit vector in the direction of v: v̂ = v / |v|. The position vector of a point A relative to the origin O is OA.

沿 v 方向的单位向量：v̂ = v / |v|。点 A 相对于原点 O 的位置向量为 OA。

Vector AB = OB – OA. The distance between two points A and B is |AB|.

向量 AB = OB – OA。两点 A 和 B 间的距离为 |AB|。

Dot product (scalar product): a · b = |a| |b| cos θ. In component form, a · b = a₁b₁ + a₂b₂ (2D) or + a₃b₃ (3D). For perpendicular vectors, a · b = 0.

点积（标量积）：a · b = |a| |b| cos θ。在分量形式中，a · b = a₁b₁ + a₂b₂（二维）或 + a₃b₃（三维）。垂直向量满足 a · b = 0。

The angle between two vectors: cos θ = (a · b) / (|a| |b|).

两向量间夹角：cos θ = (a · b) / (|a| |b|)。

10. Statistics | 统计

Mean of a set of data: x̄ = Σ x / n. For grouped data, x̄ = Σ (f x) / Σ f, where x is the class midpoint.

数据集的平均数：x̄ = Σ x / n。对于分组数据，x̄ = Σ (f x) / Σ f，其中 x 为组中点。

Median: the middle value when data are ordered; from a cumulative frequency graph, find the value at the ½ total frequency position.

中位数：数据排序后的中间值；从累积频率图中，找到总频率一半位置对应的值。

Variance and standard deviation: variance = Σ (x – x̄)² / n or Σx²/n – (Σx/n)². Standard deviation is the square root of variance.

方差与标准差：方差 = Σ (x – x̄)² / n 或 Σx²/n – (Σx/n)²。标准差是方差的平方根。

Probability: P(A) = number of favourable outcomes / total number of outcomes. For mutually exclusive events, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A) × P(B).

概率：P(A) = 有利结果数 / 总结果数。互斥事件：P(A ∪ B) = P(A) + P(B)。独立事件：P(A ∩ B) = P(A) × P(B)。

Conditional probability: P(A|B) = P(A ∩ B) / P(B).

条件概率：P(A|B) = P(A ∩ B) / P(B)。

The binomial distribution: if X ~ B(n, p), then P(X = r) = nCr p^r (1 – p)^n-r. Mean μ = np, variance σ² = np(1 – p).

二项分布：若 X ~ B(n, p)，则 P(X = r) = nCr p^r (1 – p)^n-r。均值 μ = np，方差 σ² = np(1 – p)。

Normal distribution: X ~ N(μ, σ²). The standardised variable is Z = (X – μ)/σ. Use tables to find probabilities.

正态分布：X ~ N(μ, σ²)。标准化变量为 Z = (X – μ)/σ。查表求概率。

11. Mechanics | 力学

Equations of motion for constant acceleration (suvat):

匀加速运动方程（suvat）：

v = u + at (English) / 末速度 v = 初速度 u + 加速度 a × 时间 t。
s = ut + ½ at² (English) / 位移 s = 初速度 × 时间 + ½ × 加速度 × 时间²。
s = ½ (u + v) t (English) / 位移 s = ½ × (初速度 + 末速度) × 时间 t。
v² = u² + 2as (English) / 末速度² = 初速度² + 2 × 加速度 × 位移。

Newton’s second law: F = m a (resultant force = mass × acceleration). Weight: W = m g, where g ≈ 9.8 m/s².

牛顿第二定律：F = m a（合力 = 质量 × 加速度）。重力：W = m g，其中 g ≈ 9.8 m/s²。

Momentum: p = m v. Impulse: Impulse = change in momentum = F t = m(v – u).

动量：p = m v。冲量：冲量 = 动量变化 = F t = m(v – u)。

Work done by a constant force: W = F d cos θ, where θ is the angle between the force and direction of motion. Kinetic energy: KE = ½ m v². Gravitational potential energy: GPE = m g h.

恒力做功：W = F d cos θ，θ 为力与运动方向的夹角。动能：KE = ½ m v²。重力势能：GPE = m g h。

Power: P = work done / time = F v (for constant force and velocity in the same direction).

功率：P = 做功 / 时间 = F v（当力与速度同向且恒定时）。

12. Further Formulas Provided in A Level Booklet | A Level 公式手册额外提供公式

Edexcel provides a formula booklet for exams. The following are included and should be recognised:

Edexcel 在考试中提供公式手册。以下为所包含的公式，需要正确识别使用：

Double angle formulas: sin 2θ = 2 sin θ cos θ; cos 2θ = cos² θ – sin² θ = 2 cos² θ – 1 = 1 – 2 sin² θ; tan 2θ = 2 tan θ / (1 – tan² θ).
Addition formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.
Factor formulas: sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2), etc.
Differentiation from first principles: f ‘(x) = lim_h→0 [f(x+h) – f(x)]/h.
Integration by parts: ∫ u dv = uv – ∫ v du.
Numerical methods: Newton-Raphson iteration: x_{Published by TutorHao | Mathematics Revision Series | aleveler.com
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Edexcel Mathematics: Formula Summary Handbook | Edexcel 数学：公式汇总手册