📚 IB WJEC Mathematics: Formula Handbook | IB WJEC 数学：公式汇总手册
This comprehensive formula handbook covers the essential equations, identities, and theorems required for the IB Mathematics courses (Analysis and Approaches SL/HL and Applications and Interpretation SL/HL), aligned with the rigorous standards of the WJEC examination board. It serves as a one-stop revision resource, from algebra through calculus to probability and vectors. Every formula is presented with clear annotations to support both conceptual understanding and exam preparation.
本手册汇总了 IB 数学课程（分析与方法 SL/HL、应用与解释 SL/HL）的核心公式、恒等式和定理，与 WJEC 考试局的严格要求保持一致。内容涵盖从代数到微积分、概率与向量的所有必考公式，并附有清晰注释，帮助学生巩固概念理解、高效备考。无论是日常复习还是考前速查，本手册都是不可或缺的工具。
1. Algebra Fundamentals | 代数基础
The quadratic formula solves ax² + bx + c = 0, a ≠ 0: x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one real repeated root; Δ < 0 gives no real roots (complex conjugates).
一元二次方程 ax² + bx + c = 0（a ≠ 0）的求根公式为：x = (−b ± √(b² − 4ac)) / (2a)。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（共轭复数）。
Completing the square: ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)). The vertex of the parabola y = ax² + bx + c is at (−b/(2a), c − b²/(4a)). For the sum and product of roots α and β: α + β = −b/a, αβ = c/a.
配方法：ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a))。抛物线 y = ax² + bx + c 的顶点坐标为 (−b/(2a), c − b²/(4a))。若 α 与 β 为方程的两个根，则根的和：α + β = −b/a，根的积：αβ = c/a。
The binomial expansion for (1 + x)ⁿ is (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …, valid for |x| < 1, n ∈ ℚ. The general term for n ∈ ℕ is ⁿCᵣ aⁿ⁻ʳ bʳ.
二项式展开 (1 + x)ⁿ 为 (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …，适用于 |x| < 1, n ∈ ℚ。对于正整数 n，通项公式为 ⁿCᵣ aⁿ⁻ʳ bʳ。
2. Functions and Equations | 函数与方程
A function f: A → B maps each element of its domain A to exactly one element in its codomain B. The range is the set of all output values. To find the inverse function f⁻¹(x), swap x and y in y = f(x) and solve for y, ensuring the function is one-to-one (passes horizontal line test).
函数 f: A → B 将定义域 A 中的每个元素唯一映射到值域 B 中的一个元素。值域是所有输出值的集合。要求反函数 f⁻¹(x)，将 y = f(x) 中的 x 与 y 互换并解出 y，前提是函数必须是一一映射的（通过水平线检验）。
Composite function: (f ∘ g)(x) = f(g(x)). The domain of f ∘ g is the set of x in the domain of g such that g(x) is in the domain of f. Transformations of graphs: y = f(x) + a translates vertically by a; y = f(x + a) translates horizontally by −a; y = a f(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a.
复合函数：(f ∘ g)(x) = f(g(x))，其定义域为 g 中所有使 g(x) 落入 f 定义域的 x 的集合。图像变换：y = f(x) + a 表示垂直平移 a 个单位；y = f(x + a) 表示水平左移 a 个单位（若 a>0）；y = a f(x) 垂直伸缩 a 倍；y = f(ax) 水平伸缩 1/a 倍。
Solving modulus equations: |x − a| = b ⇔ x = a ± b. The inequality |x − a| < b ⇔ −b < x − a < b. For absolute value functions, consider critical points and split into cases.
解绝对值方程：|x − a| = b ⇔ x = a ± b。不等式 |x − a| < b ⇔ −b < x − a < b。处理绝对值函数时，需找出临界点并分段讨论。
3. Exponents and Logarithms | 指数与对数
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ. For rational exponents, a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ.
指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ；(aᵐ)ⁿ = aᵐⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ；a⁰ = 1（a ≠ 0）；a⁻ⁿ = 1/aⁿ。有理指数定义：a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ。
Logarithms are the inverse of exponentials: if aˣ = b then x = logₐ b. Properties: logₐ (MN) = logₐ M + logₐ N; logₐ (M/N) = logₐ M − logₐ N; logₐ (Mᵏ) = k logₐ M. Change of base: logₐ b = logₓ b / logₓ a. Natural log: ln x = logₑ x, where e ≈ 2.71828.
对数为指数运算的逆运算：若 aˣ = b，则 x = logₐ b。运算性质：logₐ (MN) = logₐ M + logₐ N；logₐ (M/N) = logₐ M − logₐ N；logₐ (Mᵏ) = k logₐ M。换底公式：logₐ b = logₓ b / logₓ a。自然对数：ln x = logₑ x，其中 e ≈ 2.71828。
Solving exponential equations often involves taking logs: e.g., 3²ˣ⁺¹ = 20 becomes 2x + 1 = log₃ 20 = ln 20 / ln 3. For equations of the form a·bˣ = c, use logarithms to isolate x.
解指数方程通常需要取对数，例如 3²ˣ⁺¹ = 20 转化为 2x + 1 = log₃ 20 = ln 20 / ln 3。对于形如 a·bˣ = c 的方程，利用对数运算分离 x。
4. Sequences and Series | 数列与级数
Arithmetic sequence: uₙ = a + (n − 1)d, where a is the first term and d is the common difference. The sum of the first n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l = uₙ.
等差数列通项：uₙ = a + (n − 1)d，其中 a 为首项，d 为公差。前 n 项和公式：Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)，其中 l = uₙ。
Geometric sequence: uₙ = a rⁿ⁻¹, r is the common ratio. Sum of first n terms: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. For an infinite geometric series with |r| < 1, the sum to infinity S∞ = a/(1 − r).
等比数列通项：uₙ = a rⁿ⁻¹，r 为公比。前 n 项和：Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。对于满足 |r| < 1 的无穷等比级数，无穷和为 S∞ = a/(1 − r)。
Sigma notation: Σₖ₌₁ⁿ uₖ denotes the sum of terms. Arithmetic and geometric series can be expressed in sigma form. Compound interest and population growth are modelled with geometric sequences.
求和符号 Σₖ₌₁ⁿ uₖ 表示各项之和。等差与等比级数均可用求和符号表示。复利计算与种群增长模型均使用等比数列。
5. Trigonometry | 三角学
In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Pythagorean identity: sin² θ + cos² θ = 1, from which follow 1 + tan² θ = sec² θ and 1 + cot² θ = csc² θ.
直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。毕达哥拉斯恒等式：sin² θ + cos² θ = 1，由此推导出 1 + tan² θ = sec² θ 和 1 + cot² θ = csc² 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). Double angle: sin 2A = 2 sin A cos A; cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A; tan 2A = 2 tan A / (1 − tan² A).
和差角公式：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)。倍角公式：sin 2A = 2 sin A cos A；cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A；tan 2A = 2 tan A / (1 − tan² A)。
Sine rule: a / sin A = b / sin B = c / sin C = 2R. Cosine rule: a² = b² + c² − 2bc cos A, or cos A = (b² + c² − a²) / (2bc). Area of triangle: (1/2) bc sin A.
正弦定理：a / sin A = b / sin B = c / sin C = 2R。余弦定理：a² = b² + c² − 2bc cos A，或 cos A = (b² + c² − a²) / (2bc)。三角形面积公式：(1/2) ab sin C。
Radian measure: π radians = 180°. Arc length s = rθ, sector area A = ½ r² θ (θ in radians). Graphs of sin, cos, tan: amplitude, period, phase shift; e.g., y = a sin(bx + c) + d has amplitude |a|, period 2π/b.
弧度制：π 弧度 = 180°；弧长 s = rθ，扇形面积 A = ½ r² θ（θ 用弧度）。正弦、余弦、正切函数的图像：振幅、周期和相位移动；例如 y = a sin(bx + c) + d 的振幅为 |a|，周期为 2π/b。
6. Differentiation | 微分
Derivative from first principles: f'(x) = limₕ→₀ (f(x + h) − f(x)) / h. Basic derivative: d/dx (xⁿ) = n xⁿ⁻¹. Constant multiple rule: d/dx (c f(x)) = c f'(x). Sum rule: d/dx (u + v) = du/dx + dv/dx.
导数定义：f'(x) = limₕ→₀ (f(x + h) − f(x)) / h。基本导数公式：d/dx (xⁿ) = n xⁿ⁻¹。常数倍数法则：d/dx (c f(x)) = c f'(x)；和法则：d/dx (u + v) = du/dx + dv/dx。
Product rule: if y = uv, then dy/dx = u dv/dx + v du/dx. Quotient rule: if y = u/v, then dy/dx = (v du/dx − u dv/dx) / v². Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x).
乘法法则：若 y = uv，则 dy/dx = u dv/dx + v du/dx。除法法则：若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。
Derivatives 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; d/dx (aˣ) = aˣ ln a.
常见函数的导数：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；d/dx (aˣ) = aˣ ln a。
Applications: gradient of a curve at a point is f'(x₀). Equation of tangent: y − y₀ = m(x − x₀). Stationary points occur where f'(x) = 0; second derivative test classifies them: f”(x) > 0 local minimum, f”(x) < 0 local maximum.
应用：曲线在某点的斜率为 f'(x₀)。切线方程：y − y₀ = m(x − x₀)。驻点出现在 f'(x) = 0 处；二阶导数判别法：若 f”(x) > 0 为局部极小值，f”(x) < 0 为局部极大值。
7. Integration | 积分
Integration is the reverse process of differentiation. Indefinite integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x). Basic rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. ∫ x⁻¹ dx = ln |x| + C.
积分是微分的逆运算。不定积分：∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。基本法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1；∫ x⁻¹ dx = ln |x| + C。
Standard integrals: ∫ eˣ dx = eˣ + C; ∫ aˣ dx = aˣ/ln a + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C.
常见积分：∫ eˣ dx = eˣ + C；∫ aˣ dx = aˣ/ln a + C；∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C。
Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x). For definite integrals, change limits accordingly. Integration by parts: ∫ u dv = uv − ∫ v du, applied to products such as x eˣ, ln x, etc.
换元积分法：∫ f(g(x)) g'(x) dx = ∫ f(u) du，令 u = g(x)。对于定积分，相应更换积分限。分部积分法：∫ u dv = uv − ∫ v du，适用于形如 x eˣ、ln x 等乘积的积分。
The area between a curve and the x-axis from a to b is ∫ₐᵇ f(x) dx (taking absolute value where negative). Volume of revolution about the x-axis: V = π ∫ₐᵇ [f(x)]² dx.
曲线与 x 轴在区间 [a,b] 所围面积计算为 ∫ₐᵇ f(x) dx（负面积部分取绝对值）。曲线绕 x 轴旋转所得体积：V = π ∫ₐᵇ [f(x)]² dx。
8. Vectors | 向量
A vector has magnitude and direction. Notation: **a** = (a₁, a₂, a₃) or a₁**i** + a₂**j** + a₃**k**. Magnitude: |**a**| = √(a₁² + a₂² + a₃²). Unit vector in direction of **a**: **â** = **a** / |**a**|.
向量既有大小又有方向。记法：**a** = (a₁, a₂, a₃) 或 a₁**i** + a₂**j** + a₃**k**。模长：|**a**| = √(a₁² + a₂² + a₃²)。沿 **a** 方向的单位向量：**â** = **a** / |**a**|。
Scalar (dot) product: **a**·**b** = a₁b₁ + a₂b₂ + a₃b₃ = |**a**||**b**| cos θ, where θ is the angle between them. If **a** ⟂ **b**, dot product = 0. Vector projection of **a** onto **b**: proj_bs(**a**) = (**a**·**b** / |**b**|²) **b**.
数量积（点乘）：**a**·**b** = a₁b₁ + a₂b₂ + a₃b₃ = |**a**||**b**| cos θ，其中 θ 为两向量夹角。若 **a** ⟂ **b**，则点乘为 0。**a** 在 **b** 上的向量投影：proj_bs(**a**) = (**a**·**b** / |**b**|²) **b**。
Vector (cross) product (only in 3D): **a** × **b** has magnitude |**a**||**b**| sin θ and direction perpendicular to both, given by right-hand rule. Determinant form: **a** × **b** = |**i** **j** **k**; a₁ a₂ a₃; b₁ b₂ b₃|. The area of a parallelogram = |**a** × **b**|.
向量积（叉乘，仅适用于三维）：**a** × **b** 的大小为 |**a**||**b**| sin θ，方向垂直于 **a** 与 **b** 所张平面（右手定则）。行列式形式：**a** × **b** = |**i** **j** **k**; a₁ a₂ a₃; b₁ b₂ b₃|。平行四边形的面积 = |**a** × **b**|。
Equation of a line: **r** = **a** + t**d**, where **a** is a point on the line, **d** is direction vector. Equation of a plane: **r**·**n** = **a**·**n** or scalar form ax + by + cz = d. Distance from point to line/plane can be found using vector projections.
直线方程：**r** = **a** + t**d**，其中 **a** 为直线上一点，**d** 为方向向量。平面方程：**r**·**n** = **a**·**n** 或标量形式 ax + by + cz = d。点到直线或平面的距离可利用向量投影公式求得。
9. Probability and Statistics | 概率与统计
Probability basics: P(A) = n(A)/n(U), where 0 ≤ P(A) ≤ 1. Complement rule: P(A’) = 1 − P(A). For mutually exclusive events, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A) P(B). Conditional probability: P(A|B) = P(A ∩ B) / P(B).
概率基础：P(A) = n(A)/n(U)，其中 0 ≤ P(A) ≤ 1。补事件法则：P(A’) = 1 − P(A)。互斥事件：P(A ∪ B) = P(A) + P(B)。独立事件：P(A ∩ B) = P(A) P(B)。条件概率：P(A|B) = P(A ∩ B) / P(B)。
Bayes’ theorem: P(A|B) = [P(B|A) P(A)] / P(B). Probability tree diagrams and Venn diagrams are useful tools. Expectation of a discrete random variable: E(X) = Σ x P(X = x). Variance: Var(X) = E(X²) − [E(X)]².
贝叶斯定理：P(A|B) = [P(B|A) P(A)] / P(B)。概率树形图与文氏图是有效的解题工具。离散随机变量的期望值：E(X) = Σ x P(X = x)。方差：Var(X) = E(X²) − [E(X)]²。
Binomial distribution: X ~ B(n, p), P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, mean = np, variance = np(1 − p). Normal distribution: X ~ N(μ, σ²). Standardisation: Z = (X − μ) / σ ~ N(0, 1). Use symmetry and tables for probabilities.
二项分布：X ~ B(n, p)，P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，均值 = np，方差 = np(1 − p)。正态分布：X ~ N(μ, σ²)。标准化：Z = (X − μ) / σ ~ N(0, 1)。利用对称性和标准正态分布表计算概率。
Descriptive statistics: Sample mean x̄ = Σ xᵢ / n; sample variance s² = Σ (xᵢ − x̄)² / (n − 1). Linear regression: y = a + bx, where b = Sxy / Sxx and a = ȳ − b x̄. Pearson’s correlation coefficient r = Sxy / √(Sxx Syy).
描述统计学：样本均值 x̄ = Σ xᵢ / n；样本方差 s² = Σ (xᵢ − x̄)² / (n − 1)。线性回归：y = a + bx，其中 b = Sxy / Sxx，a = ȳ − b x̄。皮尔逊积矩相关系数 r = Sxy / √(Sxx Syy)。
10. Complex Numbers and Matrices | 复数与矩阵
Complex number form: z = a + bi, where i² = −1. Modulus: |z| = √(a² + b²). Argument: arg z = θ, where tan θ = b/a. Polar form: z = r (cos θ + i sin θ) = r cis θ. Euler’s formula: e^(iθ) = cos θ + i sin θ.
复数形式：z = a + bi，其中 i² = −1。模：|z| = √(a² + b²)。辐角：arg z = θ，tan θ = b/a。极坐标形式：z = r (cos θ + i sin θ) = r cis θ。欧拉公式：e^(iθ) = cos θ + i sin θ。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Used to find powers and roots. The n nth roots of unity are given by z = e^(2πk i/n) for k = 0, 1, …, n−1.
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ，用于求幂和开方。n 次单位根为 z = e^(2πk i/n)，k = 0, 1, …, n−1。
Matrices (especially for AI HL): A matrix is an array of numbers. Addition and scalar multiplication are element-wise. Matrix multiplication: AB is defined if columns of A = rows of B, with (AB)ᵢⱼ = Σ Aᵢₖ Bₖⱼ. The identity matrix I satisfies AI = IA = A. Inverse A⁻¹ exists if det(A) ≠ 0.
矩阵（主要用于 AI HL）：矩阵是一个数字阵列。加法与数乘按元素运算。矩阵乘法：若 A 的列数等于 B 的行数，则 AB 的第 i 行第 j 列元素为 (AB)ᵢⱼ = Σ Aᵢₖ Bₖⱼ。单位矩阵 I 满足 AI = IA = A。若行列式 det(A) ≠ 0，则逆矩阵 A⁻¹ 存在。
Determinant of 2×2 matrix |a b; c d| = ad − bc. For solving linear systems, write AX = B, then X = A⁻¹ B if A invertible. Eigen
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