Year 13 WJEC Mathematics: Core Knowledge Review | Year 13 WJEC 数学：核心知识点梳理

📚 Year 13 WJEC Mathematics: Core Knowledge Review | Year 13 WJEC 数学：核心知识点梳理

As Year 13 students approach their WJEC A-level Mathematics exams, a systematic review of core topics is essential. This article consolidates key concepts from Pure Mathematics, Statistics, and Mechanics, aligning with the WJEC specification. Each section pairs English explanations with Chinese translations to support bilingual learners.

Year 13 学生即将参加 WJEC A-level 数学考试，系统梳理核心知识点至关重要。本文整合了纯数学、统计学与力学中的重要概念，贴合 WJEC 考纲。每个部分均采用英中双语讲解，助力双语学习者。

1. Algebra and Functions | 代数与函数

The modulus function |x| is defined as x when x ≥ 0 and –x when x < 0. Graphs of y = |f(x)| require reflecting the negative portions in the x-axis. Solving equations like |2x – 3| = 5 involves splitting into 2x – 3 = 5 and 2x – 3 = –5.

模函数 |x| 定义为：当 x ≥ 0 时取 x，当 x < 0 时取 –x。绘制 y = |f(x)| 的图像需将 x 轴下方的部分向上翻折。解如 |2x – 3| = 5 的方程需分两种情况：2x – 3 = 5 和 2x – 3 = –5。

A function f maps each input to exactly one output. The composite function (f∘g)(x) = f(g(x)) applies g first, then f. For an inverse function f⁻¹(x) to exist, f must be one-to-one; its graph is the reflection of y = f(x) in the line y = x.

函数为每个输入指定唯一输出。复合函数 (f∘g)(x) = f(g(x)) 先作用 g 再作用 f。反函数 f⁻¹(x) 存在的条件是 f 为单射；它的图像是 y = f(x) 关于直线 y = x 的反射。

Exponential and logarithmic functions are inverses: eˣ and ln x satisfy e^{ln x} = x and ln(e^x) = x. Key log laws include ln(ab) = ln a + ln b and ln(aⁿ) = n ln a.

指数函数与对数函数互为反函数：eˣ 与 ln x 满足 e^{ln x} = x 及 ln(e^x) = x。重要的对数运算法则有 ln(ab) = ln a + ln b，ln(aⁿ) = n ln a。

2. Trigonometry | 三角学

Radian measure is essential: π radians = 180°. Arc length s = rθ, area of a sector A = ½ r²θ. These formulas only hold when θ is in radians.

弧度制十分重要：π 弧度 = 180°。弧长公式 s = rθ，扇形面积公式 A = ½ r²θ，仅当 θ 以弧度为单位时成立。

Pythagorean identity: sin²θ + cos²θ = 1. The reciprocal functions are secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ.

平方恒等式：sin²θ + cos²θ = 1。倒数函数为 secθ = 1/cosθ，cosecθ = 1/sinθ，cotθ = 1/tanθ。

Compound angle formulas: sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB. The double-angle identities sin 2A = 2 sinA cosA and cos 2A = cos²A – sin²A are direct consequences.

和角公式：sin(A ± B) = sinA cosB ± cosA sinB，cos(A ± B) = cosA cosB ∓ sinA sinB。倍角公式 sin 2A = 2 sinA cosA 与 cos 2A = cos²A – sin²A 均可由和角公式导出。

Expressing a sinθ + b cosθ in the form R sin(θ ± α) or R cos(θ ± α) simplifies equation-solving. Here R = √(a² + b²) and α is an appropriate phase angle.

将 a sinθ + b cosθ 化为 R sin(θ ± α) 或 R cos(θ ± α) 能简化方程求解。其中 R = √(a² + b²)，α 为相应的辅助角。

Inverse trigonometric functions arcsin, arccos, arctan are defined with restricted domains to ensure one-to-oneness.

反三角函数 arcsin、arccos、arctan 均通过限制定义域来保证单射性。

3. Sequences and Series | 序列与级数

An arithmetic progression has general term u_n = a + (n–1)d and sum S_n = n/2 [2a + (n–1)d] = n/2 (a + l), where l is the last term.

等差数列的通项为 u_n = a + (n–1)d，前 n 项和为 S_n = n/2 [2a + (n–1)d] = n/2 (a + l)，其中 l 为末项。

A geometric progression has u_n = ar^n–1. For r ≠ 1, S_n = a(1 – rⁿ)/(1 – r). When |r| < 1 the sum to infinity is S_∞ = a/(1 – r).

等比数列的通项为 u_n = ar^n–1。当 r ≠ 1 时，和为 S_n = a(1 – rⁿ)/(1 – r)。若 |r| < 1，无穷和 S_∞ = a/(1 – r)。

The binomial expansion for rational powers is (1 + x)ⁿ = 1 + nx + n(n–1)/2! x² + … , convergent for |x| < 1 when n is not a positive integer.

有理指数的二项展开式为 (1 + x)ⁿ = 1 + nx + n(n–1)/2! x² + … ，当 n 不是正整数时要求 |x| < 1。

Maclaurin series approximates a function near zero: f(x) = f(0) + f′(0)x + f″(0)/2! x² + f″′(0)/3! x³ + … . Standard expansions such as eˣ, sin x, cos x are particularly important.

麦克劳林级数在零点附近逼近函数：f(x) = f(0) + f′(0)x + f″(0)/2! x² + f″′(0)/3! x³ + … 。像 eˣ、sin x、cos x 的标准展开式格外重要。

4. Calculus | 微积分

The chain rule: if y = f(g(x)) then dy/dx = f′(g(x)) g′(x). The product rule: d/dx (uv) = u′v + uv′. The quotient rule: d/dx (u/v) = (u′v – uv′) / v².

链式法则：若 y = f(g(x))，则 dy/dx = f′(g(x)) g′(x)。乘法法则：d/dx (uv) = u′v + uv′。除法法则：d/dx (u/v) = (u′v – uv′) / v²。

Derivatives of key 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.

基本函数的导数：eˣ 的导数为 eˣ，ln x 的导

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