📚 Year 13 WJEC Maths: Key Formulas & Theorems | Year 13 WJEC 数学：公式定理速查手册
This quick reference guide covers essential formulas and theorems for the Year 13 WJEC A Level Mathematics course, focusing on Pure Mathematics. Each section summarises key results you need to memorise and apply in the exam. Use this handbook to consolidate your understanding of identities, derivatives, integrals, vectors and more.
这份速查手册涵盖了 Year 13 WJEC A Level 数学课程（纯数学）中的核心公式与定理。每个小节都总结了考试中必须记忆并运用的关键结论。用这本手册来巩固你对恒等式、导数、积分、向量等内容的理解。
1. Algebra and Functions | 代数与函数
Master the binomial expansion, partial fractions, and function transformations. These tools are fundamental for simplifying expressions and solving equations throughout Year 13.
掌握二项展开、分式分解和函数变换。这些工具是 Year 13 中化简表达式和求解方程的基础。
| Binomial theorem: (1+x)ⁿ = Σ⁰ⁿ nCr xr = 1 + nx + [n(n−1)/2!]x² + … , for |x| < 1, n ∈ Q. | 二项式定理：(1+x)ⁿ = Σ⁰ⁿ nCr xr = 1 + nx + [n(n−1)/2!]x² + … ，当 |x| < 1, n 为有理数。 |
| (a+bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ, expand with the same rule. | (a+bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ，按相同规则展开。 |
| ((x+a)(x+b)) = A/(x+a) + B/(x+b) (linear factors). Repeat root adds a denominator squared. | 部分分式：(px+q)/((x+a)(x+b)) = A/(x+a) + B/(x+b)（不同线性因式）。重复根需增加平方分母。 |
| Modulus function: |x| = { x, x ≥ 0; −x, x < 0. |f(x)| = k ⇔ f(x) = ±k. | 绝对值函数：|x| = { x, x ≥ 0; −x, x < 0. |f(x)| = k ⇔ f(x) = ±k。 |
| Transformations: f(x+a) translates graph by –a along x; f(ax) stretches horizontally by 1/a; a f(x) stretches vertically by a. | 变换：f(x+a) 将图像沿 x 轴平移 –a；f(ax) 水平方向拉伸 1/a；a f(x) 垂直方向拉伸 a。 |
2. Exponential and Logarithmic Functions | 指数与对数函数
The natural exponential and logarithm are central to growth models and calculus. Ensure you can switch between forms and differentiate/integrate them effortlessly.
自然指数与对数是增长模型和微积分的核心。确保你能在形式之间自由切换，并轻松进行微积分运算。
| ax = ex ln a ; logₐ x = (ln x)/(ln a). | ax = ex ln a ; logₐ x = (ln x)/(ln a)。 |
| ln(ex) = x ; eln x = x (x>0). | ln(ex) = x ; eln x = x (x>0)。 |
| d/dx (ex) = ex ; ∫ ex dx = ex + C. | d/dx (ex) = ex ; ∫ ex dx = ex + C。 |
| d/dx (ln x) = 1/x ; ∫ (1/x) dx = ln|x| + C. | d/dx (ln x) = 1/x ; ∫ (1/x) dx = ln|x| + C。 |
| d/dx (ax) = ax ln a ; ∫ ax dx = ax/(ln a) + C. | d/dx (ax) = ax ln a ; ∫ ax dx = ax/(ln a) + C。 |
3. Trigonometry – Radians and Basic Definitions | 三角函数 – 弧度制与基本定义
Radian measure simplifies calculus with trig functions. Learn the conversion, arc length, sector area, and the reciprocal functions sec, cosec and cot.
弧度制简化了三角函数的微积分运算。掌握换算、弧长、扇形面积，以及倒数函数 sec、cosec 和 cot。
| π rad = 180°; 1 rad ≈ 57.3°. | π rad = 180°; 1 rad ≈ 57.3°。 |
| Arc length: s = rθ (θ in rad). | 弧长：s = rθ（θ 单位弧度）。 |
| Sector area: A = ½ r²θ. | 扇形面积：A = ½ r²θ。 |
| sec θ = 1/cos θ ; cosec θ = 1/sin θ ; cot θ = 1/tan θ = cos θ/sin θ. | sec θ = 1/cos θ ; cosec θ = 1/sin θ ; cot θ = 1/tan θ = cos θ/sin θ。 |
| Small-angle approximations (rad): sin θ ≈ θ, cos θ ≈ 1 – θ²/2, tan θ ≈ θ. | 小角近似（弧度）：sin θ ≈ θ, cos θ ≈ 1 – θ²/2, tan θ ≈ θ。 |
4. Trigonometric Identities and Equations | 三角恒等式与方程
These identities are vital for simplifying expressions, solving equations and proving results. You must know exact values for standard angles and be fluent with the R‑formula.
这些恒等式对于化简表达式、求解方程和证明结论至关重要。你必须熟记标准角度的精确值，并能熟练运用 R 公式。
| Pythagorean identities: sin²θ + cos²θ ≡ 1 ; 1 + tan²θ ≡ sec²θ ; 1 + cot²θ ≡ cosec²θ. | 勾股恒等式：sin²θ + cos²θ ≡ 1 ; 1 + tan²θ ≡ sec²θ ; 1 + cot²θ ≡ cosec²θ。 |
| Addition formulas: sin(A±B) = sinA cosB ± cosA sinB ; cos(A±B) = cosA cosB ∓ sinA sinB. | 加法公式：sin(A±B) = sinA cosB ± cosA sinB ; cos(A±B) = cosA cosB ∓ sinA sinB。 |
| Double-angle: sin2A = 2 sinA cosA; cos2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A; tan2A = 2 tanA/(1 – tan²A). | 倍角：sin2A = 2 sinA cosA；cos2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A；tan2A = 2 tanA/(1 – tan²A)。 |
| R‑formula: a cosθ ± b sinθ = R cos(θ ∓ α) or R sin(θ ± α), where R = √a²+b², tan α = b/a (adjust quadrant). | R 公式：a cosθ ± b sinθ = R cos(θ ∓ α) 或 R sin(θ ± α)，其中 R = √a²+b², tan α = b/a（注意象限）。 |
| Exact values: memorize sin/cos/tan for 0, π/6, π/4, π/3, π/2 rad. | 特殊角值：熟记 0, π/6, π/4, π/3, π/2 弧度的 sin/cos/tan 精确值。 |
5. Differentiation | 微分
You must be able to differentiate standard functions and apply the chain, product and quotient rules. Implicit and parametric differentiation extend these techniques to linked and geometric contexts.
你必须能够对标准函数求导，并运用链式法则、乘积法则和商法则。隐函数与参数求导将这些技巧拓展到关联和几何问题中。
| d/dx (xⁿ) = n xⁿ⁻¹ (n real). | d/dx (xⁿ) = n xⁿ⁻¹ (n 为实数)。 |
| d/dx (sin x) = cos x ; d/dx (cos x) = –sin x ; d/dx (tan x) = sec² x. | d/dx (sin x) = cos x ; d/dx (cos x) = –sin x ; d/dx (tan x) = sec² x。 |
| d/dx (sec x) = sec x tan x ; d/dx (cosec x) = –cosec x cot x ; d/dx (cot x) = –cosec² x. | d/dx (sec x) = sec x tan x ; d/dx (cosec x) = –cosec x cot x ; d/dx (cot x) = –cosec² x。 |
| Chain rule: dy/dx = (dy/du)(du/dx). Product: d/dx(uv) = u'v + uv'. Quotient: d/dx(u/v) = (u'v – uv')/v². | 链式法则：dy/dx = (dy/du)(du/dx)。乘积：d/dx(uv) = u'v + uv'。商：d/dx(u/v) = (u'v – uv')/v²。 |
| t). Implicit: differentiate both sides w.r.t. x, treat y as function; collect dy/dx terms. | 参数求导：dy/dx = (dy/dt)/(dx/dt)。隐函数：方程两边对 x 求导，将 y 视作 x 的函数；合并 dy/dx 项。 |
6. Integration | 积分
Integration is the reverse of differentiation. The key methods—substitution, parts, and partial fractions—allow you to tackle a wide variety of integrals. Definite integrals give areas and volumes.
积分是微分的逆运算。关键方法——换元、分部与部分分式——使你能够处理各种各样的积分。定积分可求面积与体积。
| ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1). | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)。 |
| ∫ sin x dx = –cos x + C ; ∫ cos x dx = sin x + C ; ∫ sec² x dx = tan x + 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, u = g(x). | 换元积分：∫ f(g(x))g'(x) dx = ∫ f(u) du，其中 u = g(x)。 |
| Integration by parts: ∫ u dv/dx dx = uv – ∫ v du/dx dx. | 分部积分：∫ u dv/dx dx = uv – ∫ v du/dx dx。 |
| Area: A = ∫ₐᵇ y dx. Volume of revolution: V = π ∫ₐᵇ y² dx (about x‑axis). | 面积：A = ∫ₐᵇ y dx。旋转体体积：V = π ∫ₐᵇ y² dx（绕 x 轴）。 |
7. Vectors | 向量
Vectors in 3D describe lines and planes. The dot product measures angles and perpendiculars, while vector equations model geometric objects precisely.
三维向量可用于描述直线与平面。点积可度量角度与垂直关系，向量方程则精确地建立几何对象的模型。
| Magnitude: |a| = √x² + y² + z² for vector a = x i + y j + z k. | 模：对于向量 a = x i + y j + z k，|a| = √x² + y² + z²。 |
| Dot product: a·b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b| cos θ. | 点积：a·b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b| cos θ。 |
| Angle between vectors: cos θ = (a·b)/(|a||b|). Perpendicular if a·b = 0. | 向量夹角：cos θ = (a·b)/(|a||b|)。若 a·b = 0，则两向量垂直。 |
| Vector equation of a line: r = a + t d, where a is position on line, d is direction. | 直线的向量方程：r = a + t d，a 为直线上一点，d 为方向向量。 |
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