📚 Year 13 WJEC Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 WJEC 数学：公式定理速查手册
This handbook provides a compact summary of the essential formulas, identities, and theorems examined in the Year 13 WJEC Mathematics specification. Covering pure mathematics, statistics and mechanics, it is intended as a quick revision aid to reinforce understanding and streamline last‑minute preparation.
本手册精简总结了 Year 13 WJEC 数学考试中的必考公式、恒等式与定理，涵盖纯数、统计和力学三大模块，方便你快速查阅、巩固记忆，高效备考。
1. Algebra and Functions | 代数与函数
The discriminant of a quadratic ax2 + bx + c = 0 determines the nature of the roots.
Δ = b2 – 4ac
二次方程的判别式决定根的性质。
For a rational index n and |x| < 1, the binomial expansion converges to an infinite series.
(1 + x)n = 1 + nx + n(n–1)/2! x2 + n(n–1)(n–2)/3! x3 + …
当指数为有理数且 |x| < 1 时，二项式展开为一个无穷级数。
Partial fractions decompose a rational function into simpler terms that are easier to integrate or expand.
f(x) / [(x+a)(x+b)] = A/(x+a) + B/(x+b)
部分分式将有理函数拆分为更简单的分式，便于积分或级数展开。
The laws of logarithms convert multiplication into addition and powers into products.
logₐ(xy) = logₐx + logₐy logₐ(xk) = k logₐx logₐx = log_b x / log_b a
对数运算法则将乘法化为加法、幂运算化为乘积，并提供了换底公式。
2. Trigonometry | 三角学
Radian measure links angle, arc length and sector area directly to the radius.
π rad = 180° s = rθ A = ½ r2θ
弧度制将角度、弧长和扇形面积直接与半径关联。
The fundamental Pythagorean identities relate the trigonometric ratios and their reciprocals.
sin2θ + cos2θ = 1 1 + tan2θ = sec2θ 1 + cot2θ = cosec2θ
基本毕达哥拉斯恒等式建立了三角函数及其倒数之间的关系。
Compound‑angle formulas allow the sine, cosine and tangent of sums and differences to be expressed.
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
和差角公式将两角和或差的三角函数用单个角的三角函数表示。
When an angle is small (θ in radians), sinθ, cosθ and tanθ can be approximated by polynomials.
sinθ ≈ θ cosθ ≈ 1 – ½ θ2 tanθ ≈ θ
当角度很小（弧度制）时，三角函数可近似为简单的代数式。
The sine and cosine rules solve non‑right‑angled triangles, while the R‑formula combines a sine and cosine term.
a / sinA = b / sinB = c / sinC a2 = b2 + c2 – 2bc cosA
a sinθ ± b cosθ = R sin(θ ± α) or R cos(θ ∓ α)
正弦定理和余弦定理用于求解一般三角形；R‑公式将正弦与余弦的线性组合化为单个三角函数。
3. Differentiation | 微分
The derivative of a function measures its instantaneous rate of change. Key standard derivatives are listed below.
d/dx (xn) = n xn–1 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) = sec2x
导数衡量函数的瞬时变化率，以下是最常用的基本导数公式。
The chain rule handles composite functions; product and quotient rules deal with products and ratios.
Chain: dy/dx = dy/du · du/dx
Product: d/dx (uv) = u’v + uv’ Quotient: d/dx (u/v) = (u’v – uv’) / v2
链式法则处理复合函数，乘积法则与商法则分别用于乘积和商的求导。
Implicit differentiation avoids solving for y explicitly, and parametric equations use the parameter t.
Implicit: differentiate both sides, then collect dy/dx.
Parametric: dy/dx = (dy/dt) / (dx/dt)
隐函数求导无须显式解出 y，参数方程通过参数 t 的导数之比得到 dy/dx。
4. Integration | 积分
Integration reverses differentiation. The standard integrals are the building blocks for more complicated techniques.
∫ xn dx = xn+1/(n+1) + C (n≠–1) ∫ 1/x dx = ln|x| + C ∫ ex dx = ex + C
∫ sin x dx = –cos x + C ∫ cos x dx = sin x + C ∫ sec2x dx = tan x + C
积分是微分的逆运算，以上基本积分是所有复杂积分方法的基石。
Integration by substitution simplifies an integrand by a change of variable; integration by parts uses the product rule in reverse.
Substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du
By parts: ∫ u dv = uv – ∫ v du
换元积分法通过变量代换化简被积函数，分部积分法则反向运用乘积法则。
Definite integrals give the area under a curve, and volumes of revolution are generated by rotating the curve around an axis.
Area = ∫ab y dx Volume (x‑axis) = π ∫ab y2 dx
定积分用于计算曲线下的面积，旋转体积分通过绕坐标轴旋转曲线求得立体体积。
5. Differential Equations | 微分方程
A first‑order separable differential equation can be solved by separating the variables onto opposite sides.
dy/dx = g(x)·h(y) → ∫ 1/h(y) dy = ∫ g(x) dx
一阶可分离变量的微分方程通过将变量分离至等号两边并积分求解。
A linear first‑order ODE of the form dy/dx + P(x)y = Q(x) uses an integrating factor.
IF = e∫ P dx then y·IF = ∫ Q·IF dx
一阶线性常微分方程 dy/dx + P(x)y = Q(x) 可借助积分因子 e∫P dx 求解。
6. Vectors | 向量
A vector has magnitude and direction. Position vectors locate points relative to an origin, and free vectors can be added tip‑to‑tail.
|a| = √(a12 + a22 + a32)
向量具有大小和方向，位置向量表示点相对原点的位置，自由向量可通过三角形法则相加。
The scalar (dot) product measures how much two vectors point in the same direction and gives the angle between them.
a·b = |a||b| cosθ cosθ = (a·b) / (|a||b|)
标量积（点乘）度量两个向量的方向一致性，并可用于计算夹角。
The vector equation of a straight line is given by a point on the line and a direction vector.
r = a + λ b
直线的向量方程由直线上的一点和方向向量确定。
The shortest distance from a point to a line (or between skew lines) is found using perpendicular vectors.
点到直线（或异面直线之间）的最短距离利用垂直向量求解。
7. Complex Numbers | 复数
A complex number z = a + bi has a real part a and an imaginary part bi, with i2 = –1.
Conjugate: z* = a – bi Modulus: |z| = √(a2 + b2) Argument: arg z = θ
复数 z = a + bi 的实部为 a，虚部为 bi，并满足 i2 = –1。
The polar form and Euler’s formula connect trigonometry to exponentials.
z = r (cosθ + i sinθ) = r eiθ
极坐标形式和欧拉公式将三角表示与指数表示联系起来。
De Moivre’s theorem is used to raise complex numbers to integer powers and to find roots.
(cosθ + i sinθ)n = cos nθ + i sin nθ
z1/n = r1/n [ cos((θ+2kπ)/n) + i sin((θ+2kπ)/n) ] , k = 0,1,…,n–1
德莫弗定理用于计算复数的整数次幂以及求 n 次方根。
8. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference; its sum forms an arithmetic series.
Sn = n/2 [2a + (n–1)d]
等差数列的公差恒定，其前 n 项和公式如上。
A geometric sequence has a constant ratio; the finite sum and the infinite sum (for |r|<1) are given below.
Sn = a(1–rn)/(1–r) S∞ = a/(1–r)
等比数列的公比恒定，其有限项和及无穷递减等比级数求和公式如上。
The Maclaurin expansion expresses a smooth function as an infinite power series centred at zero.
f(x) = f(0) + f'(0)x + f”(0)/2! x2 + f”'(0)/3! x3 + …
麦克劳林展开将光滑函数在 x=0 处展开为无限幂级数。
9. Hyperbolic Functions | 双曲函数
Hyperbolic functions are defined analogously to trigonometric functions but using exponential functions.
sinh x = (ex – e–x) / 2 cosh x = (ex + e–x) / 2 tanh x = sinh x / cosh x
双曲函数通过指数函数定义，在结构上与三角函数相似。
The fundamental identity and simple derivatives mirror the trigonometric
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