📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This concise handbook covers all essential formulas, identities and theorems required for the Year 12 WJEC Further Mathematics specification. Each entry is presented bilingually to support rapid revision and deep understanding. Use this guide to reinforce your classroom learning, prepare for topic tests and build confidence for the AS examination.
本简明手册涵盖了 WJEC Year 12 进阶数学考试所需的所有核心公式、恒等式与定理。每项内容均以中英双语呈现，便于快速复习与深入理解。可用此手册巩固课堂所学、准备单元测验并为 AS 考试建立信心。
1. Complex Numbers | 复数
A complex number is written in Cartesian form as z = a + bi, where a, b ∈ ℝ and i² = −1. The real part is Re(z) = a and the imaginary part is Im(z) = b.
复数在直角坐标形式下写作 z = a + bi，其中 a, b ∈ ℝ 且 i² = −1。实部 Re(z) = a，虚部 Im(z) = b。
The complex conjugate is z* = a − bi, and the modulus is |z| = √(a² + b²).
共轭复数为 z* = a − bi，模为 |z| = √(a² + b²)。
The argument θ satisfies tan θ = b/a, with appropriate quadrant adjustment; the principal argument lies in (−π, π].
辐角 θ 满足 tan θ = b/a，并需根据象限调整；主辐角取值范围为 (−π, π]。
Multiplication and division in polar form: if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)] and z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)].
极坐标形式下的乘法与除法：若 z₁ = r₁(cos θ₁ + i sin θ₁)，z₂ = r₂(cos θ₂ + i sin θ₂)，则 z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]，z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n.
棣莫弗定理：对于整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。
The n distinct nth roots of unity are given by cos(2kπ/n) + i sin(2kπ/n), k = 0, 1, …, n−1.
n 次单位根的 n 个不同根为 cos(2kπ/n) + i sin(2kπ/n)，其中 k = 0, 1, …, n−1。
2. Matrices and Linear Transformations | 矩阵与线性变换
For two matrices to be multiplied, the number of columns in the first must equal the number of rows in the second.
两矩阵相乘时，第一个矩阵的列数必须等于第二个矩阵的行数。
The 2×2 matrix M = [a b; c d] maps the unit square to a parallelogram with area scale factor |det(M)| = |ad − bc|.
2×2 矩阵 M = [a b; c d] 将单位正方形映射为一个平行四边形，其面积比例因子为 |det(M)| = |ad − bc|。
A transformation is a rotation if its matrix has the form [cos θ −sin θ; sin θ cos θ]; reflection in the line y = (tan θ)x gives [cos 2θ sin 2θ; sin 2θ −cos 2θ].
若变换矩阵形如 [cos θ −sin θ; sin θ cos θ]，则为旋转；关于直线 y = (tan θ)x 的反射矩阵为 [cos 2θ sin 2θ; sin 2θ −cos 2θ]。
Invariant points satisfy Mx = x; invariant lines satisfy Mx = λx for some scalar λ.
不变点满足 Mx = x；不变直线满足存在标量 λ 使得 Mx = λx。
The inverse of a 2×2 matrix exists iff det(M) ≠ 0, and is given by (1/det)[d −b; −c a].
2×2 矩阵的逆存在当且仅当 det(M) ≠ 0，其公式为 (1/det)[d −b; −c a]。
3. Vector Algebra | 向量代数
The scalar (dot) product of vectors a and b is a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.
向量 a 与 b 的数量积（点积）为 a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。
The vector (cross) product a × b is perpendicular to both a and b, with magnitude |a||b| sin θ. In component form, a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).
向量积（叉积）a × b 垂直于 a 和 b，大小为 |a||b| sin θ。在分量形式下，a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)。
The area of a triangle with sides given by vectors a and b is ½|a × b|.
以向量 a、b 为两边的三角形面积为 ½|a × b|。
The vector equation of a line passing through point A with direction d is r = a + td.
经过点 A 且方向为 d 的直线的向量方程为 r = a + td。
The shortest distance from point P to line r = a + td is |(p − a) × d̂|, where d̂ is the unit direction vector.
点 P 到直线 r = a + td 的最短距离为 |(p − a) × d̂|，其中 d̂ 为单位方向向量。
4. Differentiation Techniques | 微分技巧
The derivative of xⁿ is nxⁿ⁻¹ for rational n. The chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x).
xⁿ 的导数为 nxⁿ⁻¹（n 为有理数）。链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。
Product rule: d/dx(uv) = u’v + uv’. Quotient rule: d/dx(u/v) = (u’v − uv’)/v².
乘法法则：d/dx(uv) = u’v + uv’。除法法则：d/dx(u/v) = (u’v − uv’)/v²。
Derivatives of trigonometric functions: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec² x.
三角函数的导数：d/dx(sin x) = cos x，d/dx(cos x) = −sin x，d/dx(tan x) = sec² x。
Derivatives of exponentials and logs: d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.
指数函数与对数函数的导数：d/dx(eˣ) = eˣ，d/dx(ln x) = 1/x。
Implicit differentiation: differentiate both sides of F(x, y) = 0 with respect to x, treating y as a function of x.
隐函数微分：对方程 F(x, y) = 0 两边关于 x 求导，将 y 视为 x 的函数。
5. Integration Techniques | 积分技巧
Basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C.
基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1），∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C。
Integration by substitution: ∫ f(g(x))g'(x) dx = ∫ f(u) du, where u = g(x). Definite integrals adjust limits accordingly.
换元积分法：∫ f(g(x))g'(x) dx = ∫ f(u) du，其中 u = g(x)。定积分需要相应调整积分限。
Integration by parts: ∫ u dv = uv − ∫ v du. Choose u according to LIATE (Logs, Inverse trig, Algebraic, Trig, Exponential).
分部积分法：∫ u dv = uv − ∫ v du。按 LIATE 顺序选择 u（对数、反三角函数、代数、三角、指数）。
Standard trigonometric integrals: ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C.
标准三角积分：∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C，∫ sec² x dx = tan x + C。
Partial fractions: for distinct linear factors, write algebraic fraction as sum of simpler fractions before integrating.
部分分式法：对于不同的线性因子，先将代数分式拆分为简单分式之和再进行积分。
6. Hyperbolic Functions | 双曲函数
Definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x.
定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。
Fundamental identity: cosh² x − sinh² x = 1.
基本恒等式：cosh² x − sinh² x = 1。
Derivatives: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech² x.
导数：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech² x。
Inverse functions: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) for x ≥ 1, artanh x = ½ ln((1+x)/(1−x)) for |x| < 1.
反双曲函数：arsinh x = ln(x + √(x² + 1))，arcosh x = ln(x + √(x² − 1))，x ≥ 1；artanh x = ½ ln((1+x)/(1−x))，|x| < 1。
Osborn’s rule: to obtain a hyperbolic identity from a trigonometric one, replace every product of two sines by −sinh².
奥斯本法：由三角恒等式得到双曲恒等式时，将每两个正弦的乘积替换为 −sinh²。
7. Sequences and Series | 数列与级数
Sum of first n natural numbers: Σᵢ₌₁ⁿ i = n(n+1)/2.
前 n 个自然数的和：Σᵢ₌₁ⁿ i = n(n+1)/2。
Sum of squares: Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6; sum of cubes: Σᵢ₌₁ⁿ i³ = [n(n+1)/2]².
平方和：Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6；立方和：Σᵢ₌₁ⁿ i³ = [n(n+1)/2]²。
The Maclaurin series expansion: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + ….
麦克劳林级数展开：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。
Standard Maclaurin series: eˣ = 1 + x + x²/2! + x³/3! + …, sin x = x − x³/3! + x⁵/5! − …, cos x = 1 − x²/2! + x⁴/4! − …, ln(1+x) = x − x²/2 + x³/3 − … for −1 < x ≤ 1.
标准麦克劳林级数：eˣ = 1 + x + x²/2! + x³/3! + …，sin x = x − x³/3! + x⁵/5! − …，cos x = 1 − x²/2! + x⁴/4! − …，ln(1+x) = x − x²/2 + x³/3 − …，其中 −1 < x ≤ 1。
8. Polar Coordinates | 极坐标
A point P is defined by (r, θ), where r ≥ 0 is the distance from the pole O, and θ is the anticlockwise angle from the initial line.
点 P 用 (r, θ) 表示，其中 r ≥ 0 为到极点 O 的距离，θ 为从极轴逆时针测量的角度。
Conversion formulas: x = r cos θ, y = r sin θ; r = √(x² + y²), θ = arctan(y/x) with quadrant adjustment.
坐标转换：x = r cos θ，y = r sin θ；r = √(x² + y²)，θ = arctan(y/x) 并需象限调整。
The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is A = ½ ∫ₐᵝ r² dθ.
极坐标曲线 r = f(θ) 在 θ = α 与 θ = β 之间所围面积为 A = ½ ∫ₐᵝ r² dθ。
The arc length of a polar curve is s = ∫ₐᵝ √(r² + (dr/dθ)²) dθ.
极坐标曲线的弧长为 s = ∫ₐᵝ √(r² + (dr/dθ)²) dθ。
9. Differential Equations | 微分方程
A first order separable ODE has the form dy/dx = g(x)h(y). Rewrite as ∫ 1/h(y) dy = ∫ g(x) dx.
一阶可分离常微分方程形如 dy/dx = g(x)h(y)。改写为 ∫ 1/h(y) dy = ∫ g(x) dx。
An integrating factor for dy/dx + P(x)y = Q(x) is I(x) = e^{∫ P(x) dx}. Multiply both sides by I(x) to obtain d/dx(I y) = I Q(x).
对于 dy/dx + P(x)y = Q(x)，积分因子为 I(x) = e^{∫ P(x) dx}。两边同乘 I(x) 得 d/dx(I y) = I Q(x)。
Second order homogenous linear ODE with constant coefficients: a d²y/dx² + b dy/dx + c y = 0. Auxiliary equation: a m² + b m + c = 0. Solution depends on real distinct roots m₁, m₂: y = A e^{m₁ x} + B e^{m₂ x}.
常系数二阶齐次线性微分方程：a d²y/dx² + b dy/dx + c y = 0。辅助方程：a m² + b m + c = 0。对于相异实根 m₁, m₂，解为 y = A e^{m₁ x} + B e^{m₂ x}。
For repeated root m: y = (A + B x) e^{m x}. For complex roots α ± iβ: y = e^{α x}(A cos βx + B sin βx).
重根 m 的情况：y = (A + B x) e^{m x}。共轭复根 α ± iβ：y = e^{α x}(A cos βx + B sin βx)。
10. Proof by Induction | 归纳证明
The principle of mathematical induction: to prove a statement P(n) for all positive integers n, show P(1) is true (base case), and show that if P(k) is true then P(k+1) is true (inductive step).
数学归纳法原理：要证明一个对全体正整数 n 的命题 P(n)，需验证基础情形 P(1) 为真，并证明若 P(k) 真则 P(k+1) 亦真（归纳步骤）。
Common induction formulas: ∑_{r=1}^{n} r = n(n+1)/2; ∑_{r=1}^{n} r² = n(n+1)(2n+1)/6; 2ⁿ > n² for n ≥ 5.
常见归纳公式：∑_{r=1}^{n} r = n(n+1)/2；∑_{r=1}^{n} r² = n(n+1)(2n+1)/6；当 n ≥ 5 时有 2ⁿ > n²。
For divisibility proofs, assume P(k): f(k) is divisible by d, then simplify f(k+1) − f(k) to show the divisibility carries forward.
整除性证明中，假设 P(k)：f(k) 能被 d 整除，然后化简 f(k+1) − f(k) 以表明整除性可延续。
Matrix induction: prove that Aⁿ has a specified form for all n ∈ ℕ by showing Aᵏ⁺¹ = Aᵏ A.
矩阵归纳：证明对于所有自然数 n，Aⁿ 具有特定形式，只需验证 Aᵏ⁺¹ = Aᵏ A。
Always write a conclusion: “Therefore, by mathematical induction, P(n) is true for all n ∈ ℕ.”
务必写出结论：“因此，由数学归纳法可知，对所有 n ∈ ℕ，P(n) 成立。”
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply