📚 Year 13 CIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CIE 数学：公式定理速查手册
This quick reference handbook compiles essential formulas, theorems, and standard results needed for the Year 13 CIE A-Level Mathematics (9709) course, covering Pure Mathematics 3 as well as key mechanics and statistics topics. Use it for rapid revision and exam preparation.
本速查手册汇编了 Year 13 CIE A-Level 数学 (9709) 课程所需的基本公式、定理和标准结论，涵盖纯数学 3 以及关键的力学和统计主题。可用于快速复习和备考。
1. Algebra and Exponential/Logarithmic Functions | 代数与指数对数函数
Algebraic manipulation and the laws of indices and logarithms are fundamental tools for differentiation, integration, and equation solving in A2 Mathematics.
代数运算以及指数和对数法则是 A2 数学中微分、积分和方程求解的基本工具。
Index laws for positive real bases a, b:
aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (ab)ᵐ = aᵐ bᵐ
正实数底 a、b 的指数法则。
Logarithm rules (a > 0, a ≠ 1, x, y > 0):
logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x – logₐ y, logₐ(xⁿ) = n logₐ x
对数法则（a > 0，a ≠ 1，x, y > 0）。
The change-of-base formula:
logₐ x = (log_b x) / (log_b a)
换底公式。
The natural logarithm and exponential function are inverses: e^(ln x) = x for x > 0, and ln(e^x) = x.
自然对数与指数函数互为反函数：当 x > 0 时 e^(ln x) = x，且 ln(e^x) = x。
For rational functions, partial fraction decomposition is often used before integration. Standard forms include:
1/((x+a)(x+b)) = A/(x+a) + B/(x+b), or (px+q)/((x+a)(x²+b)) = A/(x+a) + (Bx+C)/(x²+b)
对于有理函数，部分分式分解常用于积分前。标准形式如上。
2. Trigonometry | 三角学
Trigonometric identities, radian measure, and inverse trigonometric functions appear extensively in differentiation, integration, and solving equations.
三角恒等式、弧度制以及反三角函数广泛出现在微分、积分和方程求解中。
Fundamental Pythagorean identities:
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
基本毕达哥拉斯恒等式。
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)
和角与差角公式。
Double angle formulae:
sin 2A = 2 sin A cos A, cos 2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A, tan 2A = 2 tan A / (1 – tan²A)
二倍角公式。
Expressions for a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α):
a cos θ + b sin θ = R cos(θ – α) or R sin(θ + β), where R = √(a² + b²) and tan α = b/a.
将 a cos θ + b sin θ 表示为 R cos(θ ± α) 或 R sin(θ ± β) 的形式，其中 R = √(a² + b²)，tan α = b/a。
Small angle approximations (in radians):
sin θ ≈ θ, cos θ ≈ 1 – ½ θ², tan θ ≈ θ
小角近似（弧度制）。
3. Differentiation | 微分
Differentiation techniques in P3 extend the rules learned in AS and introduce parametric and implicit differentiation, as well as derivatives of exponentials, logarithms, and trigonometric functions.
P3 中的微分技巧在 AS 的基础上进行了扩展，引入了参数微分和隐函数微分，以及指数、对数和三角函数的导数。
Standard derivatives:
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
标准导数。
Chain, product, and quotient rules:
Chain: dy/dx = dy/du × du/dx; Product: d(uv)/dx = u’v + uv’; Quotient: d(u/v)/dx = (u’v – uv’)/v²
链式法则、乘积法则和商法则。
Implicit differentiation differentiates both sides of an equation F(x,y)=0 with respect to x, treating y as a function of x.
隐函数微分法对方程 F(x,y)=0 两边同时对 x 求导，将 y 视为 x 的函数。
Parametric differentiation: if x = f(t), y = g(t), then
dy/dx = (dy/dt) / (dx/dt)
参数微分：如果 x = f(t)，y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。
4. Integration | 积分
Integration in P3 covers reverse differentiation, substitution, integration by parts, and applications such as finding areas and volumes of revolution.
P3 的积分涵盖逆微分、换元积分、分部积分，以及求面积和旋转体体积等应用。
Standard integrals (with constant of integration +C):
∫ 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
标准积分（含积分常数 +C）。
Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x). For definite integrals, change limits accordingly.
换元积分法：∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。对于定积分需要相应地改变积分上下限。
Integration by parts:
∫ u dv = uv – ∫ v du, or ∫ u (dv/dx) dx = uv – ∫ v (du/dx) dx
分部积分法。
Using the reverse chain rule for forms ∫ f'(x)/f(x) dx = ln|f(x)| + C.
利用逆链式法则处理形如 ∫ f'(x)/f(x) dx = ln|f(x)| + C 的积分。
5. Differential Equations | 微分方程
First-order differential equations are solved by separating variables or using an integrating factor. These appear in modelling contexts.
一阶微分方程可通过分离变量法或积分因子法求解，常用于建模情境中。
Separation of variables: if dy/dx = f(x)g(y), then
∫ 1/g(y) dy = ∫ f(x) dx
分离变量法：若 dy/dx = f(x)g(y)，则 ∫ 1/g(y) dy = ∫ f(x) dx。
For a linear first-order ODE dy/dx + P(x)y = Q(x), the integrating factor is
I(x) = e^(∫ P(x) dx)
对于一阶线性常微分方程 dy/dx + P(x)y = Q(x)，积分因子为 I(x) = e^(∫ P(x) dx)。
Then the solution is given by
y I(x) = ∫ I(x) Q(x) dx + C
则解为 y I(x) = ∫ I(x) Q(x) dx + C。
6. Vectors | 向量
Vectors in three dimensions are used to describe lines, planes, and distances, and the scalar product is the central tool for angles and perpendicularity.
三维向量用于描述直线、平面和距离，而数量积（点积）是处理角度与垂直问题的核心工具。
For vectors a = (x₁, y₁, z₁) and b = (x₂, y₂, z₂):
Magnitude: |a| = √(x₁² + y₁² + z₁²); Scalar product: a·b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b| cos θ
向量的模和数量积。
Equation of a straight line in vector form:
r = a + t b, where a is a point on the line and b is the direction vector.
直线的向量方程：r = a + t b，其中 a 是直线上一点，b 为方向向量。
Equation of a plane:
r·n = a·n, or in Cartesian form ax + by + cz = d, where n = (a,b,c) is normal.
平面方程：r·n = a·n，或笛卡尔形式 ax + by + cz = d，其中 n = (a,b,c) 为法向量。
Distance from a point (x₁, y₁, z₁) to the plane ax + by + cz = d:
Distance = |ax₁ + by₁ + cz₁ – d| / √(a² + b² + c²)
点到平面的距离公式。
7. Complex Numbers | 复数
Complex numbers extend the real number system, and their polar and exponential forms simplify multiplication, division, powers, and roots.
复数扩展了实数系统，其极坐标形式和指数形式可简化乘除、乘方和开方运算。
A complex number z = x + iy has modulus r = √(x² + y²) and argument θ = arg(z). The polar form is
z = r (cos θ + i sin θ)
复数 z = x + iy 的模 r = √(x²+y²)，辐角 θ = arg(z)。极坐标形式为 z = r (cos θ + i sin θ)。
Multiplication and division in polar form:
z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)], z₁/z₂ = r₁/r₂ [cos(θ₁–θ₂) + i sin(θ₁–θ₂)]
极坐标形式的乘法与除法。
De Moivre’s theorem:
[r(cos θ + i sin θ)]ⁿ = rⁿ (cos nθ + i sin nθ) for integer n.
棣莫弗定理。
The exponential form is z = r e^(iθ), where e^(iθ) = cos θ + i sin θ. This leads to:
e^(iπ) + 1 = 0
指数形式为 z = r e^(iθ)，且 e^(iθ) = cos θ + i sin θ。由此可得欧拉恒等式 e^(iπ) + 1 = 0。
The n-th roots of a complex number are given by
z^(1/n) = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0,1,…,n–1
复数的 n 次方根公式。
8. Numerical Methods | 数值方法
Numerical methods provide approximate solutions to equations and definite integrals when algebraic methods fail. The trapezium rule and Newton-Raphson method are key.
当代数方法无法求解时，数值方法可为方程和定积分提供近似解。梯形法则和牛顿−拉弗森法是关键。
Newton-Raphson iterative formula for finding a root of f(x) = 0:
xₙ₊₁ = xₙ – f(xₙ) / f'(xₙ)
求方程 f(x)=0 根的牛顿−拉弗森迭代公式。
Fixed-point iteration: rearrange f(x)=0 as x = g(x), then iterate xₙ₊₁ = g(xₙ). Convergence requires |g'(x)| < 1 near the root.
不动点迭代：将 f(x)=0 改写为 x = g(x)，然后迭代 xₙ₊₁ = g(xₙ)。收敛性要求在根附近 |g'(x)| < 1。
The trapezium rule for approximating ∫ₐᵇ f(x) dx with n strips of width h = (b–a)/n:
∫ₐᵇ f(x) dx ≈ ½ h [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)], where yᵢ = f(xᵢ)
梯形法则近似定积分，其中 n 个等宽条带，宽度 h = (b–a)/n。
9. Mechanics Essentials (M2) | 力学必备公式（M2）
Mechanics topics in Year 13 often include projectile motion, work-energy principles, momentum, and circular motion. These formulas are crucial for problem solving.
Year 13 的力学主题通常包括抛射体运动、功能原理、动量和圆周运动。这些公式对解题至关重要。
Constant acceleration equations (suvat):
v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½ (u+v)t
匀加速运动方程（suvat 公式）。
Work done by a constant force: W = F s cos θ, where θ is the angle between force and displacement. Kinetic energy: KE = ½ mv². Gravitational potential energy: GPE = mgh.
恒力做的功 W = F s cos θ，动能 KE = ½ mv²，重力势能 GPE = mgh。
Impulse–momentum principle: Impulse = change in momentum = m(v – u). For a constant force: F Δt = m(v – u).
冲量−动量原理：冲量 = 动量的变化 = m(v – u)。恒力作用下 F Δt = m(v – u)。
Uniform circular motion: centripetal acceleration a = v² / r = r ω², centripetal force F = mv² / r = m r ω². Angular velocity ω = θ/t = 2π/T.
匀速圆周运动：向心加速度 a = v²/r = r ω²，向心力 F = mv²/r = m r ω²。角速度 ω = θ/t = 2π/T。
10. Statistics Essentials (S2) |
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