Year 13 CCEA Maths: Formula & Theorem Quick Reference | Year 13 CCEA 数学：公式定理速查手册

📚 Year 13 CCEA Maths: Formula & Theorem Quick Reference | Year 13 CCEA 数学：公式定理速查手册

This comprehensive quick-reference guide distils the crucial formulas and theorems required for CCEA Year 13 Mathematics. It spans pure mathematics, mechanics, and statistics, providing a concise yet thorough revision aid. Each entry is presented with clear notation and paired with its Chinese translation to support bilingual learners.

本速查手册浓缩了 CCEA Year 13 数学所需的关键公式与定理，涵盖纯数学、力学与统计，提供简洁全面的复习帮助。每个条目均以清晰符号呈现，并配有中文翻译，便于双语学习者使用。

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1. Algebraic Techniques | 代数技巧

Binomial Theorem: (a + b)ⁿ = ∑ⁿ_{r=0} C(n, r) aⁿ⁻ʳ bʳ, where C(n, r) = n! / (r!(n − r)!).

二项式定理：(a + b)ⁿ = ∑ⁿ_{r=0} C(n, r) aⁿ⁻ʳ bʳ，其中二项式系数 C(n, r) = n! / (r!(n − r)!).

Partial Fractions: For linear factors, f(x) / [(x + a)(x + b)] = A/(x + a) + B/(x + b). For a repeated linear factor (x + a)², use A/(x + a) + B/(x + a)². For an irreducible quadratic factor ax² + bx + c, include a term (Cx + D)/(ax² + bx + c).

部分分式：对于线性因式，f(x) / [(x + a)(x + b)] = A/(x + a) + B/(x + b)。对于重线性因式 (x + a)²，使用 A/(x + a) + B/(x + a)²。对于不可约二次因式 ax² + bx + c，添加项 (Cx + D)/(ax² + bx + c)。

Remainder Theorem: If a polynomial f(x) is divided by (x − a), the remainder is f(a).

余数定理：若多项式 f(x) 除以 (x − a)，余数为 f(a)。

Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

因子定理：(x − a) 是 f(x) 的因式，当且仅当 f(a) = 0。

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2. Functions and Graphs | 函数与图像

Domain and Range: The domain of a function is the set of all possible input values (x) for which the function is defined. The range is the set of all output values (y) that the function can produce.

定义域与值域：函数的定义域是使函数有定义的所有可能输入值 (x) 的集合。值域是函数能产生的所有输出值 (y) 的集合。

Composite Functions: (f ∘ g)(x) = f(g(x)). The domain of f ∘ g consists of x in the domain of g such that g(x) is in the domain of f.

复合函数：(f ∘ g)(x) = f(g(x))。f ∘ g 的定义域由满足 g(x) 属于 f 的定义域的 x 组成。

Inverse Functions: f⁻¹(x) is the inverse of f(x) if f⁻¹(f(x)) = x for all x in the domain. Graphically, the inverse is a reflection in the line y = x.

反函数：若对所有定义域中的 x 有 f⁻¹(f(x)) = x，则 f⁻¹(x) 为 f(x) 的反函数。其图像关于直线 y = x 对称。

Transformations of Graphs:

• f(x) + a: vertical translation by a (up if a > 0).
• f(x + a): horizontal translation by −a (left if a > 0).
• af(x): vertical stretch by factor a (if a > 1, stretch; if 0 < a < 1, compression).
• f(ax): horizontal stretch by factor 1/a (if a > 1, compression; if 0 < a < 1, stretch).
• −f(x): reflection in the x-axis.
• f(−x): reflection in the y-axis.

图像变换：

• f(x) + a：垂直平移 a（a > 0 时向上）。
• f(x + a)：水平平移 −a（a > 0 时向左）。
• af(x)：垂直伸缩因子 a（a > 1 伸长；0 < a < 1 压缩）。
• f(ax)：水平伸缩因子 1/a（a > 1 压缩；0 < a < 1 伸长）。
• −f(x)：关于 x 轴对称。
• f(−x)：关于 y 轴对称。

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3. Trigonometry | 三角学

Radian Measure: π radians = 180°. Arc length s = rθ, sector area A = ½ r² θ.

弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½ r² θ。

Fundamental Identities:

sin² θ + cos² θ = 1    1 + tan² θ = sec² θ    1 + cot² θ = cosec² θ

基本恒等式：

sin² θ + cos² θ = 1    1 + tan² θ = sec² θ    1 + cot² θ = cosec² θ

Compound Angle Formulas:

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)

和差角公式：

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)

Double Angle Formulas:

sin 2θ = 2 sin θ cos θ

cos 2θ = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ

tan 2θ = 2 tan θ / (1 − tan² θ)

二倍角公式：

sin 2θ = 2 sin θ cos θ

cos 2θ = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ

tan 2θ = 2 tan θ / (1 − tan² θ)

R-formula: a sin θ + b cos θ = R sin(θ ± α) or R cos(θ ∓ α), where R = √(a² + b²) and tan α = b/a (with appropriate quadrant for α).

R 公式：a sin θ + b cos θ = R sin(θ ± α) 或 R cos(θ ∓ α)，其中 R = √(a² + b²)，α 满足 tan α = b/a（并依据象限确定 α）。

Solving Trigonometric Equations: Use identities to reduce to a single trig function, then find principal values and add multiples of the period (2π or π for tan).

解三角方程：利用恒等式化为单一三角函数，求出主值后加上周期的整数倍（正弦、余弦周期 2π，正切周期 π）。

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4. Exponentials and Logarithms | 指数与对数

Index Laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(m/n) = ⁿ√aᵐ.

指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a^(m/n) = ⁿ√aᵐ。

Logarithm Laws: logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, logₐ xⁿ = n logₐ x, logₐ 1 = 0, logₐ a = 1.

对数律：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x − logₐ y，logₐ xⁿ = n logₐ x，logₐ 1 = 0，logₐ a = 1。

Change of Base: logₐ b = (log_c b) / (log_c a). In particular, ln x = log_e x and log₁₀ x = log x.

换底公式：logₐ b = (log_c b) / (log_c a)。特别地，ln x = log_e x，log₁₀ x = log x。

Natural Exponential and Logarithm: e

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