📚 Year 10 SQA Advanced Mathematics: Quick Reference Formula & Theorem Handbook | SQA 进阶数学：公式定理速查手册
This handbook provides a concise summary of essential formulas, theorems, and techniques for the Year 10 SQA Advanced Mathematics course. It is designed as a quick reference for revision, covering algebra, trigonometry, calculus, vectors, and more. Every section presents key results in a clear, bilingual format to support both understanding and exam preparation.
本手册总结了 Year 10 SQA 进阶数学课程的核心公式、定理和技巧，是一份用于复习的快速参考。内容涵盖代数、三角学、微积分、向量等多个板块，每节均以中英双语清晰呈列关键结论，助力理解与备考。
1. Algebra – Quadratic Equations | 代数 - 二次方程
The standard form of a quadratic equation is ax² + bx + c = 0, with a ≠ 0. Solutions are given by the quadratic formula.
二次方程的标准形式为 ax² + bx + c = 0，a ≠ 0。其解由二次公式给出。
x = ( –b ± √b² – 4ac ) / 2a
The discriminant D = b² – 4ac determines the nature of the roots: D > 0 ⇒ two distinct real roots; D = 0 ⇒ one repeated real root; D < 0 ⇒ no real roots.
判别式 D = b² – 4ac 决定根的性质：D > 0 ⇒ 两个不等实根；D = 0 ⇒ 一个重实根；D < 0 ⇒ 无实根。
the square transforms ax² + bx + c into the form a(x + p)² + q, where p = b/(2a) and q = c – (b²/(4a)).
配方法将 ax² + bx + c 化为 a(x + p)² + q，其中 p = b/(2a)，q = c – (b²/(4a))。
The sum of roots is –b/a and the product of roots is c/a for ax² + bx + c = 0.
对于 ax² + bx + c = 0，两根之和为 –b/a，两根之积为 c/a。
2. Algebra – Polynomials and Factor Theorem | 代数 - 多项式与因式定理
The Factor Theorem states: For a polynomial P(x), if P(a) = 0, then (x – a) is a factor of P(x). Conversely, if (x – a) is a factor, then P(a) = 0.
因式定理：对于多项式 P(x)，若 P(a) = 0，则 (x – a) 是 P(x) 的一个因式；反之亦然。
The Remainder Theorem: When P(x) is divided by (x – a), the remainder is P(a).
余式定理：P(x) 除以 (x – a) 的余式等于 P(a)。
Synthetic division can be used to efficiently divide a polynomial by a linear factor, helping to factorise cubic or higher-degree polynomials.
综合除法可用于高效地将多项式除以一次因式，帮助分解三次或更高次多项式。
l (often testing integer divisors of the constant term), then use factorisation or the quadratic formula for the reduced polynomial.
解 P(x) = 0 时，可通过试根（常测试常数项整除数）找到一个根，然后对降次后的多项式使用因式分解或二次公式。
3. Functions and Graph Transformations | 函数与图像变换
A function f(x) can be transformed in several ways. The graph of y = f(x) + k translates vertically by k units; y = f(x + h) translates horizontally by –h units.
函数 f(x) 可通过多种方式变换。y = f(x) + k 使图像垂直平移 k 个单位；y = f(x + h) 使图像水平平移 –h 个单位。
Reflections: y = –f(x) reflects in the x‑axis; y = f(–x) reflects in the y‑axis.
对称变换：y = –f(x) 关于 x 轴对称；y = f(–x) 关于 y 轴对称。
Stretches: y = a f(x) stretches vertically by factor a; y = f(kx) compresses horizontally by factor 1/k when k>1.
伸缩变换：y = a f(x) 垂直方向伸缩 a 倍；y = f(kx) 水平方向压缩为原来的 1/k（当 k>1 时）。
Recognising the basic shapes of y = x², y = x³, y = 1/x, y = √x, y = eˣ, y = ln x helps in applying transformations.
熟记 y = x², y = x³, y = 1/x, y = √x, y = eˣ, y = ln x 的基本图像有助于应用变换。
4. Trigonometry – Right‑Angle Triangles & Identities | 三角学 – 直角三角形与恒等式
In a right‑angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
在直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Pythagoras’ theorem: a² + b² = c², where c is the hypotenuse.
勾股定理：a² + b² = c²，其中 c 为斜边。
Fundamental trig identities: sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.
基本三角恒等式：sin²θ + cos²θ = 1 以及 tan θ = sin θ / cos θ。
Exact values for common angles (in degrees):
常见角的精确值（角度）：
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
5. Trigonometry – Non‑Right‑Angle Triangles | 三角学 – 非直角三角形
For any triangle with sides a, b, c and opposite angles A, B, C, the Sine Rule states:
对于任意三角形，边 a, b, c 对应角 A, B, C，正弦定理为：
a / sin A = b / sin B = c / sin C
The Cosine Rule connects the sides and an included angle:
余弦定理关联三边及其夹角：
a² = b² + c² – 2bc cos A
The area of a triangle can be found using Area = ½ ab sin C, where a and b are two sides and C is the included angle.
三角形面积可用公式 面积 = ½ ab sin C，其中 a, b 为两边，C 为其夹角。
The ambiguous case (SSA) can yield two possible triangles when using the Sine Rule; check the supplement of the angle if necessary.
使用正弦定理时，边边角情形可能产生两解；必要时需检验角的补角是否也构成有效三角形。
6. Exponentials and Logarithms | 指数与对数
Laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁰ = 1. For rational exponents, a1/n = ⁿ√a.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ/aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1。有理指数 a1/n = ⁿ√a。
The logarithm function is the inverse of the exponential: if y = bˣ, then x = log_b y. The natural logarithm ln x has base e.
对数是指数的逆运算：若 y = bˣ，则 x = log_b y。自然对数 ln x 以 e 为底。
Key log rules: log_b (xy) = log_b x + log_b y, log_b (x/y) = log_b x – log_b y, log_b (xⁿ) = n log_b x.
对数运算法则：log_b (xy) = log_b x + log_b y，log_b (x/y) = log_b x – log_b y，log_b (xⁿ) = n log_b x。
Change of base formula: log_b a = ln a / ln b or log_b a = log_c a / log_c b.
换底公式：log_b a = ln a / ln b 或 log_b a = log_c a / log_c b。
7. Differentiation – Rules and Applications | 微分 – 法则与应用
The derivative f'(x) gives the gradient of the curve y = f(x). The power rule: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹.
导数 f'(x) 表示曲线 y = f(x) 的梯度。幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。
Constant multiple rule: d/dx [c f(x)] = c f'(x). Sum rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
常数倍法则：d/dx [c f(x)] = c f'(x)。和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
Product rule: d/dx [u v] = u'v + u v'. Quotient rule: d/dx [u/v] = (u'v – u v') / v².
乘积法则：d/dx [u v] = u'v + u v'。商法则：d/dx [u/v] = (u'v – u v') / v²。
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) ⋅ g'(x).
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) ⋅ g'(x)。
| f(x) | f'(x) |
|---|---|
| sin x | cos x |
| cos x | –sin x |
| eˣ | eˣ |
| ln x | 1/x |
The equation of a tangent at (a, f(a)) is y – f(a) = f'(a)(x – a).
在点 (a, f(a)) 处的切线方程为 y – f(a) = f'(a)(x – a)。
8. Integration – Indefinite and Definite Integrals | 积分 – 不定积分与定积分
Integration is the reverse process of differentiation. The indefinite integral of xⁿ: ∫ xⁿ dx = (1/(n+1)) xⁿ⁺¹ + C, n ≠ –1.
积分是微分的逆运算。xⁿ 的不定积分：∫ xⁿ dx = (1/(n+1)) xⁿ⁺¹ + C, n ≠ –1。
Standard integrals: ∫ 1/x dx = ln |x| + C, ∫ eˣ dx = eˣ + C, ∫ sin x dx = –cos x + C, ∫ cos x dx = sin x + C.
标准积分：∫ 1/x dx = ln |x| + C，∫ eˣ dx = eˣ + C，∫ sin x dx = –cos x + C，∫ cos x dx = sin x + C。
The definite integral from a to b, ∫_ab f(x) dx sents the signed area under the curve. It is evaluated using the Fundamental Theorem: if F'(x) = f(x), then ∫_ab f(x) dx = F(b) – F(a).
从 a 到 b 的定积分 ∫_ab f(x) dx 表示曲线下的有向面积，通过基本定理计算：若 F'(x) = f(x)，则 ∫_ab f(x) dx = F(b) – F(a)。
on, use the reverse of the chain rule, e.g. ∫ (ax + b)ⁿ dx = (1/(a(n+1))) (ax + b)ⁿ⁺¹ + C.
涉及线性代换时，逆向使用链式法则，例如 ∫ (ax + b)ⁿ dx = (1/(a(n+1))) (ax + b)ⁿ⁺¹ + C。
9. Vectors – Magnitude, Dot Product | 向量 – 模长与点积
A vector u is often written in component form: u = (u₁, u₂) in 2D, or u = (u₁, u₂, u₃) in 3D. The magnitude (length) is |u| = √u₁² + u₂² + ….
向量常以分量形式表示：2D 中 u = (u₁, u₂)，3D 中 u = (u₁, u₂, u₃)。其模长（长度）为 |u| = √u₁² + u₂² + …。
The dot product (scalar product) of two vectors is u · v = u₁v₁ + u₂v₂ (+ u₃v₃). It is also given by u · v = |u| |v| cos θ, where θ is the angle between them.
两向量的点积（数量积）为 u · v = u₁v₁ + u₂v₂ (+ u₃v₃)，也可表示为 u · v = |u| |v| cos θ，θ 为向量间的夹角。
Two non‑zero vectors are perpendicular if and only if u · v = 0. The angle θ can be found from cos θ = (u · v) / (|u| |v|).
两非零向量垂直当且仅当 u · v = 0。夹角 θ 可由 cos θ = (u · v) / (|u| |v|) 求得。
Position vectors describe points relative to the origin. The vector from A to B is AB = b – a, where a, b are the position vectors of A and B.
位置向量描述点相对于原点的位置。由 A 到 B 的向量为 AB = b – a，其中 a, b 是 A 和 B 的位置向量。
10. Statistics – Measures and Probability | 统计 – 度量和概率
For a data set x₁, x₂, …, xₙ, the mean (average) is μ = (Σ xᵢ) / n. The (population) standard deviation measures spread: σ = √[ Σ (xᵢ – μ)² / n ].
对于数据集 x₁, x₂, …, xₙ，均值（平均数）为 μ = (Σ xᵢ) / n。（总体）标准差衡量离散程度：σ = √[ Σ (xᵢ – μ)² / n ]。
The sample standard deviation uses n – 1 in the denominator. Quartiles and the interquartile range (IQR = Q₃ – Q₁) summarise spread and outliers.
样本标准差分母用 n – 1。四分位数和四分位距 (IQR = Q₃ – Q₁) 用于概括分布与异常值。
vent A: P(A) = (number of favourable outcomes) / (total outcomes). For independent events, P(A ∩ B) = P(A) ⋅ P(B).
事件 A 的概率：P(A) = 有利结果数 / 总结果数。对于独立事件，P(A ∩ B) = P(A) ⋅ P(B)。
Mutually exclusive events: P(A ∪ B) = P(A) + P(B). For non‑mutually exclusive events, subtract P(A ∩ B).
互斥事件：P(A ∪ B) = P(A) + P(B)。非互斥事件需减去 P(A ∩ B)。
Conditional probability: P(A | B) = P(A ∩ B) / P(B).
条件概率：P(A | B) = P(A ∩ B) / P(B)。
11. Geometry – Circle Theorems and Equations | 几何 – 圆定理与方程
The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – b)² = r².
圆心 (a, b)、半径 r 的圆的方程为 (x – a)² + (y – b)² = r²。
The tangent to a circle at a point is perpendicular to the radius at that point. To find the tangent equation, use the gradient of the radius and the point.
圆上一点的切线垂直于该点处的半径。求切线方程时，利用半径的斜率和该点坐标。
Key circle theorems: the angle at the centre is twice the angle at the circumference subtended by the same arc; angles in the same segment are equal; the angle in a semicircle is 90°.
重要圆定理：同弧所对的圆心角是圆周角的二倍；同弓形内的圆周角相等；半圆上的圆周角是 90°。
Length of an arc: s = rθ (θ in radians). Area of a sector: A = ½ r²θ. Conversions: π rad = 180°.
弧长公式：s = rθ（θ 以弧度计）。扇形面积：A = ½ r²θ。弧度与角度换算：π rad = 180°。
12. Sequences and Series – Arithmetic and Geometric | 数列与级数 – 等差与等比
An arithmetic sequence has a common difference d: Uₙ = a + (n – 1)d, where a is the first term. Sum of the first n terms: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l) with l the last term.
等差数列有公差 d：Uₙ = a + (n – 1)d，a 为首项。前 n 项和：Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，l 为末项。
A geometric sequence has a common ratio r: Uₙ = a rⁿ⁻¹. Sum of first n terms (r ≠ 1): Sₙ = a (1 – rⁿ) / (1 – r).
等比数列有公比 r：Uₙ = a rⁿ⁻¹。前 n 项和 (r ≠ 1)：Sₙ = a (1 – rⁿ) /
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