📚 Year 10 CAIE Maths: Formula & Theorem Quick Reference Handbook | Year 10 CAIE 数学：公式定理速查手册
Welcome to your Year 10 CAIE Mathematics formula and theorem quick-reference handbook. This guide consolidates all the essential facts you need for algebra, geometry, trigonometry, statistics, and more, aligned with the Cambridge IGCSE syllabus. Use it for rapid revision and homework support.
欢迎使用 Year 10 CAIE 数学公式定理速查手册。本手册整合了代数、几何、三角学、统计学等所有与剑桥 IGCSE 课程一致的关键知识点，方便你快速复习和完成作业。
1. Algebraic Expansion and Factorisation | 代数展开与因式分解
The distributive law is the foundation for expanding brackets: a(b + c) = ab + ac. Multiply the term outside by each term inside.
分配律是展开括号的基础：a(b + c) = ab + ac。将括号外的项乘以括号内的每一项。
When expanding a product of two binomials, apply the FOIL method: (a + b)(c + d) = ac + ad + bc + bd.
展开两个二项式的乘积时，应用 FOIL 法则：(a + b)(c + d) = ac + ad + bc + bd。
Special products include perfect squares: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b².
特殊乘积包括完全平方：(a + b)² = a² + 2ab + b² 和 (a – b)² = a² – 2ab + b²。
The difference of two squares factorises as: a² – b² = (a + b)(a – b). Factorising by a common factor reverses the distributive law: ab + ac = a(b + c).
平方差可分解为：a² – b² = (a + b)(a – b)。提公因式法是分配律的逆运算：ab + ac = a(b + c)。
2. Linear Equations and Inequalities | 一次方程与不等式
To solve a linear equation of the form ax + b = c, isolate x: x = (c – b)/a.
要求解形如 ax + b = c 的一次方程，将 x 分离：x = (c – b)/a。
Always perform the same operation on both sides. When dealing with fractions, multiply through by the common denominator.
始终在等式两边执行相同运算。如果遇到分数，两边同乘公分母以消去分母。
For inequalities, the rules remain the same, except if you multiply or divide by a negative number, the inequality sign reverses.
对于不等式，解法规则相同，但当两边乘或除以负数时，不等号方向需反转。
Solid circles on a number line indicate ≤ or ≥, while open circles indicate < or >. The solution set can be written in interval notation or on a number line.
数轴上实心圆点表示 ≤ 或 ≥，空心圆点表示 < 或 >。解集可用区间表示或绘制在数轴上。
3. Simultaneous Equations | 联立方程组
Simultaneous equations can be solved by elimination (adding/subtracting equations to cancel one variable) or by substitution (solving one equation for a variable and substituting into the other).
联立方程组可通过消元法（将两方程相加减以消去一个变量）或代入法（从一个方程解出一个变量并代入另一个方程）求解。
For example, to solve 2x + 3y = 8 and 3x – 2y = –1, you can multiply the first by 2 and the second by 3, then add to eliminate y.
例如，求解 2x + 3y = 8 和 3x – 2y = –1，可将第一个方程乘以 2、第二个乘以 3，然后相加消去 y。
The intersection point of the two lines represents the unique solution. If the lines are parallel, there is no solution; if they coincide, there are infinitely many solutions.
两条直线的交点表示唯一解。若两直线平行，则无解；若两直线重合，则有无穷多解。
4. Sequences | 数列
An arithmetic (linear) sequence has a constant difference d between terms. The nth term is given by: uₙ = a + (n – 1)d, where a is the first term.
等差（线性）数列的相邻项之差 d 为常数。第 n 项的通项公式为：uₙ = a + (n – 1)d，其中 a 是首项。
To find the nth term of a linear sequence from a table of values, determine the common difference d and adjust to match the first term: uₙ = dn + c where c = a – d.
若从数值表求线性数列的通项公式，先确定公差 d，再调整匹配首项：uₙ = dn + c，其中 c = a – d。
The recursive definition is: uₙ₊₁ = uₙ + d, with a given u₁. For a quadratic sequence, the second differences are constant, but Year 10 primarily focuses on linear patterns.
递推定义为：uₙ₊₁ = uₙ + d，并已知 u₁。对于二次数列，二次差为常数，但 Year 10 主要关注线性规律。
Special sequences include square numbers n², cube numbers n³, and triangular numbers ½n(n+1).
特殊数列包括平方数 n²、立方数 n³ 和三角形数 ½n(n+1)。
5. Coordinate Geometry and Straight Lines | 坐标几何与直线
The gradient (slope) of a line joining points (x₁, y₁) and (x₂, y₂) is: m = (y₂ – y₁) / (x₂ – x₁).
连接点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率（梯度）为：m = (y₂ – y₁) / (x₂ – x₁)。
The equation of a straight line is often written as y = mx + c, where m is the gradient and c is the y-intercept.
直线方程通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
The midpoint of a segment is: ((x₁ + x₂)/2, (y₁ + y₂)/2). Parallel lines have equal gradients; perpendicular lines have gradients that multiply to –1: m₁ × m₂ = –1.
线段的中点坐标为：((x₁ + x₂)/2, (y₁ + y₂)/2)。平行线的斜率相等；垂直线的斜率之积为 –1：m₁ × m₂ = –1。
To find the equation from two points, first find m, then substitute one point into y = mx + c to solve for c.
已知两点求直线方程，先求 m，再将其代入 y = mx + c 解出 c。
6. Geometry: Angles and Polygons | 几何：角度与多边形
Angles on a straight line sum to 180°, and vertically opposite angles are equal. Corresponding and alternate angles are equal when a transversal cuts parallel lines.
直线上的角度之和为 180°，对顶角相等。一条截线穿过平行线时，同位角相等，内错角相等。
In any triangle, the interior angles sum to 180°, and an exterior angle equals the sum of the two opposite interior angles.
任意三角形的内角和为 180°，一个外角等于与其不相邻的两个内角之和。
For an n-sided polygon: sum of interior angles = (n – 2) × 180°, and sum of exterior angles (one at each vertex) is always 360°.
对于 n 边形：内角和 = (n – 2) × 180°，外角和（每个顶点取一个外角）始终为 360°。
Basic circle theorems: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is 90°; angles in the same segment are equal.
基本圆定理：圆心角是圆周角的两倍；半圆上的圆周角为 90°；同弧所对的圆周角相等。
7. Perimeter, Area and Volume | 周长、面积与体积
The perimeter is the distance around a shape. Key area formulas are summarised below:
周长是图形边界的总长度。关键面积公式总结如下：
| Shape | Area (English) | 面积 (中文) |
|---|---|---|
| Rectangle | length × width | 长 × 宽 |
| Parallelogram | base × height | 底 × 高 |
| Triangle | ½ × base × height | ½ × 底 × 高 |
| Trapezium | ½(a + b)h | ½(a + b)h |
| Circle | πr² | πr² |
For circles, the circumference is C = 2πr = πd. The arc length is (θ/360) × 2πr and the sector area is (θ/360) × πr².
对于圆，周长 C = 2πr = πd。弧长公式为 (θ/360) × 2πr，扇形面积公式为 (θ/360) × πr²。
Volume of a prism = area of cross-section × length. For a cylinder, V = πr²h. Common volumes: cube s³, cuboid lwh.
棱柱的体积 = 横截面积 × 长度。圆柱体积 V = πr²h。常见体积：立方体 s³，长方体 lwh。
8. Pythagoras’ Theorem and Trigonometry | 毕达哥拉斯定理与三角函数
In a right-angled triangle, Pythagoras’ theorem states: a² + b² = c², where c is the hypotenuse.
在直角三角形中，毕达哥拉斯定理（勾股定理）为：a² + b² = c²，其中 c 是斜边。
The three trigonometric ratios relate an acute angle θ to the sides: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
三个三角函数比将锐角 θ 与边关联：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Exact trigonometric values for key angles:
关键角的精确三角函数值：
| θ | sin θ
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