📚 Year 8 OCR Further Maths: Formula & Theorem Quick Reference Handbook | Year 8 OCR 进阶数学：公式定理速查手册
This quick-reference handbook compiles all essential formulas and theorems required for the Year 8 OCR Further Maths course. Use it as a handy revision tool to reinforce your understanding of algebra, geometry, trigonometry, sequences, statistics, and more. Each section provides clear formula statements along with concise explanations, helping you to solve problems efficiently and accurately.
本速查手册汇编了 Year 8 OCR 进阶数学课程所需的所有重要公式与定理。你可以把它作为便捷的复习工具，巩固代数、几何、三角学、数列、统计等知识的理解。每个部分均提供清晰的公式表述与简明解释，帮助你高效、准确地解题。
1. Algebra Foundations | 代数基本公式
The commutative laws allow you to change the order of addition or multiplication without affecting the result. They form the basis for rearranging terms in algebraic expressions.
a + b = b + a and a × b = b × a
交换律允许你改变加法或乘法的顺序而不影响结果。它们是重新排列代数表达式中各项的基础。
The associative laws state that when adding or multiplying three or more numbers, the grouping does not change the outcome. This is particularly useful when simplifying expressions with multiple terms.
(a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
结合律指出，在三个或更多数的加法或乘法中，分组方式不会改变结果。这在化简含有多个项的式子时特别有用。
The distributive law connects multiplication and addition, enabling you to expand brackets or factorise expressions. It is one of the most frequently used properties in algebra.
a(b + c) = ab + ac
分配律将乘法与加法联系起来，使你能够展开括号或因式分解表达式。它是代数中最常用的性质之一。
Index laws govern how to simplify expressions involving powers. The product rule adds exponents when multiplying like bases, while the quotient rule subtracts exponents when dividing.
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)
指数律支配着含有幂的表达式如何化简。相乘时指数相加，相除时指数相减。
The power of a power rule requires multiplying the exponents. Any non-zero number raised to the power of zero equals one, and negative exponents indicate reciprocals.
(aᵐ)ⁿ = aᵐⁿ a⁰ = 1 (a ≠ 0) a⁻ⁿ = 1 / aⁿ
幂的幂法则需要将指数相乘。任何非零数的零次幂等于1，负指数则表示倒数。
2. Expanding and Factorising | 展开与因式分解
Expanding a binomial squared uses the distributive law twice. The square of a sum results in the sum of the squares plus twice the product of the two terms.
(a + b)² = a² + 2ab + b²
二项式的平方需要两次运用分配律。两个数之和的平方等于各自平方之和加上两数乘积的两倍。
Similarly, the square of a difference subtracts twice the product, but the sign of the middle term flips.
(a – b)² = a² – 2ab + b²
类似地，两个数之差的平方减去两倍乘积，中间项的符号变为负号。
The difference of two squares is a crucial factorisation pattern. It expresses a subtraction of two perfect squares as a product of a sum and a difference, allowing rapid simplification.
a² – b² = (a + b)(a – b)
平方差是一个关键的因式分解形式。它将两个完全平方数的差表示为和与差的乘积，可以快速化简。
th the constant term equal to the product a × b and the linear coefficient equal to a + b.
(x + a)(x + b) = x² + (a + b)x + ab
展开形如 (x+a)(x+b) 的两个一次二项式，会得到一个二次表达式，其中常数项等于 a × b，一次项系数等于 a + b。
When factorising, always look for a common factor first. Taking out the highest common factor simplifies the expression and often reveals a recognisable pattern.
ax + ay = a(x + y)
进行因式分解时，始终先寻找公因式。提取最大公因式可简化表达式，并常常能揭示出可识别的模式。
3. Quadratic Equations | 二次方程
A quadratic equation in standard form is written with one side equal to zero. The coefficients a, b, and c are constants, with a never equal to zero.
ax² + bx + c = 0
标准形式的二次方程一侧等于零。系数 a、b 和 c 为常数，且 a 永远不等于零。
If the quadratic expression can be factorised into the product of two linear brackets, the solutions are found by setting each bracket to zero. This is the simplest method for solving quadratics with integer roots.
(x + p)(x + q) = 0 → x = –p or x = –q
如果二次表达式可因式分解为两个一次括号的乘积，令每个括号等于零即可求得解。这是求解具有整数根的二次方程最简单的方法。
The quadratic formula provides a universal solution for any quadratic equation, including those that cannot be factorised easily. The discriminant b² – 4ac inside the square root determines the nature of the roots.
x = (–b ± √b² – 4ac) / (2a)
二次公式为任意二次方程提供通用解法，包括那些不易因式分解的方程。根号内的判别式 b²–4ac 决定了根的类型。
4. Coordinate Geometry of Straight Lines | 直线坐标几何
The gradient (slope) of a straight line measures its steepness and direction. It is calculated as the vertical change divided by the horizontal change between any two distinct points on the line.
Gradient m = (y₂ – y₁) / (x₂ – x₁)
直线的梯度（斜率）度量其倾斜程度和方向。它等于线上任意两个不同点之间的垂直变化除以水平变化。
The equation of a straight line in slope-intercept form makes it easy to identify the gradient and the point where the line crosses the y‑axis.
y = mx + c
斜截式直线方程能让你很容易地识别出梯度以及直线与 y 轴的交点。
The distance between two points comes directly from Pythagoras’ theorem. It is the square root of the sum of squared horizontal and vertical differences.
d = √[(x₂ – x₁)² + (y₂ – y₁)²]
两点间的距离直接源于毕达哥拉斯定理。它是水平差与垂直差平方和的平方根。
The midpoint formula averages the x‑coordinates and the y‑coordinates of the two endpoints, giving the point exactly halfway between them.
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
中点公式将两个端点的 x 坐标和 y 坐标分别取平均，得到位于它们正中间的点。
5. Pythagoras’ Theorem | 毕达哥拉斯定理
Pythagoras’ theorem applies only to right‑angled triangles. It relates the lengths of the three sides, stating that the square of the hypotenuse equals the sum of the squares of the other two sides.
a² + b² = c² (where c is the hypotenuse)
毕达哥拉斯定理仅适用于直角三角形。它联系三条边的长度：斜边的平方等于两条直角边的平方和。
Common Pythagorean triples like 3‑4‑5 and 5‑12‑13 are whole‑number sets that satisfy the theorem and appear frequently in problems. Recognising them can save valuable time.
3² + 4² = 5² | 5² + 12² = 13²
常见的毕达哥拉斯三数组，如 3‑4‑5 和 5‑12‑13，是满足定理的整数组合，在题目中频繁出现。识别它们可以节省宝贵时间。
6. Basic Trigonometry | 三角学基础
In a right‑angled triangle, the three trigonometric ratios relate an acute angle to the ratios of two sides. These definitions are the foundation for solving unknown sides and angles.
sin θ = opposite / hypotenuse | cos θ = adjacent / hypotenuse | tan θ = opposite / adjacent
在直角三角形中，三个三角函数比将锐角与两条边的比联系起来。这些定义是求解未知边和角的基础。
Special angles yield exact trigonometric values that you are expected to memorise. They frequently appear in non‑calculator exam questions.
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | √3/3 | 1 | √3 | undefined |
特殊角有精确的三角函数值，需要牢记。它们经常出现在不允许使用计算器的考题中。
The fundamental Pythagorean identity is a direct consequence of the theorem and the definitions of sine and cosine. It is useful for proving other trigonometric relations and checking answers.
sin²θ + cos²θ = 1
基本的毕达哥拉斯恒等式是毕达哥拉斯定理与正弦、余弦定义的直接推论。它在证明其他三角关系及检查答案时很有用。
7. Area, Volume & Surface Area | 面积、体积与表面积
The area of a triangle is half the product of its base and perpendicular height. This formula works for any triangle, provided you use the height measured at a right angle to the chosen base.
Area = ½ × base × height
三角形的面积等于底与垂直高度乘积的一半。只要所选的高与底垂直，此公式适用于任何三角形。
A trapezium (trapezoid) has one pair of parallel sides. Its area is found by averaging the two parallel lengths and multiplying by the distance between them.
Area = ½ (a + b) h (a and b are parallel sides)
梯形有一组对边平行。其面积等于两平行边长度的平均值乘以它们之间的距离。
For a circle, the circumference and area are both expressed in terms of the radius and π.
Circumference = 2πr or πd | Area = πr²
对于圆，周长和面积都用半径和 π 来表示
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