📚 Review Exercise 3: Trigonometry, Vectors and Calculus | 复习练习3：三角、向量与微积分
This revision article covers the core topics grouped in Review Exercise 3 of the Edexcel AS/A Level Pure Mathematics course. It brings together trigonometric ratios, trigonometric equations, vectors, differentiation, integration, exponentials and logarithms. Each section gives the key facts you need, followed by a short worked example or exam-style reminder.
本篇复习文章涵盖 Edexcel AS/A Level 纯数学课程中复习练习 3 的核心主题，包括三角比、三角方程、向量、微分、积分、指数与对数。每一节先给出关键结论，再配以简短例题或考试风格提示。
1. Trigonometric Ratios and Graphs | 三角比与图像
cent. You must know exact values for 30°, 45° and 60°, such as sin 30° = 1/2, cos 45° = 1/√2 and tan 60° = √3.
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。必须牢记 30°、45° 和 60° 的精确值，例如 sin 30° = 1/2，cos 45° = 1/√2，tan 60° = √3。
re periodic. Sine and cosine have period 360° or 2π radians, while tangent has period 180° or π radians. Use the graphs to identify all solutions in a given interval.
y = sin θ、y = cos θ 和 y = tan θ 的图像具有周期性。正弦和余弦的周期为 360° 或 2π 弧度，正切的周期为 180° 或 π 弧度。利用图像可找出给定区间内的所有解。
ine is positive in the first and second quadrants, the solutions are θ = 30° and θ = 150°.
例题提示：在 0° ≤ θ ≤ 360° 内解 sin θ = 0.5。因为正弦在第一、第二象限为正，所以解为 θ = 30° 和 θ = 150°。
Common mistake: the calculator angle mode must match the question. If the question uses radians, set the calculator to radian mode before evaluating trigonometric values.
常见错误：计算器的角度模式必须与题目一致。如果题目使用弧度制，在计算三角函数值之前应先将计算器设为弧度模式。
2. Trigonometric Identities and Equations | 三角恒等式与方程
cos²θ = 1. These allow you to rewrite trigonometric equations in terms of one ratio only.
最常用的两个恒等式是 tan θ = sin θ / cos θ 和 sin²θ + cos²θ = 1。它们能把三角方程改写为只含一个三角比的形式。
s the variable. Factorise first, then solve each linear trigonometric equation separately.
如果出现二次方程，例如 2 cos²θ − cos θ − 1 = 0，可把 cos θ 当作变量来处理。先因式分解，再分别解每个线性三角方程。
Always give answers in the required interval and in the required unit. Do not mix degrees and radians unless a question specifically asks for conversion.
务必按题目要求的区间和单位给出答案。除非题目特别要求换算，否则不要混用角度制和弧度制。
Common mistake: when squaring both sides of a trigonometric equation, you may create extra solutions. Always substitute your answers back into the original equation to check them.
常见错误：对三角方程两边平方时可能会产生增根。必须把答案代回原方程进行检验。
3. Vectors in Two Dimensions | 二维向量
A vector can be written as ai + bj or as a column vector. The magnitude of ai + bj is √a² + b². Direction is usually measured from the positive i-axis.
向量可写成 ai + bj 或列向量形式。ai + bj 的模长为 √a² + b²。方向通常从正 i 轴开始测量。
To add or subtract vectors, combine the i-components and j-components separately. Multiplication by a scalar changes the magnitude and, if the scalar is negative, reverses the direction.
向量加减时，分别合并 i 分量和 j 分量。乘以标量会改变模长；若标量为负，方向会反向。
A unit vector has magnitude 1. For example, if a = 3i + 4j, then |a| = 5, so the unit vector in the direction of a is (3/5)i + (4/5)j.
单位向量的模长为 1。例如，若 a = 3i + 4j，则 |a| = 5，因此沿 a 方向的单位向量为 (3/5)i + (4/5)j。
Common mistake: direction and magnitude are separate pieces of information. Two vectors can have the same direction but different magnitudes, and vice versa.
常见错误：方向与模长是两个不同的信息。两个向量可以方向相同但模长不同，也可以模长相同但方向不同。
4. Differentiation: Rules and Tangents | 微分：法则与切线
xⁿ⁻¹. It works for any real n, including negative and fractional powers, so rewrite roots
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