📚 Edexcel Maths: Trigonometry – Essential Revision Guide | Edexcel 数学：三角函数 核心考点精讲
Trigonometry is a cornerstone of Edexcel Mathematics, weaving through Pure, Mechanics and Statistics. This comprehensive guide distils the essential concepts, from basic right-triangle ratios to advanced compound-angle identities, and is tailored for both IGCSE and A-Level learners. Mastery of these topics will sharpen your problem-solving skills across equations, graphs, identities and real-world modelling.
三角函数是 Edexcel 数学的基石，贯穿纯数、力学和统计。这本综合指南浓缩了从基本直角三角形比到高级复合角恒等式的核心概念，为 IGCSE 和 A-Level 学生量身定制。掌握这些主题将提升你在方程、恒等式、图像以及实际建模中的解题能力。
1. Trigonometric Ratios in Right Triangles | 直角三角形中的三角比
The three primary trigonometric ratios are sine (sin), cosine (cos) and tangent (tan). In a right-angled triangle containing an acute angle θ, sin θ is the ratio of the opposite side to the hypotenuse, cos θ is the ratio of the adjacent side to the hypotenuse, and tan θ is the ratio of the opposite side to the adjacent side.
三个主要的三角比是正弦 (sin)、余弦 (cos) 和正切 (tan)。在含有锐角 θ 的直角三角形中，sin θ 是对边与斜边之比，cos θ 是邻边与斜边之比，tan θ 是对边与邻边之比。
H cos θ = A / H tan θ = O / A tan θ = sin θ ÷ cos θ
These relationships are often memorised with the acronym SOH CAH TOA, where each letter block links a ratio to its two sides. When one side and one acute angle are known, or when two sides are known, the remaining quantities can always be found.
这些关系通常用记忆口诀 SOH CAH TOA 来记住，每个字母组把某个比与其两条边对应起来。只要已知一条边加一个锐角，或者已知两条边，其余的量总能求出。
- Finding a side: choose the ratio containing the known side and the required side, then rearrange. | 求边：选择同时含已知边与所求边的比，再移项求解。
- Finding an angle: use the inverse function sin⁻¹, cos⁻¹ or tan⁻¹. | 求角：使用反函数 sin⁻¹、cos⁻¹ 或 tan⁻¹。
On the unit circle, the coordinates (cos θ, sin θ) define the cosine and sine of the angle measured anticlockwise from the positive x-axis, which extends the definitions to any real angle, not just acute ones.
在单位圆上，坐标 (cos θ, sin θ) 定义了从正 x 轴逆时针起测得的角 θ 的余弦和正弦，从而把定义扩展到任意实角，而不仅限于锐角。
2. Exact Values of Special Angles | 特殊角的精确值
Exact trigonometric values for 0°, 30°, 45°, 60° and 90° must be committed to memory, because they form the basis for many non-calculator exam questions. Rather than rote learning, derive them from two standard triangles: an equilateral triangle of side 2 split into two right triangles, and an isosceles right triangle with legs of length 1.
必须牢记 0°、30°、45°、60° 和 90° 的精确三角函数值，因为它们是许多无计算器考题的基础。与其死记硬背，不如从两个标准三角形推导：边长为 2 的等边三角形对半分开，以及直角边长为 1 的等腰直角三角形。
| θ (deg) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3 / 2 | 1 / √3 |
| 45° | 1 / √2 | 1 / √2 | 1 |
| 60° | √3 / 2 | ½ | √3 |
| 90° | 1 | 0 | undefined / 未定义 |
e the cosine is zero there. Exam questions frequently ask for answers in exact surd form, so a decimal such as 0.866 will usually lose the accuracy mark.
注意其对称性：正弦列恰好与余弦列反向，而 tan θ = sin θ ÷ cos θ 在 90° 处失效，因为那里余弦为零。考题常要求用精确根式作答，所以像 0.866 这样的近似小数通常会丢掉精度分。
3. Radians, Arc Length and Sector Area | 弧度、弧长与扇形面积
s are converted by multiplying by π ÷ 180 and radians by 180 ÷ π. Radian measure is compulsory in calculus, because the derivative of sin x is only cos x when x is measured in radians.
1 弧度是圆弧长度等于半径时，该弧在圆心所张的角。整圈换算关系为 π 弧度 = 180°，因此度转弧度乘以 π ÷ 180，弧度转度乘以 180 ÷ π。弧度制在微积分中是必需的，因为只有当 x 用弧度表示时，sin x 的导数才等于 cos x。
arc length l = r θ sector area A = ½ r² θ (θ in radians)
d when θ is in radians. These are useful in A-Level mechanics for pendulums and small oscillations, and appear in the Edexcel specification as small-angle approximations.
对于小角度，当 θ 用弧度表示时有近似式 sin θ ≈ θ、tan θ ≈ θ 和 cos θ ≈ 1 − θ² ÷ 2。这些式子在 A-Level 力学中处理单摆与小振动时十分有用，也是 Edexcel 考纲中明确列出的小角近似。
- Always check whether the question demands radians or degrees before substituting. | 代入前务必检查题目要求弧度还是角度。
- chord length, combine l = rθ with the cosine rule in the triangle formed by the two radii. | 求弦长时，把 l = rθ 与两条半径所成三角形中的余弦定理结合使用。
4. The Sine and Cosine Rules | 正弦定理与余弦定理
For any triangle, not necessarily right-angled, the sine rule links each side to the sine of its opposite angle, and the cosine rule generalises Pythagoras to include the angle between two known sides.
对于任意三角形（不必是直角三角形），正弦定理把每条边与其对角的正弦联系起来，余弦定理则把勾股定理推广到包含两条已知边夹角的情形。
a ÷ sin A = b ÷ sin B = c ÷ sin C
a² = b² + c² − 2 b c cos A area = ½ a b sin C
Use the sine rule when you have a matching side-and-angle pair; use the cosine rule when you have two sides plus the included angle, or all three sides. The ambiguous case of the sine rule arises when you are given two sides and a non-included angle, and both an acute and an obtuse solution may exist.
当已知一组「边与其对角」的配对时用正弦定理；当已知两边及其夹角，或已知三边时用余弦定理。正弦定理的歧义情形出现在已知两边及其中一边的对角时，此时可能同时存在一个锐角解和一个钝角解。
- Proof idea: drop a perpendicular and compare two expressions for its height. | 证明思路：作一条高，用两种方式表示它的长度并比较。
- Area formula: area = ½ a b sin C requires the included angle, not any angle. | 面积公式 area = ½ a b sin C 必须使用夹角，而不是任意角。
5. Core Identities: Pythagorean, Quotient and Reciprocal | 核心恒等式：平方、商数与倒数关系
The Pythagorean identity is the single most useful tool in trigonometry, and it is simply Pythagoras' theorem applied to the unit circle. Dividing it by cos²θ or sin²θ generates the two companion identities involving secant and cosecant.
平方关系恒等式是三角函数中最有用的工具，它本质上就是勾股定理应用于单位圆的结果。把它分别除以 cos²θ 或 sin²θ，便得到两个涉及正割与余割的伴随恒等式。
sin²θ + cos²θ = 1 tan θ = sin θ ÷ cos θ
1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ
Be careful not to confuse the reciprocal notation with the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹, which mean something entirely different.
倒数三角比为 sec θ = 1 ÷ cos θ、cosec θ = 1 ÷ sin θ 和 cot θ = 1 ÷ tan θ。注意不要把倒数记号与反函数 sin⁻¹、cos⁻¹、tan⁻¹ 混淆，它们表示完全不同的含义。
- To prove an identity, work on one side only until it matches the other. | 证明恒等式时，只从一边入手变形，直到与另一边相同。
- Convert everything to sin and cos when you are stuck. | 卡住时，把所有式子都化成 sin 和 cos。
6. Compound, Double and Half Angle Formulae | 复合角、倍角与半角公式
The compound-angle formulae express the sine, cosine and tangent of a sum or difference. They are given in the Edexcel formula booklet, but you must know how to apply them fluently in both directions.
复合角公式给出两角之和或差的正弦、余弦与正切。Edexcel 会在公式册中提供这些公式，但你必须能在两个方向上熟练运用。
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)
the double-angle formulae, from which the half-angle and the power-reducing forms follow. The three equivalent versions of cos 2θ are especially important, because each one is convenient in a different context.
令 A = B = θ 便立即得到倍角公式，再由它推出半角公式与降幂形式。cos 2θ 的三个等价形式尤为重要，因为它们在各自不同的情境中最为方便。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1
tan 2θ = 2 tan θ ÷ (1 − tan²θ) sin²θ = (1 − cos 2θ) ÷ 2
The last identity is the standard route for integrating sin²θ, since it replaces a squared term with a linear combination of cos 2θ that can be integrated directly.
最后一个恒等式是积分 sin²θ 的标准路径，因为它把平方项换成 cos 2θ 的线性组合，可以直接积分。
7. The Harmonic Form R sin(θ ± α) | 辅助角形式 R sin(θ ± α)
Any expression of the form a sin θ + b cos θ can be written as a single sine or cosine function, which makes finding maxima, minima and solutions far easier.
任何形如 a sin θ + b cos θ 的表达式都可以写成单一的正弦或余弦函数，这使求最大值、最小值以及解方程都容易得多。
a sin θ + b cos θ = R sin(θ + α), R = √a² + b², tan α = b ÷ a
Once written in harmonic form, the maximum value is R and the minimum is −R, occurring where the bracket equals 90° and 270° respectively. This is a very common exam question, and examiners usually award marks separately for R, for α and for the final statement.
写成辅助角形式后，最大值为 R，最小值为 −R，分别对应括号内角等于 90° 与 270° 的情形。这是极常见的考题，评分时通常对 R、对 α 以及对最终结论分别给分。
- Match the sign pattern carefully: for R cos(θ − α) the coefficients swap roles. | 要仔细匹配符号模式：写成 R cos(θ − α) 时，两个系数的角色互换。
- Give α in the correct quadrant, not just the principal value from the calculator. | α 必须落在正确的象限，而不是只写计算器给出的主值。
8. Graphs of Trigonometric Functions | 三角函数的图像
−1 ≤ y ≤ 1, while tan θ has period 180° (or π) and vertical asymptotes wherever cos θ = 0.
sin θ 与 cos θ 的图像是周期为 360°（或 2π）、振幅为 1、值域为 −1 ≤ y ≤ 1 的波形；而 tan θ 的周期为 180°（或 π），并在 cos θ = 0 处有竖直渐近线。
| Function / 函数 | Period / 周期 | Range / 值域 |
|---|---|---|
| y = sin θ | 360° or 2π | −1 ≤ y ≤ 1 |
| y = cos θ | 360° or 2π | −1 ≤ y ≤ 1 |
| y = tan θ | 180° or π | all real values / 全体实数 |
Transformations follow the standard rules. Writing y = a sin(bθ + c) + d gives amplitude |a| rtical shift of d. Sketching these quickly and accurately is worth easy marks in the exam.
图像变换遵循标准规则。写成 y = a sin(bθ + c) + d 时，振幅为 |a|，周期为 360° ÷ b（或 2π ÷ b），水平平移 −c ÷ b，竖直平移 d。快速准确地画出草图能轻松拿到分数。
- Mark the key points: intercepts, maxima, minima and asymptotes. | 标出关键点：截距、极大值、极小值与渐近线。
- Use radians on the axis labels whenever the question is in radians. | 题目用弧度时，坐标轴标注也应使用弧度。
9. Solving Trigonometric Equations | 解三角方程
lution comes from the calculator, and the remaining ones come from symmetry within the given interval.
大多数三角方程有无穷多个解，因此题目通常限定区间，例如 0° ≤ θ < 360°。第一个解来自计算器，其余的解则由给定区间内的对称性得到。
The CAST diagram, or equivalently a sketch of the graph, tells you where each ratio is positive. Both sin θ and cos θ have two solutions per full turn, while tan θ has one solution per 180°, so its pattern repeats twice as often.
CAST 图（或等价地画出函数图像）告诉你每个三角比在哪些象限为正。sin θ 和 cos θ 在整圈内各有两解，而 tan θ 每 180° 有一解，所以它的重复频率是前两者的两倍。
sin θ = k → θ = sin⁻¹k or 180° − sin⁻¹k
= k → θ = ±cos⁻¹k tan θ = k → θ = tan⁻¹k + 180°n
e the interval rather than the equation: if 0° ≤ θ < 360°, then 0° ≤ 2θ < 720°, which produces four solutions instead of two.
对于关于 sin θ 的二次方程，先因式分解，再把每个因式当作独立的线性方程处理。对于 sin 2θ = ½ 这类方程，应变换区间而不是变换方程：若 0° ≤ θ < 360°，则 0° ≤ 2θ < 720°，于是得到四个解而非两个。
- Never divide by sin θ or cos θ; factorise instead so you keep every solution. | 绝不要两边同时除以 sin θ 或 cos θ；应因式分解，以免丢解。
- Check each answer against the stated interval. | 逐一核对答案是否落在给定区间内。
10. Inverse Trigonometric Functions | 反三角函数
The inverse functions sin⁻¹, cos⁻¹ and tan⁻¹ recover an angle from a ratio. Because the original functions are periodic they are not one-to-one, so each inverse is defined only on a restricted domain, called the principal range.
反函数 sin⁻¹、cos⁻¹ 和 tan⁻¹ 由三角比反求角度。由于原函数是周期函数、并非一一对应，因此每个反函数都只在受限的定义域（称为主值范围）上定义。
| Function / 函数 | Domain / 定义域 | Principal range / 主值范围 |
|---|---|---|
| sin⁻¹x | −1 ≤ x ≤ 1 | −90° ≤ y ≤ 90° |
| cos⁻¹x | −1 ≤ x ≤ 1 | 0° ≤ y ≤ 180° |
| tan⁻¹x | all real x / 全体实数 | −90° < y < 90° |
over the principal interval. Remember that the output of an inverse function is always an angle, so it must be given in the correct unit.
典型的 A-Level 题要求画出 y = sin⁻¹x 的图像，它其实就是 y = sin x 在主区间内关于直线 y = x 的镜像。请记住反函数的输出永远是角度，因此必须使用正确的角度单位。
11. Differentiation and Integration of Trigonometric Functions | 三角函数的微积分
In radians, the derivatives of sine and cosine follow a neat cycle, and every integration result is simply the reverse of that cycle. Getting the signs right is where most marks are lost.
在弧度制下，正弦与余弦的导数构成一个简洁的循环，而每个积分结果只是该循环的逆过程。符号出错正是丢分最多的地方。
sin x) ÷ dx = cos x d(cos x) ÷ dx = −sin x d(tan x) ÷ dx = sec²x
x dx = −cos x + c ∫ cos x dx = sin x + c ∫ sec²x dx = tan x + c
With the chain rule, the same results extend to linear arguments: the derivative of sin(ax + b) is a cos(ax + b), and the integral of sin(ax + b) is −cos(ax + b) ÷ a. For
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