📚 IB & Edexcel Mathematics: Trigonometry Key Points | IB & Edexcel 数学：三角函数考点精讲
Trigonometry forms a cornerstone of both the IB and Edexcel Mathematics syllabi. It bridges geometric intuition with algebraic manipulation, covering angle measurement, circular functions, identities, equation-solving, and real-world applications. A strong command of trigonometric principles is essential for success in examinations and for further study in calculus, vectors and physics.
三角函数是 IB 与 Edexcel 数学大纲的核心基石，它将几何直观与代数操作联系起来，涵盖角度测量、圆函数、恒等变换、方程求解及实际应用。牢固掌握三角学原理，不仅对考试成功至关重要，也是后续学习微积分、向量与物理的必要前提。
1. Why Trigonometry Matters: Syllabus Map | 三角学为何重要：大纲定位
Trigonometry is unusual because it appears in almost every strand of both specifications. In IB, it is a core topic in Analysis and Approaches (AA) and in Applications and Interpretation (AI), and it reappears inside complex numbers, vectors and calculus. In Edexcel, it dominates Pure Mathematics 1 and 2 at AS level and is extended in Pure Mathematics 3 and 4, and again in Further Pure Mathematics through inverse functions, secant, cosecant and cotangent.
三角学的特殊之处在于它几乎出现在两份大纲的每一条主线中。在 IB 中，它是 Analysis and Approaches（AA）与 Applications and Interpretation（AI）的核心专题，并会再次出现在复数、向量与微积分里。在 Edexcel 中，它主导 AS 阶段的 Pure Mathematics 1 与 2，并在 Pure Mathematics 3 与 4 中延伸，还会在 Further Pure Mathematics 中通过反函数、正割、余割与余切再次出现。
Typical weighting and appearance:
常见的考查比重与出现方式：
- IB AA HL: Paper 1 and Paper 2 questions on identities, equations and radians; Paper 3 investigations.
- IB AI SL/HL: modelling periodic behaviour, sine rule and cosine rule in real contexts.
- Edexcel AS: exact values, graphs, simple equations, sine and cosine rules.
- Edexcel A2: reciprocal functions, compound and double angles, R sin(θ+ α) style formulations, integration by substitution.
- IB AA HL：Paper 1 与 Paper 2 的恒等式、方程与弧度题，以及 Paper 3 探究题。
- IB AI SL/HL：周期现象的建模，真实情境中的正弦定理与余弦定理。
- Edexcel AS：特殊角精确值、图像、简单方程、正弦与余弦定理。
- Edexcel A2：倒数函数、和角与二倍角、辅助角形式，以及换元积分。
The topic rewards precision. Most marks are lost not through conceptual failure but through incorrect mode on the calculator, incomplete solution sets, or dropped units.
这一专题极其考验严谨性。失分大多并非源于概念不懂，而是计算器模式错误、解集不完整或单位遗漏。
2. Radian Measure & Arc Length | 弧度制与弧长
Radian measure is the standard unit of angular measurement in advanced mathematics. One radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. Because the circumference of a circle is 2πr, a full turn is 2π radians.
弧度制是高等数学中角度的标准计量单位。一弧度定义为：当圆弧长度等于半径时，该弧所对的圆心角的大小。由于圆周长为 2πr，因此一整圈为 2π 弧度。
The essential conversion is:
基本换算关系为：
π rad = 180° so 1 rad = 180/π ° ≈ 57.3°
Arc length is then simply the product of radius and angle, provided the angle is measured in radians:
当角度以弧度表示时，弧长就是半径与圆心角的乘积：
l = r θ
All advanced work, especially calculus with trigonometric functions, assumes angles are in radians unless specified otherwise. The derivative of sin x is cos x only in radians; in degrees an unwanted factor of π/180 appears. Always check that your calculator is set to radians before attempting any question involving π, arc length or calculus.
所有高等数学内容，尤其是含三角函数的微积分，除非另有说明，一律默认角度以弧度计。只有当 x 以弧度计时，sin x 的导数才是 cos x；若用角度制，会多出 π/180 这个因子。在做任何涉及 π、弧长或微积分的题目之前，务必确认计算器处于弧度模式。
3. Sector Area & Angular Speed | 扇形面积与角速度
Once radian measure is accepted, the area of a sector follows from the same proportionality that gives arc length. Since a full circle of area πr² corresponds to 2π radians, a sector of angle θ has area:
接受弧度制之后，扇形面积可由与弧长相同的比例关系推出。由于面积为 πr² 的整圆对应 2π 弧度，圆心角为 θ 的扇形面积为：
A = ½ r² θ = ½ r l
the perimeter of a sector is l + 2r, not simply l, because the two radii must be included.
这两个形式可以互换使用，考官常给出只含弧长的条件，迫使你使用 A = ½ r l。注意扇形周长是 l + 2r 而非仅 l，因为必须加上两条半径。
Angular speed connects radians to kinematics. If a point rotates through angle θ in time t, then:
角速度把弧度与运动学联系起来。若某点在一段时间 t 内转过角 θ，则：
ω = θ / t (rad s⁻¹) and v = r ω
ans by multiplying by 2π, convert minutes to seconds by dividing by 60, then apply v = rω.
典型题目会给出车轮每分钟转数，要求求出边缘某点的线速度。先把转数乘以 2π 化为弧度，再把分钟除以 60 化为秒，最后代入 v = rω。
4. The Unit Circle & Exact Values | 单位圆与特殊角精确值
The unit circle is the conceptual engine behind every trigonometric identity. For a point on the circle of radius 1 at angle θ measured anticlockwise from the positive x-axis, the coordinates are (cos θ, sin θ). This immediately gives the fundamental Pythagorean identity and the definition of tangent.
单位圆是一切三角恒等式背后的概念引擎。对于半径为 1 的圆上、从 x 轴正方向逆时针量得角 θ 的一点，其坐标为 (cos θ, sin θ)。由此立刻得到基本的平方恒等式与正切的定义。
sin²θ + cos²θ = 1 tan θ = sin θ / cos θ
Because the circle is symmetric, signs of the functions repeat in a fixed pattern usually memorised as CAST or ASTC: all positive in the first quadrant, sine only in the second, tangent only in the third, cosine only in the fourth.
由于圆具有对称性，各函数的符号按固定规律循环，通常记为 CAST 或 ASTC：第一象限全为正，第二象限只有正弦为正，第三象限只有正切为正，第四象限只有余弦为正。
Exact values for the standard angles must be recalled instantly:
标准角的精确值必须做到立即反应：
| θ | 0 | π/6 | π/4 | π/3 | π/2 |
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
5. Graphs, Periodicity & Symmetry | 图像、周期性与对称性
from memory over at least two periods, marking intercepts, maxima, minima and asymptotes.
画图与读图是每年必考内容。你应当能在至少两个周期内凭记忆画出 y = sin x、y = cos x 与 y = tan x，并标出截距、极大值、极小值与渐近线。
- y = sin x: period 2π, amplitude 1, odd function, passes through (0, 0).
- y = cos x: period 2π, amplitude 1, even function, maximum at (0, 1).
- eriod π, no amplitude, asymptotes at x = π/2 + nπ, strictly increasing between asymptotes.
- y = sin x：周期 2π，振幅 1，奇函数，过点 (0, 0)。
- y = cos x：周期 2π，振幅 1，偶函数，在 (0, 1) 处取极大值。
- y = tan x：周期 π，无振幅，渐近线位于 x = π/2 + nπ，在各渐近线之间严格递增。
Symmetry properties follow directly from the graphs and are used constantly when finding second solutions:
对称性质直接来自图像，在求第二解时被频繁使用：
sin(π − θ) = sin θ cos(−θ) = cos θ tan(π + θ) = tan θ
= sin θ, which explains why the sine and cosine graphs are the same curve shifted by π/2.
一组有用的补充关系是 sin(π/2 − θ) = cos θ 与 cos(π/2 − θ) = sin θ，这解释了为何正弦与余弦图像本质上是同一条曲线平移了 π/2。
6. Transformations of Trigonometric Graphs | 三角函数图像的变换
The general sinusoidal form links graph work to modelling, and both IB and Edexcel expect fluency in reading the parameters directly from a sketch or an equation.
一般形式的正弦函数把图像与建模联系起来，IB 与 Edexcel 都要求能够从图像或方程中直接读出各参数。
y = a sin(b(x − c)) + d
- |a| is the amplitude, and a negative value reflects the graph in the x-axis.
- b determines the period: period = 2π / b for sin and cos, π / b for tan.
- c is the horizontal translation (phase shift), in radians.
- d is the vertical translation, so the principal axis is y = d.
- |a| 为振幅，a 为负时图像关于 x 轴翻转。
- b 决定周期：sin 与 cos 的周期为 2π / b，tan 的周期为 π / b。
- c 为水平平移（相位），单位为弧度。
- d 为竖直平移，因此主轴线为 y = d。
For modelling questions, identify the maximum and minimum values first. Their average gives d, half their difference gives |a|, and the horizontal distance between successive maxima gives the period, hence b.
在建模题中，先找出最大值与最小值。二者的平均值给出 d，差值的一半给出 |a|，相邻两个最大值之间的水平距离给出周期，从而求出 b。
7. Fundamental & Compound Angle Identities | 基本恒等式与和角公式
Identity work is the backbone of the subject. Beyond the Pythagorean identity and the definition of tangent, you should manipulate algebraic fractions of trigonometric expressions confidently, and be ready for the reciprocal functions where they appear in Further or Pure 3 content.
恒等变形是这一学科的骨架。除平方恒等式与正切定义之外，你还需要熟练处理三角表达式构成的代数分式，并为进阶内容中出现的倒数函数做好准备。
1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ
The compound angle formulae are printed in both formula booklets, but you must know when to reach for them:
和角公式在两份公式表中都会给出，但你必须清楚何时使用它们：
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)
Classic uses include simplifying expressions such as sin 75° as sin(45° + 30°), proving identities, and solving equations that mix two different angle multiples.
经典用法包括：把 sin 75° 写成 sin(45° + 30°) 进行化简、证明恒等式，以及求解同时含有两个不同角倍数项的方程。
8. Double Angle & Auxiliary Angle Forms | 二倍角与辅助角形式
A = B, but they appear so often that they must be memorised in all their rearranged forms.
二倍角公式是令 A = B 时和角公式的特例，但由于出现频率极高，其各种变形都必须牢记。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan 2θ = 2 tan θ / (1 − tan²θ)
The three forms of cos 2θ are not interchangeable decorations: choosing 2cos²θ − 1 or 1 − 2sin²θ is what converts an equation into a quadratic in a single function. When you see sin²x and cos 2x in the same equation, replace sin²x using 1 − 2sin²x logic in reverse.
cos 2θ 的三种形式并非可以随意替换的装饰：选择 2cos²θ − 1 或 1 − 2sin²θ，正是把方程化为关于单一函数的二次方程的关键。当方程中同时出现 sin²x 与 cos 2x 时，应反向利用 1 − 2sin²θ 的关系替换 sin²x。
The auxiliary angle technique converts a sum of sine and cosine into a single shifted sine wave:
辅助角技巧把正弦与余弦之和转化为一个单一的平移正弦波：
a sin θ + b cos θ = R sin(θ + α), R = √a² + b², tan α = b / a
ximum and minimum values of expressions such as 3 sin θ + 4 cos θ, whose maximum is R = 5.
这是求 3 sin θ + 4 cos θ 之类表达式最值问题的标准方法，其最大值为 R = 5。
9. Solving Trigonometric Equations | 三角方程的求解
Almost every paper contains at least one equation requiring you to find all solutions in a given interval. The examiner is testing whether you understand the infinite family of solutions, not merely whether you can press the inverse function key.
几乎每份试卷都至少有一道要求在给定区间内求出全部解的方程题。考官考查的是你是否理解解的无限族，而不仅仅是你是否会按反函数键。
sin θ = k → θ = arcsin k + 2nπ or θ = π − arcsin k + 2nπ
cos θ = k → θ = ± arccos k + 2nπ
tan θ = k → θ = arctan k + nπ
A reliable routine avoids almost all errors:
一套可靠的流程可以避免几乎所有错误：
- nto the form sin θ = k, cos θ = k or tan θ = k, or factorise into a product equal to zero.
- argument is 2x or x + π/3, temporarily set u = argument and find the full interval for u.
- Use the principal value from the calculator plus symmetry or the general solution to list every value inside the interval.
- Convert back to x by dividing or subtracting, and check each answer lies in the stated range.
- 整理成 sin θ = k、cos θ = k 或 tan θ = k 的形式，或分解为乘积等于零。
- 若自变量为 2x 或 x + π/3，可先设 u 为该自变量，并求出 u 的完整区间。
- 用计算器给出主值，再借助对称性或通解列出区间内的全部解。
- 通过除法或减法换回 x，并逐一核对答案是否落在题目给定范围内。
sin θ, and for those where dividing both sides by cos θ destroys solutions when cos θ = 0.
要警惕以 sin θ 为变量的二次方程，以及在两侧同时除以 cos θ 时会因 cos θ = 0 而丢失解的情况。
10. Sine Rule, Cosine Rule & Triangle Area | 正弦定理、余弦定理与三角形面积
Non-right-angled triangles are handled by two rules and one area formula, all of which appear on every formula sheet.
非直角三角形由两条定理与一个面积公式处理，它们都会出现在公式表上。
a / sin A = b / sin B = c / sin C
a² = b² + c² − 2bc cos A
Area = ½ a b sin C
Choose the sine rule when you are matching a side with its opposite angle; choose the cosine rule when you have three sides, or two sides and the included angle. The cosine rule is essentially the Pythagorean theorem with a correction term that accounts for the angle not being 90°.
当边与其对角配对出现时选用正弦定理；当已知三边，或已知两边及其夹角时选用余弦定理。余弦定理本质上是勾股定理加上一个修正项，用来补偿夹角不是 90° 的情况。
a non-included angle, the equation sin B = b sin A / a can produce two valid values for B,
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图