📚 IB CCEA Maths: Trigonometry - Key Exam Points | IB CCEA 数学：三角函数考点精讲
Trigonometry is a cornerstone of the IB CCEA Mathematics curriculum, encompassing radian measure, trigonometric graphs, identities, equation-solving, and calculus applications. Mastery of these topics is essential for success in both the Pure and Applied components, where a single lost mark often comes from a sign slip or a forgotten general solution rather than from a genuine gap in understanding. This guide consolidates the key concepts, common pitfalls, and exam techniques you need for high-scoring answers.
三角函数是 IB CCEA 数学课程的核心支柱，涵盖弧度度量、三角函数图像、恒等式、方程求解以及微积分应用。牢固掌握这些内容对纯数与数学应用题部分都至关重要——很多时候丢分并非因为知识漏洞，而是符号写错或忘记通解形式。本指南全面总结关键概念、常见易错点与应试技巧，助你在考试中脱颖而出。
1. Radian Measure and Degree Conversion | 弧度制与角度弧度转换
Radian measure is defined so that an angle subtended by an arc of length r on a circle of radius r is 1 radian. The full revolution of 360° equals 2π radians. Using radians is mandatory in calculus and simplifies many formulae.
弧度制的定义为：在半径为 r 的圆上，长度为 r 的弧所对的圆心角是 1 弧度。一整圈 360° 等同于 2π 弧度。在微积分中必须使用弧度制，它也能简化许多公式。
π rad = 180°
exact form (e.g. π/3) unless a decimal approximation is requested. Common conversions worth memorising are 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2 and 120° = 2π/3.
将角度转换为弧度，乘以 π/180；将弧度转换为角度，乘以 180/π。除非题目要求近似值，否则弧度应保留精确形式（如 π/3）。值得熟记的常见换算包括 30° = π/6、45° = π/4、60° = π/3、90° = π/2、120° = 2π/3。
Arc length is given by s = rθ and sector area by A = ½ r²θ, where θ must be in radians. The area of the corresponding segment is found by subtracting the triangle area ½ r² sin θ from the sector area.
弧长公式为 s = rθ，扇形面积公式为 A = ½ r²θ，其中 θ 必须以弧度为单位。对应的弓形面积等于扇形面积减去三角形面积 ½ r² sin θ。
2. The Unit Circle and Trigonometric Ratios | 单位圆与三角函数比
nates of a point on the circle are (cos θ, sin θ). Consequently, tan θ = sin θ / cos θ = y / x, provided cos θ ≠ 0.
单位圆（半径为 1，圆心在原点）为任意从正 x 轴逆时针量起的角度 θ 定义了正弦和余弦。圆周上点的坐标为 (cos θ, sin θ)。因此，tan θ = sin θ / cos θ = y / x，前提是 cos θ ≠ 0。
The signs of the three ratios in each quadrant follow the CAST (or ASTC) pattern: All positive in the first quadrant, Sine positive in the second, Tangent positive in the third, and Cosine positive in the fourth. Writing the quadrant diagram at the top of your answer sheet takes ten seconds and prevents a large number of sign errors.
三个三角比在各象限的正负号遵循 CAST（或 ASTC）规律：第一象限全为正，第二象限正弦为正，第三象限正切为正，第四象限余弦为正。在答题纸顶部快速画出象限图只需十秒，却能避免大量符号错误。
(−θ) = cos θ. In radians the same results read sin(π − θ) = sin θ and cos(π − θ) = −cos θ.
对称性由单位圆直接导出：sin(180° − θ) = sin θ，cos(180° − θ) = −cos θ，sin(−θ) = −sin θ，cos(−θ) = cos θ。用弧度表示即 sin(π − θ) = sin θ，cos(π − θ) = −cos θ。
3. Exact Values for Special Angles | 特殊角的精确值
Exact values for the special angles must be instant recall: 0°, 30°, 45°, 60°, 90° and their radian equivalents. Examiners frequently require an exact surd answer, so a decimal from the calculator will lose marks even when it is numerically correct.
特殊角的精确值必须做到条件反射式记忆：0°、30°、45°、60°、90° 及其弧度值。考官经常要求精确根式答案，因此即使用计算器算出的十进制数值正确，也会失分。
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Derive these values from the two standard right triangles rather than memorising them blindly: the 45° triangle with legs 1, 1 and hypotenuse √2, and the 30-60-90 triangle which is half of an equilateral triangle. Once you can draw them in five seconds, the table becomes self-checking.
建议从两个标准直角三角形推导这些值，而不是盲目死记：边长为 1、1、斜边 √2 的 45° 三角形，以及等边三角形的一半——30-60-90 三角形。只要能五秒内画出它们，整张表就可以自我校验。
4. Graphs of Trigonometric Functions and Transformations | 三角函数图像与变换
e period 2π and range [−1, 1]; tangent has period π with vertical asymptotes at x = π/2 + nπ and an unbounded range.
必须能画出给定区间上 y = sin x、y = cos x、y = tan x 的草图，标注截距、极大值、极小值和渐近线。正弦与余弦的周期为 2π，值域为 [−1, 1]；正切的周期为 π，在 x = π/2 + nπ 处有垂直渐近线，值域无界。
For the general form y = a sin(bx + c) + d, the amplitude is |a|, the period is 2π/|b| sually split between these four parameters, so state each one explicitly before sketching.
对于一般形式 y = a sin(bx + c) + d，振幅为 |a|，周期为 2π/|b|，相位平移为 −c/b，垂直平移为 d。余弦函数规则完全相同。在 IB 或 CCEA 题目中，分数通常按这四个参数分配，因此作草图前应逐一明确写出。
, so a cosine curve is simply a sine curve shifted left by π/2. This single fact explains most of the apparent differences between the two graphs and is a useful shortcut when matching a graph to an equation in a multiple-choice section.
另外要牢记 cos x = sin(x + π/2)，即余弦曲线就是正弦曲线向左平移 π/2。这一事实解释了两者大部分的表面差异，在选择题中根据图像匹配方程时是极好的捷径。
5. Fundamental Identities | 基本恒等式
The Pythagorean identity sin²θ + cos²θ ≡ 1 is the most heavily used identity in the whole syllabus. Dividing it by cos²θ gives 1 + tan²θ ≡ sec²θ, and dividing by sin²θ gives 1 + cot²θ ≡ cosec²θ. All three are needed for proof questions and for converting an equation into a solvable form.
勾股恒等式 sin²θ + cos²θ ≡ 1 是整个大纲中使用频率最高的恒等式。两边同除以 cos²θ 得 1 + tan²θ ≡ sec²θ，同除以 sin²θ 得 1 + cot²θ ≡ cosec²θ。这三个恒等式在证明题以及把方程化为可解形式时都必不可少。
Alongside them you need the quotient identities tan θ ≡ sin θ / cos θ and cot θ ≡ ec θ is not the same as sin⁻¹θ, and that sec θ is not cos⁻¹θ: this notation confusion is a classic source of lost marks.
与它们配套的还有商数恒等式 tan θ ≡ sin θ / cos θ 与 cot θ ≡ cos θ / sin θ，以及倒数定义 sec θ = 1/cos θ、cosec θ = 1/sin θ、cot θ = 1/tan θ。切记 cosec θ 不等于 sin⁻¹θ，sec θ 也不等于 cos⁻¹θ：这种记号混淆是典型的失分点。
When proving an identity, work from the more complicated side towards the simpler one, convert everything into sine and cosine, and never move terms across the equals sign. Start from the LHS, manipulate, and finish the line with the LHS expression equal to the RHS expression you were given.
证明恒等式时，应从较复杂的一侧向较简单的一侧化简，把所有函数都换成正弦与余弦，并且绝不要把项跨等号移动。应从左边开始变形，最后一行写出左边表达式等于题目给出的右边表达式。
6. Compound Angle and Double Angle Formulae | 复合角与二倍角公式
The compound angle formulae are sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. Note carefully that the cosine expansion reverses the sign, which is the single most common slip in this topic.
复合角公式为 sin(A ± B) = sin A cos B ± cos A sin B 与 cos(A ± B) = cos A cos B ∓ sin A sin B。特别注意余弦展开式中符号会相反，这是本主题最常见的失误。
For the tangent, tan(A ± B) = (tan A ± tan B) / (1 ∓ d cos 2A = cos²A − sin²A = 1 − 2 sin²A = 2 cos²A − 1, with tan 2A = 2 tan A / (1 − tan²A).
正切方面，tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。在复合角公式中令 A = B 即得二倍角结果：sin 2A = 2 sin A cos A，cos 2A = cos²A − sin²A = 1 − 2 sin²A = 2 cos²A − 1，以及 tan 2A = 2 tan A / (1 − tan²A)。
Choosing the right version of cos 2A is a skill in itself. If the rest of the expression contains only sines, use 1 − 2 sin²A; if it contains only cosines, use 2 cos²A − 1. This simple decision often turns an unmanageable equation into a quadratic in a single trigonometric function.
选择 cos 2A 的哪一种形式本身就是一项技巧：若式中其余部分只含正弦，用 1 − 2 sin²A；若只含余弦，用 2 cos²A − 1。这一简单判断常常能把无从下手的方程变成单个三角函数的二次方程。
7. Solving Trigonometric Equations | 三角函数方程求解
ion of the form sin θ = k, the principal value is θ = arcsin k and the general solution is θ = nπ + (−1)ⁿ arcsin k. For cos θ = k we have θ = 2nπ ± arccos k, and for tan θ = k, θ = nπ + arctan k, where n is any integer.
对于 sin θ = k 形式的方程，主值为 θ = arcsin k，通解为 θ = nπ + (−1)ⁿ arcsin k。对于 cos θ = k，θ = 2nπ ± arccos k；对于 tan θ = k，θ = nπ + arctan k，其中 n 为任意整数。
award marks for every root inside it. Work out the number of solutions first from the period: a factor of 3x triples the number of roots. Missing a solution is the fastest way to lose a method mark even when the algebra is flawless.
实际考试中多数题目会指定区间，例如 0 ≤ θ < 2π，并对区间内每一个根给分。先根据周期判断解的个数：若自变量是 3x，则根的个数变为三倍。即使代数运算完全正确，漏解也是丢分最快的方式。
Equations that mix functions should be reduced using the identities from Section 5, producing either a quadratic in sin θ or cos θ or an equation in a single tangent. Where an expression of the form a sin θ + b cos θ appears, rewrite it as R sin(θ + α) with R = √a² + b² and tan α = b/a, paying attention to the correct quadrant for α.
若方程中混有不同函数，应利用第 5 节的恒等式化为关于 sin θ 或 cos θ 的二次方程，或化为只含正切的方程。出现 a sin θ + b cos θ 形式时，改写为 R sin(θ + α)，其中 R = √a² + b²、tan α = b/a，并注意 α 所在的正确象限。
8. Solving Triangles: Sine Rule, Cosine Rule and Area | 解三角形：正弦与余弦定理及面积
For any triangle, the sine rule states a / sin A = b / sin B = c / sin C and the cosine rule states a² = b² + c² − 2bc cos A. Use the sine rule when you have a matching side-and-angle pair; use the cosine rule when you have three sides or two sides and the included angle.
对任意三角形，正弦定理为 a / sin A = b / sin B = c / sin C，余弦定理为 a² = b² + c² − 2bc cos A。已知一组边角对应关系时用正弦定理；已知三边或两边及夹角时用余弦定理。
The area of a triangle is ½ ab sin C = k has two solutions in the range 0° to 180°, and only one of them may fit the given triangle.
三角形面积为 ½ ab sin C，这一公式也是第 1 节弓形面积计算的基础。用正弦定理求角时要注意歧义情形：sin θ = k 在 0° 到 180° 内有两个解，而其中可能只有一个符合题设三角形。
Bearings, elevation and depression problems are simply triangle problems in disguise. Sketch the situation, mark all known angles in degrees, and only convert to radians if the question explicitly demands it. Keeping a unit label on every angle prevents the most damaging unit mix-ups.
方位角、仰角与俯角问题本质上就是解三角形问题。先画示意图，把所有已知角用角度标出，只有题目明确要求时才转换为弧度。给每个角标注单位，可避免最严重的单位混用错误。
9. Differentiation and Integration of Trigonometric Functions | 三角函数的微分与积分
In radians only, d/dx (sin x) = cos x and d/dx (cos x) = −sin x. The minus sign matters. For tangent, d/dx (tan x) = sec²x = 1 / cos²x. If the argument is linear, apply the chain rule: d/dx sin(ax + b) = a cos(ax + b).
只有在弧度制下才有 d/dx (sin x) = cos x 与 d/dx (cos x) = −sin x，那个负号极为关键。正切的导数为 d/dx (tan x) = sec²x = 1 / cos²x。若自变量为线性形式，则使用链式法则：d/dx sin(ax + b) = a cos(ax + b)。
Integration reverses the process: ∫ sin x dx = −cos x + C and ∫ cos x dx = sin x + C, with ∫ sec²x dx = tan x + C + b) dx = (1/a) sin(ax + b) + C. Forgetting the constant C is a routine loss of one mark.
积分是其逆过程：∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C，以及 ∫ sec²x dx = tan x + C。若自变量为线性形式，则除以系数：∫ cos(ax + b) dx = (1/a) sin(ax + b) + C。忘记常数 C 是常见的白丢一分。
Typical examination tasks include finding stationary points of a trigonometric curve, determining whether they are maxima or minima via the second derivative, and evaluating definite integrals to obtain an area. Always check whether the required region lies above or below the x-axis before writing down the integral.
典型考题包括求三角函数曲线的驻点、用二阶导数判断极大或极小，以及计算定积分求面积。写积分式之前，务必判断所围区域位于 x 轴上方还是下方。
10. Inverse Trigonometric Functions | 反三角函数
functions are defined on restricted domains so that they remain one-to-one. For y = arcsin
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