IB WJEC Mathematics: Trigonometric Functions Key Points | IB WJEC 数学：三角函数考点精讲

📚 IB WJEC Mathematics: Trigonometric Functions Key Points | IB WJEC 数学：三角函数考点精讲

Trigonometric functions form a cornerstone of both the IB Diploma programme (Analysis & Approaches and Applications & Interpretation) and the WJEC A-level Mathematics specification. This article carefully distills the essential concepts, identities, graph transformations and problem-solving strategies that appear repeatedly in examinations. By mastering radian measure, the unit circle, key identities and equation-solving techniques, you will build a solid foundation for tackling everything from pure trigonometry questions to calculus and vectors later in the course.

三角函数是IB文凭课程（分析与方法、应用与解释）以及WJEC A-level数学的共同基石。本文系统梳理了考试中反复出现的核心概念、恒等式、图像变换以及解题策略。通过掌握弧度制、单位圆、关键恒等式和方程求解技巧，你将建立起牢固的基础，以便后续攻克纯三角问题，乃至微积分与向量等更高阶的内容。

1. Radian Measure and Arc Length | 弧度制与弧长

In both IB and WJEC, the radian is the natural unit for measuring angles. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. The conversion is 180° = π rad. The arc length s for a sector of radius r and angle θ rad is given by s = rθ, and the area of the sector is A = ½ r²θ. These formulae are only valid when θ is in radians. Be careful to switch your calculator mode appropriately, especially when dealing with trigonometric derivatives and integrals.

在IB和WJEC课程中，弧度是度量角的自然单位。1弧度是圆中弧长等于半径时所对的圆心角。换算关系为180° = π rad。半径为r、圆心角为θ rad的扇形，弧长公式为s = rθ，扇形面积公式为A = ½ r²θ。这些公式只在θ为弧度时成立。尤其在处理三角函数的导数和积分时，务必确保计算器模式设置正确。

2. Unit Circle Definitions | 单位圆定义

The unit circle extends trigonometric definitions to all real numbers. For a point (x, y) on the circle x² + y² = 1, cosθ = x and sinθ = y, where θ is the angle measured anticlockwise from the positive x‑axis. Tanθ is then y/x. This model immediately gives the signs of the functions in each quadrant (All Students Take Calculus) and explains why sine and cosine oscillate between −1 and 1. It also allows exact values for angles like 0, π/6, π/4, π/3, π/2 and their multiples to be read directly from the special triangles.

单位圆将三角函数的定义扩展到所有实数。对于圆 x² + y² = 1 上的点 (x, y)，cosθ = x，sinθ = y，其中θ是从正x轴逆时针度量的角，tanθ则为 y/x。这个模型直接给出了各象限内三角函数的符号（All Students Take Calculus规律），并解释了正弦和余弦的值域为何在 −1 到 1 之间。对于0, π/6, π/4, π/3, π/2及其倍数等特殊角，其精确值也可以从特殊三角形中直接读取。

3. Graphs of Sine, Cosine and Tangent | 正弦、余弦、正切图像

The graphs y = sin x, y = cos x and y = tan x reveal amplitude, period, symmetry and asymptotes. Sine and cosine have period 2π, amplitude 1, and are continuous. Cosine is an even function, sine is odd. The tangent graph has period π, vertical asymptotes at x = π/2 + kπ, and is also odd. Understanding these parent graphs is essential before applying transformations. In exams you may be asked to sketch these functions, label intersections and key features, or deduce equations from given graphs.

y = sin x、y = cos x 和 y = tan x 的图像揭示了振幅、周期、对称性和渐近线。正弦和余弦的周期为2π，振幅为1，且处处连续。余弦为偶函数，正弦为奇函数。正切图像的周期是π，在 x = π/2 + kπ 处存在垂直渐近线，同样是奇函数。在应用图像变换之前，理解这些原始图像至关重要。考试中可能要求你画出函数草图、标注交点与关键特征，或根据给定图像推导函数解析式。

4. Transformations of Trigonometric Graphs | 三角函数图像变换

General forms such as y = a sin(bx + c) + d combine stretches, translations and reflections. |a| is the amplitude (vertical stretch); the period is 2π/|b| for sine and cosine, and π/|b| for tangent; c/b represents a horizontal shift (phase shift), and d is the vertical translation. You must be able to identify these parameters from a graph or a real‑world context, and conversely sketch a transformed function accurately. Examiners frequently test the correct ordering of transformations: horizontal compressions and shifts are especially tricky, so work inside the argument carefully.

形如 y = a sin(bx + c) + d 的一般式涵盖了伸缩、平移和反射变换。|a| 代表振幅（纵向伸缩）；正弦和余弦的周期为 2π/|b|，正切周期为 π/|b|；c/b 表示水平位移（相位移动），d 为纵向平移。你必须能够从图像或实际情境中识别这些参数，反之也要能准确绘制变换后的函数图像。考官经常考查变换顺序的正确性：水平压缩和平移尤其容易出错，因此处理括号内的自变量时要格外当心。

5. Trigonometric Identities | 三角恒等式

Mastering key identities allows simplification of expressions and proof of statements. The fundamental Pythagorean identity is

sin²θ + cos²θ = 1

. From this, dividing by cos²θ yields 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ. The quotient identities are tanθ = sinθ/cosθ and cotθ = cosθ/sinθ. Reciprocal identities: secθ = 1/cosθ, cscθ = 1/sinθ, cotθ = 1/tanθ. IB and WJEC also require the compound angle (addition) and double‑angle identities: sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB, and tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB). The double‑angle formulas emerge by setting A = B.

掌握关键恒等式有助于化简表达式和证明命题。基本的毕达哥拉斯恒等式为

sin²θ + cos²θ = 1

。将其两边除以 cos²θ 可得 1 + tan²θ = sec²θ，除以 sin²θ 则得 1 + cot²θ = csc²θ。商数关系为 tanθ = sinθ/cosθ 和 cotθ = cosθ/sinθ。倒数关系：secθ = 1/cosθ，cscθ = 1/sinθ，cotθ = 1/tanθ。IB和WJEC同样要求掌握和差公式与倍角公式：sin(A ± B) = sinA cosB ± cosA sinB，cos(A ± B) = cosA cosB ∓ sinA sinB，以及 tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)。令 A = B 即可得到倍角公式。

6. Solving Trigonometric Equations | 解三角函数方程

Trigonometric equations appear in almost every exam. The typical strategy: use identities to rewrite the equation in terms of a single trigonometric function; solve the resulting algebraic equation for that function; then find all solutions within the given domain using CAST diagrams or the unit circle, remembering to add the period of the function. For quadratic forms like 2sin²x − sinx − 1 = 0, let u = sinx, solve for u, then solve for x. Always check that solutions satisfy the original equation, especially when squaring or using reciprocal identities. Be meticulous with radian measure unless the question specifies degrees.

三角函数方程几乎出现在每份试卷中。典型解题策略为：利用恒等式将方程化为只含一个三角函数的形式；求解所得代数方程；然后借助CAST图或单位圆在指定定义域内求出所有解，并牢记加上函数的周期。对于 2sin²x − sinx − 1 = 0 一类的二次型，可设 u = sinx 求解，再反求出 x。务必检查解是否满足原方程，尤其是在实施平方或使用倒数关系时。除非题目明确使用角度制，否则要严谨使用弧度制。

7. Inverse Trigonometric Functions | 反三角函数

The inverse functions arcsin, arccos and arctan (or sin⁻¹, cos⁻¹, tan⁻¹) are defined by restricting the domains of sine, cosine and tangent to ensure they are one‑to‑one. arcsin has domain [−1, 1] and principal range [−π/2, π/2]; arccos has domain [−1, 1] and principal range [0, π]; arctan has domain ℝ and principal range (−π/2, π/2). When solving equations like sinx = −0.7, the calculator gives the principal value, but you must find additional solutions using symmetry. Understand the relationship between a function and its inverse, and be prepared to sketch or read graphs of these inverses.

反三角函数 arcsin、arccos 和 arctan（或记作 sin⁻¹、cos⁻¹、tan⁻¹）是通过限制正弦、余弦和正切的定义域以保证其一一对应关系而定义的。arcsin的定义域为[−1, 1]，主值区间为[−π/2, π/2]；arccos的定义域为[−1, 1]，主值区间为[0, π]；arctan的定义域为ℝ，主值区间为(−π/2, π/2)。当解方程 sinx = −0.7 时，计算器只会给出主值，你需要利用对称性求出其余解。要理解函数与其反函数之间的关系，并做好绘制或阅读这类反函数图像的准备。

8. Compound and Double Angle Formulae in Depth | 复合角与倍角公式深入

These formulae are heavily assessed for proof and simplification. Apart from the basic identities, the double‑angle cosine has three variants:

cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

. This is vital for integrating sin²θ and cos²θ, and for solving equations. The product‑to‑sum and sum‑to‑product identities occasionally appear in IB HL and WJEC pure maths. For example, sinA + sinB = 2 sin((A+B)/2) cos((A−B)/2). Make flashcards for these and practise deriving one formula from another; this builds fluency and confidence when faced with an unfamiliar trig expression.

这些公式在证明与化简中考查极多。除了基本恒等式，余弦的倍角公式有三种变体：

cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

。这对积分 sin²θ 和 cos²θ 以及解方程至关重要。在IB高级水平（HL）和WJEC纯数学中，还会偶尔出现积化和差与和差化积公式。例如，sinA + sinB = 2 sin((A+B)/2) cos((A−B)/2)。建议制作记忆卡片并练习由一个公式推导出另一个；这能培养你在面对陌生三角表达式时的熟练度和自信心。

9. Applications to Periodic Phenomena | 周期现象应用

Modelling temperatures, tides, sound waves and pendulums often uses sinusoidal functions. A typical problem gives average value (vertical shift), amplitude, period and sometimes phase shift. You must construct a function of the form f(t) = A sin(B(t − C)) + D, where the period is 2π/B. The IB Applications & Interpretation course and WJEC applied units place special emphasis on interpreting the parameters in context, differentiating the model to find rates of change, and making predictions. Always check the domain of the variable and think critically about whether the model remains valid beyond the data range.

对温度、潮汐、声波和单摆等周期性现象进行建模时，常使用正弦型函数。典型问题会给出平均值（纵向平移量）、振幅、周期，有时还有相位移动。你需要构造形如 f(t) = A sin(B(t − C)) + D 的函数，其中周期为 2π/B。IB“应用与解释”课程以及WJEC的应用单元特别强调在语境中解释参数、对模型求导以找出变化率，以及做出预测。务必检查变量的定义域，并批判性地思考模型在数据范围之外是否仍然有效。

10. Exact Values and Technology Use | 特殊值记忆与计算器/技术使用

Both IB and WJEC expect you to know exact trigonometric values for 0°, 30°, 45°, 60°, 90° (and their radian equivalents) without a calculator. These come from the 45°‑45°‑90° and 30°‑60°‑90° triangles. Remember that sin30° = 1/2, cos30° = √3/2, tan45° = 1, and so on. In the non‑calculator parts of exams, you must present exact answers using surds and π. When using a GDC or scientific calculator, understand how to store intermediate values to avoid rounding errors, and how to solve equations graphically or with the solver. However, never rely on the calculator to do the thinking—examiners design questions to test understanding, not button‑pushing.

IB和WJEC都要求你在不使用计算器的情况下记忆并准确写出0°、30°、45°、60°、90°（及其弧度等效值）的三角函数精确值。这些值来源于45°‑45°‑90° 和 30°‑60°‑90°的两种特殊三角形。记住 sin30° = 1/2，cos30° = √3/2，tan45° = 1 等。在考试的不可用计算器部分，你必须用根号和π给出精确答案。当使用图形计算器（GDC）或科学计算器时，要知道如何存储中间值以避免舍入误差，以及如何通过图像或求解器功能解方程。然而，永远不要依赖计算器代替思考——考官设计题目的目的是测试你的理解力，而非按键操作。

11. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Students frequently forget to consider all quadrants when solving equations, or miss the negative sign when the angle is reflected. Another classic error is applying sin(A+B) = sinA + sinB—this is false. Similarly, forgetting that tanx has period π leads to missing solutions. When solving on a calculator, ensure the mode (radians/degrees) matches the question. Drawn sketches often lack labels for axes, intercepts or asymptotes. Finally, in proof questions, always start from one side and transform it into the other, rather than manipulating both sides simultaneously unless using an equivalence approach.

学生经常在解方程时忘记考虑所有象限，或者在角度对称变换时漏掉负号。另一个典型错误是误认为 sin(A+B) = sinA + sinB ——这是错误的。类似地，忘记 tanx 的周期是π会导致漏解。使用计算器时，务必确保模式（弧度/角度）与题目一致。手绘草图常常缺少坐标轴标记、截距或渐近线。最后，在证明题中，始终从等式的一侧入手，将其变换为另一侧，而不要同时操作两侧，除非采用等价推导的方式。

12. Revision Checklist and Exam Tips | 复习清单与应试技巧

Before the examination, tick off these items: know the exact values table by heart; be able to derive all Pythagorean and compound angle identities; practise solving linear and quadratic trig equations within a fixed interval; sketch transformed sine, cosine and tangent graphs showing all key features; revisit arcsin, arccos, arctan domains and ranges; review real‑life modelling questions; and attempt at least three past‑paper questions under timed conditions. During the exam, show clear working, label solutions on a diagram, and always substitute your answers back into the original equation to verify correctness.

考试前，请逐项核对以下清单：熟记特殊值表格；能自行推导所有毕达哥拉斯恒等式与和差公式；练习在给定区间内求解线性和二次三角方程；绘制包含所有关键特征的变换后的正弦、余弦和正切图像；复习 arcsin、arccos、arctan 的定义域和值域；回顾实际建模问题；并在限时条件下至少完成三道历年真题。考试过程中，要展示清晰的步骤，在示意图上标注答案，并始终将答案代回原方程进行检验。

Published by TutorHao | Trigonometric Functions Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB WJEC Mathematics: Trigonometric Functions Key Points | IB WJEC 数学：三角函数 考点精讲