📚 IB Maths Topic 3: Geometry and Trigonometry | IB数学第三单元：几何与三角学（SL与HL）
Geometry and trigonometry form a foundational pillar of the IB Mathematics curriculum, bridging the gap between spatial reasoning and algebraic manipulation. In Topic 3, both SL and HL students explore radian measure, trigonometric functions and their graphs, identities, and the solution of triangles using sine and cosine rules. HL learners extend these ideas with reciprocal functions, compound angle formulas, double angle identities, inverse trigonometric functions, and more complex equations. This article condenses the essential concepts, complete with worked ideas and examples, to serve as a revision guide.
几何与三角学是IB数学课程的基石，连接着空间推理与代数运算。在第三单元中，SL与HL学生都将探索弧度制、三角函数及其图像、恒等式以及运用正弦和余弦定理解三角形。HL学生还会进一步学习倒数三角函数、复合角公式、倍角恒等式、反三角函数及更复杂的三角方程。本文浓缩了核心概念，配有求解思路和示例，可作为复习备考的指引。
1. Understanding Radian Measure and the Unit Circle | 理解弧度制与单位圆
Radian measure provides a natural link between an angle and the length of an arc on a circle. One radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. This gives the fundamental conversion π rad = 180°. The unit circle, centred at the origin with radius 1, allows sine and cosine to be defined for any real angle: cosθ is the x‑coordinate and sinθ is the y‑coordinate of a point on the circle.
弧度制在角度与圆的弧长之间建立了自然的联系。1弧度定义为半径长度的弧所对的圆心角。由此得到核心转换关系 π弧度 = 180°。单位圆以原点为圆心、半径为1，使得正弦和余弦可对任意实角定义：单位圆上一点的横坐标为 cosθ，纵坐标为 sinθ。
Working in radians simplifies calculus and makes arc length and sector area formulas particularly elegant: arc length l = rθ and sector area A = ½r²θ, where θ is in radians and r is the radius. This is heavily used in both SL and HL problems, especially when relating angular velocity to linear distances.
使用弧度制可以简化微积分运算，并使弧长与扇形面积公式格外简洁：弧长 l = rθ，扇形面积 A = ½r²θ，其中 θ 以弧度为单位，r 为半径。这一点在 SL 与 HL 的题目中都会频繁用到，尤其在处理角速度与线距离的关系时。
2. Trigonometric Ratios and Special Angles | 三角函数比与特殊角
In a right‑angled triangle, the three primary trigonometric ratios are defined as sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse and tanθ = opposite/adjacent. The mnemonic SOH‑CAH‑ , these ratios can be extended using the unit circle, where tanθ = sinθ / cosθ (cosθ ≠ 0).
在直角三角形中，三个基本三角函数比的定义为 sinθ = 对边/斜边、cosθ = 邻边/斜边 以及 tanθ = 对边/邻边。SOH‑CAH‑TOA 口诀有助于记忆。对于任意角，这些比值可通过单位圆推广，其中 tanθ = sinθ / cosθ（cosθ ≠ 0）。
Exact values for key angles must be memorised; they are frequently tested without a calculator. The table below summarises the most important ones.
必须熟记特殊角的精确值，因为考试中经常不允许使用计算器。下表总结了最重要的几个值。
| Angle (degrees) | Angle (radians) | sinθ | cosθ | tanθ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | ½ | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 |
| 60° | π/3 | √3/2 | ½ | √3 |
| 90° | π/2 | 1 | 0 | undefined |
Note how these ratios can be derived from the isosceles right triangle (45°) and the 30‑60‑90 triangle, giving their exact surd forms. Being fluent with these values makes solving trigonometric equations and identities much smoother.
注意这些比值可从等腰直角三角形（45°）和30‑60‑90三角形推出，得到精确的根式形式。熟练掌握这些值能让你在解三角方程和证明恒等式时更加流畅。
3. Graphs of Trigonometric Functions | 三角函数图像
The sine and cosine functions are periodic with period 2π, while the tangent function has period π. The graph of y = sin x oscillates between –1 and 1, passing through the origin; y = cos x starts at 1 when x = 0 and shifts by π/2. The tangent graph has vertical asymptotes at x = π/2 + kπ and passes through the origin with increasing steepness.
正弦和余弦函数为周期函数，周期为 2π；正切函数的周期为 π。y = sin x 的图像在 –1 与 1 之间振荡，且过原点；y = cos x 在 x = 0 时从 1 开始，与正弦曲线相差 π/2 的相位。正切图像在 x = π/2 + kπ 处存在竖直渐近线，在原点附近穿过并逐渐变陡。
Transformations of the form y = a sin(bx + c) + d change the amplitude |a|, period 2π/|b|, horizontal shift –c/b, and vertical shift d. Being able to sketch these quickly, and to identify the effect of each parameter, is an essential skill for both SL and HL exams.
形如 y = a sin(bx + c) + d 的变换会改变振幅 |a|、周期 2π/|b|、水平位移 –c/b 以及竖直位移 d。能够迅速绘制草图并识别各参数的作用，是SL和HL考试的必备技能。
4. Trigonometric Identities – SL Core | 三角恒等式（SL核心）
The most frequently used identity is the Pythagorean identity: sin²θ + cos²θ ≡ 1. From this, two variants can be derived by dividing by cos²θ or sin²θ: 1 + tan²θ ≡ sec²θ and 1 + cot²θ ≡ csc²θ (the latter two are mainly used in HL). Another key SL identity is the quotient identity: tanθ ≡ sinθ / cosθ.
最常用的恒等式是毕达哥拉斯恒等式：sin²θ + cos²θ ≡ 1。由此可除以 cos²θ 或 sin²θ 导出两种变体：1 + tan²θ ≡ sec²θ 和 1 + cot²θ ≡ csc²θ（后两个主要用于HL）。另一个SL核心恒等式是商数恒等式：tanθ ≡ sinθ / cosθ。
These identities are used to simplify expressions, prove other identities, and solve equations. A typical SL problem might ask to prove that sinθ cosθ / (1 − sin²θ) = tanθ. The strategy involves replacing the denominator using the Pythagorean identity and then applying the quotient identity.
这些恒等式用于化简表达式、证明其他恒等式以及解方程。一个典型的SL题目可能要求证明 sinθ cosθ / (1 − sin²θ) = tanθ，解题思路是利用平方恒等式替换分母，再运用商数恒等式。
5. Solving Basic Trigonometric Equations | 解基本三角方程
To solve sin x = k, cos x = k or tan x = k within a specified domain (often 0 ⩽ x ⩽ 2π), first find the principal value using inverse functions or known special angles. Then use the symmetry properties of the unit circle to find all solutions. For sine, if α is the principal value, the general solutions are x = α + 2kπ and x = π − α + 2kπ; for cosine, x = ±α + 2kπ; for tangent, x = α + kπ.
在指定区间内（通常 0 ⩽ x ⩽ 2π）求解 sin x = k、cos x = k 或 tan x = k，首先利用反函数或已知特殊角找出主值，然后利用单位圆的对称性找出所有解。对于正弦，若主值为 α，则通解为 x = α + 2kπ 和 x = π − α + 2kπ；余弦为 x = ±α + 2kπ；正切为 x = α + kπ。
When the equation involves squared trig functions, it is often helpful to use an identity to reduce it to a single function. For example, 2 sin²x + 3 cos x = 3 can be transformed into a quadratic in cos x using sin²x = 1 − cos²x.
当方程含有三角函数的平方时，常借用恒等式将其化为单一函数的方程。例如，2 sin²x + 3 cos x = 3 可利用 sin²x = 1 − cos²x 转化为关于 cos x 的二次方程。
6. Sine Rule, Cosine Rule and Triangle Area | 正弦定理、余弦定理与三角形面积
For any triangle ABC with sides a opposite A, b opposite B, c opposite C, the sine rule states a/sin A = b/sin B = c/sin C. It is used when given two angles and a side (AAS or ASA) or two sides and a non‑included angle (SSA), though the ambiguous case must be checked for the latter.
对于任意三角形 ABC，边 a 对角 A，b 对角 B，c 对角 C，正弦定理为 a/sin A = b/sin B = c/sin C。当已知两角一边（AAS或ASA）或两边及一个非夹角（SSA）时使用，后者需注意可能产生多解（歧义情况）。
The cosine rule is most useful for SAS or SSS cases: a² = b² + c² − 2bc cos A. The area of a triangle can be found using Area = ½ab sin C, which is especially handy when two sides and the included angle are known.
余弦定理最适用于已知两边及其夹角（SAS）或三边（SSS）的情形：a² = b² + c² − 2bc cos A。三角形面积可用 面积 = ½ab sin C 计算，尤其在已知两边及其夹角时极为方便。
These rules are also essential for solving bearings problems and for finding lengths and angles in real‑world contexts. In HL, students may also be asked to prove the sine and cosine rules from first principles.
这些定理也是解决方位角问题以及在实际情境下求长度与角度的关键。在HL课程中，学生还可能被要求由基本原理出发证明正弦定理和余弦定理。
7. Three‑Dimensional Geometry Problems | 三维几何问题
Many IB questions extend 2D trigonometry into three dimensions, involving cuboids, pyramids, wedges and prisms. The key is to identify the relevant right‑angled triangles within the 3D figure and apply Pythagoras' theorem together with trigonometric ratios. Commonly required skills include finding the angle between a line and a plane, or the angle between two planes (dihedral angle).
许多IB题目将二维三角学延伸至三维空间，涉及长方体、棱锥、楔形和棱柱。关键在于在三维图形中识别出相关的直角三角形，并运用勾股定理结合三角函数比。常见的技能包括求线面角或两平面的夹角（二面角）。
Always draw a clear diagram, label all given lengths and angles, and break down the 3D problem into a series of 2D right‑angle sketches. For example, to find the angle between the main diagonal of a cuboid and its base
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