📚 Trigonometry Key Points for IGCSE Edexcel Maths | IGCSE Edexcel 数学：三角函数考点精讲
Trigonometry is a fundamental topic in the IGCSE Edexcel Mathematics syllabus, linking geometry, algebra, and practical problem-solving. Mastering trigonometric ratios, identities, rules for non‑right triangles, and graph interpretation will give you confidence in both the Foundation and Higher tier papers. This article revisits every essential point, with clear explanations in both English and Chinese to support your revision.
三角函数是 IGCSE Edexcel 数学的核心主题，它将几何、代数与实际应用联系在一起。掌握三角比、恒等式、非直角三角形的法则以及图像解读，能让你在基础卷和进阶卷中都有充足的信心。本文重温每一个重要考点，提供清晰的中英双语解释，帮助你巩固复习。
1. Trigonometric Ratios in Right-Angled Triangles | 直角三角形中的三角比
In any right‑angled triangle, the three sides relative to an acute angle θ are named the opposite (opp), adjacent (adj), and hypotenuse (hyp). The three basic ratios are defined as: sin θ = opp/hyp, cos θ = adj/hyp, and tan θ = opp/adj. A simple way to remember them is ‘SOH CAH TOA’.
在任意直角三角形中，相对于锐角 θ 的三条边分别称为对边 (opp)、邻边 (adj) 和斜边 (hyp)。三个基本三角比定义为：sin θ = 对/斜，cos θ = 邻/斜，tan θ = 对/邻。一个简单的记忆口诀是 ‘SOH CAH TOA’。
2. Exact Values of Special Angles | 特殊角的精确值
You must memorise the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°. These often appear without a calculator and are derived from special right triangles (half‑equilateral and isosceles right triangles). The key values are summarised below.
你必须熟记 0°、30°、45°、60° 和 90° 的 sin、cos 和 tan 精确值。这些常在不使用计算器的题目中出现，它们来自特殊的直角三角形（半等边三角形与等腰直角三角形）。关键值总结如下。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 (or √3/3) |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
3. Using a Calculator for Trigonometric Functions | 使用计算器求三角函数
For non‑special angles, you will use the sin, cos and tan buttons on your calculator. Always ensure your calculator is in degree mode (deg) for IGCSE questions. To find an angle from a known ratio, use the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹. For example, if sin θ = 0.7, then θ = sin⁻¹(0.7) ≈ 44.4° (to 1 d.p.).
对于非特殊角，你将使用计算器上的 sin、cos 和 tan 按键。务必确保计算器处于度数模式 (deg) 以便解答 IGCSE 题目。若要根据已知的比值求角度，使用反函数 sin⁻¹、cos⁻¹ 和 tan⁻¹。例如，若 sin θ = 0.7，则 θ = sin⁻¹(0.7) ≈ 44.4°（保留一位小数）。
4. Finding Sides and Angles in Right Triangles | 求直角三角形的边长与角度
To find a missing side, label the triangle with opp, adj and hyp relative to the known acute angle. Choose the ratio that links the known side and the unknown side, set up an equation, then solve. To find an acute angle, use the inverse ratio after writing the equation. Always write the working step‑by‑step and round sensibly.
求未知边长时，先根据已知锐角标记对边、邻边和斜边。选择能将已知边和未知边联系起来的三角比，列出方程，然后求解。求锐角时，在列出方程后使用反三角比。务必一步步写出过程，并合理取整。
5. Angles of Elevation and Depression | 仰角与俯角
These angles are formed between the horizontal line of sight and the line to an object. An angle of elevation looks upward from the observer; an angle of depression looks downward. The angle of depression from A to B equals the angle of elevation from B to A because they are alternate angles in parallel lines.
仰角和俯角是水平视线与指向物体的连线之间的夹角。仰角是从观察者向上看，俯角是向下看。从 A 看 B 的俯角等于从 B 看 A 的仰角，因为它们是平行线中的内错角。
6. 3D Problems and Bearings | 三维问题与方位角
In a 3D shape, you often need to identify right triangles within planes. The angle between a line and a plane is found by dropping a perpendicular. Bearings are measured clockwise from North, always given as three digits (e.g. 060°). They often combine with right‑angled trigonometry to find distances.
在三维图形中，你常需在平面内找出直角三角形。直线与平面的夹角可通过作垂线求得。方位角是从正北顺时针度量的，始终以三位数字表示（例如 060°）。方位角常与直角三角形三角法结合求距离。
7. Trigonometric Identities | 三角恒等式
IGCSE candidates are expected to know two key identities: sin²θ + cos²θ ≡ 1, and tan θ ≡ sin θ / cos θ (provided cos θ ≠ 0). These identities are useful for proving relationships and for solving equations where one function appears in terms of another. For example, if sin θ = 0.6, you can find cos θ = √(1 – 0.6²) = 0.8.
IGCSE 考生需掌握两个关键恒等式：sin²θ + cos²θ ≡ 1，以及 tan θ ≡ sin θ / cos θ（前提 cos θ ≠ 0）。这些恒等式可用于证明关系以及将方程化为单一函数求解。例如，已知 sin θ = 0.6，可求 cos θ = √(1 – 0.6²) = 0.8。
8. Solving Trigonometric Equations | 三角方程的求解
For simple equations like sin θ = 0.5 in the range 0° ≤ θ ≤ 360°, first find the acute principal angle using sin⁻¹ (here 30°). Then use the symmetry of the sine wave: sine is positive in the first and second quadrants, so solutions are θ = 30° and 180° – 30° = 150°. Always check the range and include all solutions.
对于简单的方程，如 sin θ = 0.5 在 0° ≤ θ ≤ 360° 范围内，先用 sin⁻¹ 求出锐角主值（此处为 30°）。然后利用正弦曲线的对称性：正弦在第一和第二象限为正，因此解为 θ = 30° 和 180° – 30° = 150°。务必检查范围并给出全部解。
9. Sine Rule for Non‑Right Triangles | 非直角三角形的正弦定理
When a triangle is not right‑angled, the sine rule connects sides and their opposite angles: a / sin A = b / sin B = c / sin C. Use it when you know two angles and any side (AAS/ASA) or two sides and a non‑included angle (SSA). Be careful with the ambiguous case, where two different triangles may be possible for given SSA.
当三角形不是直角三角形时，正弦定理将边及其对角联系起来：a / sin A = b / sin B = c / sin C。当你已知两角及任一边 (AAS/ASA) 或已知两边及一对角 (SSA) 时使用。注意 SSA 的歧义情况，此时可能有两个不同的三角形。
10. Cosine Rule for Non‑Right Triangles | 非直角三角形的余弦定理
The cosine rule is used when the sine rule cannot be applied easily: a² = b² + c² – 2bc cos A. It works for SAS (two sides and the included angle) to find the third side, and for SSS (three sides) to find an angle by reshaping to cos A = (b² + c² – a²) / (2bc).
当正弦定理不易使用时采用余弦定理：a² = b² + c² – 2bc cos A。适用于已知两边及其夹角 (SAS) 求第三边，以及已知三边 (SSS) 求角，此时可变形为 cos A = (b² + c² – a²) / (2bc)。
11. Area of a Triangle Using Sine | 利用正弦求三角形面积
If two sides and the included angle are known, the area of any triangle can be found directly without height: Area = ½ ab sin C. This formula is especially useful in non‑right triangles and appears frequently on IGCSE papers. Make sure the angle is between the two given sides.
若已知两边及其夹角，任何三角形的面积都可不依赖高度直接求得：面积 = ½ ab sin C。此公式对非直角三角形特别有用，在 IGCSE 试卷中出现频繁。确保所取角是已知两边之间的角。
12. Graphs of Trigonometric Functions | 三角函数图像与性质
You should be able to sketch y = sin x, y = cos x and y = tan x for 0° ≤ x ≤ 360°. The sine and cosine graphs are waves with period 360° and amplitude 1; sin starts at 0, cos starts at 1. The tangent graph has a period of 180° and vertical asymptotes at 90°, 270°, etc. Recognising these shapes helps solve equations graphically and understand symmetry.
你应能画出 y = sin x、y = cos x 和 y = tan x 在 0° ≤ x ≤ 360° 的图像。正弦与余弦图像是周期为 360°、振幅为 1 的波形；sin 从 0 开始，cos 从 1 开始。正切图像周期为 180°，在 90°、270° 等处有垂直渐近线。认识这些图像有助于用图像法解方程并理解对称性。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导