IGCSE Mathematics: Trigonometry Key Concepts | IGCSE 数学：三角函数考点精讲

📚 IGCSE Mathematics: Trigonometry Key Concepts | IGCSE 数学：三角函数考点精讲

Trigonometry is a vital topic in IGCSE Mathematics, linking geometry, algebra and real‑world problem‑solving. Whether you are taking the Core or Extended paper, a solid grasp of trigonometric ratios, rules for any triangle, graphs and simple identities can boost your confidence and your final grade. This article walks through every major concept you need to know, with clear explanations and exam‑ready tips.

三角函数是 IGCSE 数学中连接几何、代数与现实问题解决的重要课题。无论你参加 Core 还是 Extended 考试，牢固掌握三角比、任意三角形的定理、函数图像和基本恒等式都能提升你的信心和最终成绩。本文梳理了你需要掌握的每一个核心概念，并配有清晰的解释和应试技巧。

1. Trigonometric Ratios in Right Triangles | 直角三角形中的三角比

In a right‑angled triangle, the three sides are labelled relative to a chosen acute angle θ: the opposite is the side facing θ, the adjacent is the side next to θ (but not the hypotenuse), and the hypotenuse is the longest side, opposite the right angle. The three basic trigonometric ratios are defined as:

在一个直角三角形中，三条边根据选定的锐角 θ 来命名：对边是面对 θ 的边，邻边是与 θ 相邻（但不是斜边）的边，斜边是最长边，对应直角。三个基本三角比定义为：

sin θ = opposite / hypotenuse

sin θ = 对边 / 斜边

cos θ = adjacent / hypotenuse

cos θ = 邻边 / 斜边

tan θ = opposite / adjacent

tan θ = 对边 / 邻边

A helpful mnemonic is SOH CAH TOA. These ratios only apply to right‑angled triangles and allow you to find unknown sides or angles when you know at least two other pieces of information (sides or acute angles).

一个有用的记忆口诀是 SOH CAH TOA。这些比仅适用于直角三角形，当你至少知道另外两个信息（边或锐角）时，就可以求出未知的边或角。

2. Special Angles: 30°, 45°, 60° | 特殊角：30°、45°、60°

Some angles appear so often that you should memorise their exact trigonometric values. They can be derived from an equilateral triangle of side 2 (split into two 30°‑60°‑90° triangles) and an isosceles right triangle with legs 1 (45°‑45°‑90°). The values are:

有些角度出现得非常频繁，你应该记住它们的精确三角函数值。这些值可以由边长为 2 的等边三角形（分成两个 30°‑60°‑90° 三角形）和直角边为 1 的等腰直角三角形（45°‑45°‑90°）推导出来。具体数值如下：

Angle θ	sin θ	cos θ	tan θ
30°	½	√3 / 2	1 / √3
45°	1 / √2	1 / √2	1
60°	√3 / 2	½	√3

These exact values are often required in IGCSE questions, especially when the final answer must be left in surd form or when solving equations without a calculator.

IGCSE 题目经常要求使用这些精确值，特别是当最终答案需要保留根号形式或在不用计算器解方程时。

3. Solving Right‑Angled Triangles | 解直角三角形

To solve a right‑angled triangle means to find all unknown sides and angles. You use the three trigonometric ratios together with the fact that the two acute angles add up to 90°. There are two common scenarios:

解直角三角形就是求出所有未知的边和角。你需要利用三个三角比以及两个锐角互余（和为 90°）这一事实。常见情况有两种：

Given one side and one acute angle: Choose the ratio that links the known side, the unknown side and the given angle. For example, if you know the hypotenuse and an angle, you can find the opposite side using opposite = hypotenuse × sin θ.

已知一边和一锐角：选择连接已知边、未知边和已知角的三角比。例如，若已知斜边和一个角，可以用 对边 = 斜边 × sin θ 求出对边。

Given two sides: Use the appropriate inverse trigonometric function. If you know the opposite and adjacent sides, then θ = tan⁻¹(opposite/adjacent). Always check that your answer is sensible – the largest side is opposite the largest angle.

已知两边：使用相应的反三角函数。若已知对边和邻边，则 θ = tan⁻¹(对边/邻边)。务必检查答案是否合理——最大的边应对应最大的角。

Remember to set your calculator to degree mode (D or DEG). Many exam mistakes come from leaving it in radian mode.

记得把计算器设置为角度模式（D 或 DEG）。许多考试失误都是因为留在了弧度模式。

4. Angles of Elevation and Depression | 仰角与俯角

When solving real‑world problems, angles are often measured from the horizontal. The angle of elevation is the angle up from the horizontal to an object above. The angle of depression is the angle down from the horizontal to an object below.

在解决实际问题时，角度常常从水平线量起。仰角是从水平线向上看向上方物体所形成的角。俯角是从水平线向下看向下方物体所形成的角。

The key fact is that the angle of depression from point A to point B is equal to the angle of elevation from B to A, because they are alternate interior angles formed by parallel horizontal lines.

关键事实是：从点 A 看点 B 的俯角等于从点 B 看点 A 的仰角，因为它们是由平行水平线形成的内错角。

Draw a clear right‑angled triangle, label the given lengths and the unknown height or distance, then apply SOH CAH TOA. Many questions involve a person on a cliff looking at a boat, or two buildings of different heights.

画一个清晰的直角三角形，标出已知长度和未知高度或距离，然后应用 SOH CAH TOA。很多题目涉及站在悬崖上望船或两座高度不同的建筑物。

5. Bearings and Trigonometry | 方位角与三角函数

Bearings are used in navigation and are always measured clockwise from North, given as a three‑digit angle (e.g. 045°, 225°). Trigonometry often combines bearings with distances to find the shortest path or the position of an object.

方位角用于导航，总是从北开始顺时针测量，并以三位数的角度表示（如 045°、225°）。三角函数常与方位角结合，用来求最短路径或物体的位置。

A typical question: “A ship sails 8 km on a bearing of 120° from A to B, then 6 km on a bearing of 030° from B to C. Find the distance and bearing of C from A.” Start by drawing the North lines and the distances, then resolve into right‑angled components using sine and cosine. You may need to use the sine rule or cosine rule in non‑right‑angled situations, but many IGCSE bearing problems can be broken into right triangles.

典型题目：”一艘船从 A 出发，沿方位角 120° 航行 8 公里到达 B，然后从 B 沿方位角 030° 航行 6 公里到达 C。求 C 到 A 的距离和方位角。” 首先画出指北线和各段距离，然后利用正弦和余弦分解成直角分量。对于非直角三角形可能需要用正弦定理或余弦定理，但许多 IGCSE 方位角问题可以拆分成直角三角形。

6. Sine Rule | 正弦定理

For any triangle (not just right‑angled), the Sine Rule states:

对任意三角形（不限于直角三角形），正弦定理 指出：

a / sin A = b / sin B = c / sin C

or, equivalently:

或等价形式：

sin A / a = sin B / b = sin C / c

This rule is used when you know either two angles and one side (AAS or ASA) or two sides and a non‑included angle (SSA). In the SSA case, be careful – there may be an ambiguous case where two different triangles are possible. The IGCSE Extended syllabus expects you to recognise when the ambiguous case arises and to find both solutions if asked.

正弦定理适用于已知两角一边（AAS 或 ASA），或者已知两边及一边的对角（SSA）的情况。在 SSA 情况下要小心——可能出现 多解情形，即可能有两个不同的三角形。IGCSE Extended 课程要求你识别何时出现多解情形，并在需要时求出两个解。

7. Cosine Rule | 余弦定理

The Cosine Rule is essential when the sine rule cannot be applied directly. There are two main forms:

当正弦定理无法直接使用时，余弦定理 至关重要。它有两种主要形式：

a² = b² + c² − 2bc cos A

(to find a side when you know the other two sides and the included angle), and

（已知两边及其夹角时求第三边），以及

cos A = (b² + c² − a²) / (2bc)

(to find an angle when all three sides are known). The cosine rule applies to any triangle, and it is particularly useful in SAS (side‑angle‑side) and SSS (side‑side‑side) scenarios.

（已知三边求一个角）。余弦定理适用于任意三角形，在 SAS（边角边）和 SSS（边边边）情形下尤其有用。

In the IGCSE exam, you often choose between the sine rule and cosine rule by looking at what information is given. If you have a full pair (side and its opposite angle), use the sine rule. If you have three sides or two sides and the angle between them, the cosine rule is the way to go.

在 IGCSE 考试中，通常通过审题中的已知信息来选择正弦定理还是余弦定理。如果你有完整的一对（一条边及其对角），用正弦定理。如果你有三条边或两边及其夹角，就用余弦定理。

8. Area of a Triangle (½ ab sin C) | 三角形面积（½ ab sin C）

When you know two sides of a triangle and the angle between them, the area is given by:

当你知道三角形的两边及其夹角时，面积公式为：

Area = ½ × a × b × sin C

This formula works for any triangle, not only right‑angled ones. It is particularly handy in problems involving non‑right‑angled triangles, where the base × height ÷ 2 approach is not straightforward.

这个公式适用于任意三角形，不仅仅是直角三角形。在涉及非直角三角形的问题中它特别方便，因为此时底乘高除以 2 的方法并不直接。

Sometimes an exam question will ask you to find an unknown angle given the area and two sides. Simply rearrange the formula: sin C = (2 × Area) / (a × b), then use the inverse sine.

有时考题会给出面积和两边，要求找未知角。只需将公式变形：sin C = (2 × 面积) / (a × b)，然后使用反正弦函数。

9. Graphs of Trigonometric Functions | 三角函数图像

For the Extended syllabus you need to understand the graphs of y = sin x, y = cos x and y = tan x for 0° ≤ x ≤ 360°.

对于 Extended 课程，你需要理解 y = sin x、y = cos x 和 y = tan x 在 0° ≤ x ≤ 360° 范围内的图像。

y = sin x starts at 0, rises to 1 at 90°, falls to 0 at 180°, goes down to −1 at 270° and returns to 0 at 360°. The graph is a smooth wave, symmetric about 90° and 270°.

y = sin x 从 0 开始，在 90° 升到 1，在 180° 下降到 0，在 270° 降到 −1，在 360° 回到 0。图像是平滑的波形，关于 90° 和 270° 对称。

y = cos x starts at 1, falls to 0 at 90°, goes to −1 at 180°, rises to 0 at 270° and returns to 1 at 360°. It is the same wave shifted left by 90°.

y = cos x 从 1 开始，在 90° 降到 0，在 180° 降到 −1，在 270° 升到 0，在 360° 回到 1。它与 sin 是相同的波形，只是向左平移了 90°。

y = tan x has vertical asymptotes at 90° and 270°, where it is undefined. It passes through 0 at 0° and 180°, and its period is 180°.

y = tan x 在 90° 和 270° 处有垂直渐近线，在这些点函数无定义。它在 0° 和 180° 处通过 0，周期为 180°。

Be prepared to sketch these graphs, label key points and solve simple equation‑as‑graphs questions, e.g. “solve sin x = 0.5 for 0° ≤ x ≤ 360°”.

准备好画出这些函数的草图，标注关键点，并解答简单的图像解方程问题，例如“解 sin x = 0.5，x 范围 0° ≤ x ≤ 360°”。

10. Trigonometric Identities and Equations | 三角恒等式与方程

The IGCSE Extended course introduces one fundamental identity:

IGCSE Extended 课程引入了一个基本恒等式：

sin²θ + cos²θ = 1

and the definition:

以及定义：

tan θ = sin θ / cos θ

These are used to simplify expressions or to solve trigonometric equations. For example, given sin θ = 0.6 and θ is acute, you can find cos θ using the identity (cos θ = √(1 − sin²θ) = 0.8) and then tan θ = 0.6/0.8 = 0.75.

它们用来化简表达式或解三角方程。例如，已知 sin θ = 0.6 且 θ 为锐角，你可以用恒等式求出 cos θ（cos θ = √(1 − sin²θ) = 0.8），然后 tan θ = 0.6/0.8 = 0.75。

When solving equations like 2 sin x = 1, first isolate sin x (sin x = ½), then find the principal angle (30°) and use the graph or CAST diagram to find all solutions in the given interval, e.g. x = 30°, 150°.

在解如 2 sin x = 1 的方程时，首先分离 sin x（sin x = ½），然后求主值（30°），再利用图像或 CAST 图找出给定区间内的所有解，例如 x = 30°, 150°。

Always check the domain and remember that squaring both sides can sometimes introduce extraneous solutions – test your final answers in the original equation.

务必检查定义域，并记住两边平方有时会引入增根——将最终答案代回原方程检验。

11. 3D Trigonometry | 三维三角函数

3D trigonometry problems involve objects such as pyramids, cuboids and wedges. The main challenge is to identify right‑angled triangles in different planes of the 3D shape.

三维三角函数问题涉及棱锥、长方体和楔形等物体。主要难点在于识别出三维图形不同平面内的直角三角形。

Often you need to find the length of a line that joins two vertices not on the same face. The trick is to use Pythagoras first in one plane to find a diagonal, then use that diagonal in a second right triangle to find the required length or angle. A typical question: “Find the angle between the base diagonal and the sloping edge of a pyramid.”

通常你需要找出连接不在同一面上的两个顶点的线段长度。技巧是先用勾股定理在一个平面内求出对角线，再将这条对角线用于第二个直角三角形以求出所需的长度或角度。典型问题：”求金字塔的底面对角线与侧棱之间的夹角。”

Draw a clear net or separate diagram for each relevant triangle. Label the edges and heights clearly, and apply SOH CAH TOA or the sine/cosine rules as appropriate.

为每个相关的三角形画一个清晰的展开图或单独示意图。清楚地标出棱和高，并视情况应用 SOH CAH TOA 或正/余弦定理。

Mastering IGCSE trigonometry is all about recognising the type of problem, picking the right tool (SOH CAH TOA, sine rule, cosine rule, area formula) and executing the calculations accurately. Practice past‑paper questions, keep your working neat, and never forget to check the ambiguity of the sine rule when given SSA. With these key concepts under your belt, you can approach any trig question with confidence.

掌握 IGCSE 三角函数的关键在于识别题目类型、选择正确的工具（SOH CAH TOA、正弦定理、余弦定理、面积公式）并精确进行计算。多做历年真题，保持卷面整洁，在遇到 SSA 情况时永远不要忘记检查正弦定理的多解性。把这些核心概念掌握好，你就能充满信心地应对任何三角函数题目了。

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IGCSE Mathematics: Trigonometry Key Concepts | IGCSE 数学：三角函数考点精讲