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

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

Trigonometry is a central topic in the IGCSE CIE Mathematics syllabus, testing your ability to work with angles, triangles and periodic functions. From right‑angled triangle ratios to the sine and cosine rules, trigonometric graphs and 3D applications, a solid grasp of these concepts will earn a significant share of marks. This revision guide covers every essential point you need to master for the examination.

三角函数是 IGCSE CIE 数学大纲的核心主题，考查学生对角度、三角形和周期函数的掌握程度。从直角三角形中的边角比到正弦定理和余弦定理，再到三角函数的图像及三维应用，牢固掌握这些概念将为考试赢得大量分数。本精讲覆盖了你必须掌握的每个关键考点。

1. Right‑Angled Trigonometry | 直角三角形中的三角函数

The three primary trigonometric ratios are defined on a right‑angled triangle: sin, cos and tan. Label the hypotenuse (H), opposite (O) and adjacent (A) relative to the angle of interest, then use SOH CAH TOA.

三个基本的三角函数定义于直角三角形中：正弦、余弦和正切。相对于所考虑的角，标记斜边（H）、对边（O）和邻边（A），然后使用 SOH CAH TOA 口诀。

sin θ = O / H, cos θ = A / H, tan θ = O / A

sin θ = 对边 / 斜边， cos θ = 邻边 / 斜边， tan θ = 对边 / 邻边

When given two sides, you can find an angle using the inverse functions sin⁻¹, cos⁻¹, tan⁻¹. Always check that your calculator is in degree mode. In IGCSE problems, answers are usually required to one decimal place or to the nearest degree.

已知两边时，可以使用反函数 sin⁻¹、cos⁻¹、tan⁻¹ 求角度。务必检查计算器已设定为度（degree）模式。在 IGCSE 考题中，答案通常需精确到一位小数或取整度数。

2. Exact Values for Special Angles | 特殊角的精确值

The CIE syllabus expects you to know the exact trigonometric values of 0°, 30°, 45°, 60° and 90° without a calculator. These are tested frequently in non‑calculator papers.

CIE 大纲要求你无需计算器就能写出 0°、30°、45°、60° 和 90° 的精确三角函数值。在非计算器试卷中，这些值的考查非常频繁。

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	½	√3
90°	1	0	undefined

These values derive from the standard 45° right‑angled triangle (isosceles) and the 30‑60‑90 triangle obtained by halving an equilateral triangle. Knowing them instantly lets you simplify expressions and solve equations without hesitation.

这些数值源于标准的 45° 直角三角形（等腰）和由等边三角形对半得到的 30‑60‑90 三角形。熟练记住它们能让你毫不犹豫地化简表达式和求解方程。

3. Trigonometric Graphs | 三角函数的图像

The graphs of y = sin x, y = cos x and y = tan x for 0° ≤ x ≤ 360° are essential. The sine graph starts at 0, rises to 1 at 90°, returns to 0 at 180°, drops to −1 at 270° and reaches 0 again at 360°. The cosine graph starts at 1, falls to 0 at 90°, reaches −1 at 180°, goes back to 0 at 270° and finishes at 1. The tangent graph has vertical asymptotes at 90° and 270°, crossing the x‑axis at 0°, 180° and 360°.

掌握 y = sin x、y = cos x 和 y = tan x 在 0° ≤ x ≤ 360° 的图像至关重要。正弦图像从 0 出发，在 90° 升至 1，180° 回到 0，270° 降至 −1，360° 再回到 0。余弦图像从 1 开始，在 90° 降到 0，180° 达 −1，270° 回升至 0，360° 回到 1。正切图像在 90° 和 270° 有垂直渐近线，在 0°、180° 和 360° 与 x 轴相交。

You must be able to sketch these graphs, identify key points, and interpret transformations such as y = a sin(bx) + c. The period of y = sin bx and y = cos bx is 360°/b, while the amplitude is given by |a|.

你必须能描绘这些图像、识别关键点，并解释诸如 y = a sin(bx) + c 的变换。y = sin bx 和 y = cos bx 的周期为 360°/b，振幅由 |a| 决定。

4. Solving Trigonometric Equations | 三角方程的求解

IGCSE equations usually ask for all solutions within 0° ≤ x ≤ 360°. After finding the principal value (calculator or exact value), use the symmetry of the graphs to locate additional solutions.

IGCSE 方程通常要求找出 0° ≤ x ≤ 360° 内的所有解。求得主值（通过计算器或精确值）后，利用图像的对称性定位其他解。

For sin x, if x is one solution, the other is 180° − x. For cos x, the second solution is 360° − x. For tan x, add or subtract 180° from the principal value because the period is 180°. Always list solutions in ascending order.

对于 sin x，若 x 是一个解，则另一个解为 180° − x。对于 cos x，第二个解是 360° − x。对于 tan x，由于其周期为 180°，只需在主值上加或减 180°。请始终按升序列出答案。

5. Bearings and Angles of Elevation/Depression | 方位角与仰角、俯角

Bearings are measured clockwise from north and always given as three‑figure numbers (e.g. 035°, 210°). Many trigonometry problems require drawing a diagram with a north line and applying right‑angled triangle ratios to find distances or bearings.

方位角从正北方顺时针方向测量，并用三位数字表示（如 035°、210°）。许多三角函数题需要绘制包含指北线的示意图，并运用直角三角形边角比求出距离或方位角。

Angle of elevation is the angle above the horizontal when looking at an object; angle of depression is the angle below the horizontal. These angles are equal when viewed from opposite points (alternate angles on parallel lines).

仰角是观察物体时视线在水平线上的角度；俯角是视线在水平线下的角度。当从相对两点观察时，由于平行线上的内错角相等，这两个角度相等。

6. Sine Rule | 正弦定理

The sine rule is used for non‑right‑angled triangles when you know either two angles and one side (AAS) or two sides and a non‑included angle (SSA). Ensure angles are paired with their opposite sides.

正弦定理用于非直角三角形，适用情形包括已知两角及一边（AAS）或两边及一对角（SSA）。务必确保角与其对边相对应。

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

In the ambiguous case (SSA), there may be two possible angles for the unknown angle because sin θ = sin(180° − θ). Always check whether both solutions are valid given the triangle’s angle sum.

在模糊情况（SSA）下，未知角可能有两个可能的取值，因为 sin θ = sin(180° − θ)。根据三角形内角和，必须检验两个解是否都成立。

7. Cosine Rule | 余弦定理

Use the cosine rule when you have two sides and the included angle (SAS) or all three sides (SSS). It is the extension of Pythagoras’ theorem.

当已知两边及夹角（SAS）或已知三边（SSS）时使用余弦定理。它是勾股定理的推广。

a² = b² + c² − 2bc cos A

To find an angle when all three sides are known, rearrange to cos A = (b² + c² − a²) / (2bc). This formula is heavily examined, especially for finding the largest angle opposite the longest side.

当已知三边求角时，变形为 cos A = (b² + c² − a²) / (2bc)。该公式考查频繁，常用于求最长边所对的最大角。

8. Area of a Triangle | 三角形的面积公式

Besides the standard ½ × base × height, the trigonometric area formula is crucial when two sides and the included angle are known.

除了标准的 ½×底×高，当已知两边及夹角时，三角面积公式便极为关键。

Area = ½ ab sin C

面积 = ½ ab sin C

Here C is the angle between sides a and b. This formula also helps in problems where the area is given and you need to find an unknown side or angle.

此处的 C 是边 a 和 b 的夹角。在给定面积求未知边或角的题目中，这个公式也能派上用场。

9. 3D Trigonometry | 三维三角函数

Three‑dimensional problems often involve identifying right‑angled triangles within a cuboid, pyramid or prism. You may need to find the angle between a line and a plane, or the angle between two planes.

三维问题常常涉及在长方体、棱锥或棱柱内识别直角三角形。你可能需要求直线与平面的夹角，或两个平面之间的夹角。

Draw a clear 2D net or extract the relevant triangle from the solid. Use Pythagoras’ theorem and basic trig ratios repeatedly to find missing lengths before calculating the required angle. The angle between a line and a plane is the angle the line makes with its projection on the plane.

画出清晰的二维展开图或从立体中抽出相关三角形。先用勾股定理和基本三角比求出缺失的边长，再计算所要求的角。直线与平面的夹角就是该直线与其在平面上的投影之间的夹角。

10. Trigonometric Identities | 三角恒等式

For the extended syllabus, you must know and be able to use two fundamental identities: the Pythagoras identity and the relationship between sine and cosine of complementary angles.

对于拓展课程，你必须掌握并能运用两个基本恒等式：勾股恒等式以及互余角的正弦与余弦关系。

sin² θ + cos² θ = 1

sin(90° − θ) = cos θ

These identities are useful for simplifying expressions, proving other relationships, and solving equations without a calculator. For example, if sin θ = 0.6, you can quickly find cos θ = √(1 − 0.6²) = 0.8.

这些恒等式可用于化简表达式、证明其他关系以及在无计算器的情况下解方程。例如，若 sin θ = 0.6，你能快速求得 cos θ = √(1 − 0.6²) = 0.8。

11. Problem‑Solving Strategies | 解题策略

Many IGCSE trigonometry questions combine multiple techniques. Read the question carefully, sketch a labelled diagram, and identify which rules apply in sequence. Always round only at the final step to maintain accuracy.

许多 IGCSE 三角函数题综合了多种技巧。仔细读题，画出带标注的示意图，并依序判断应用哪些定理。始终在最后一步才进行舍入，以保持解题精度。

Common extended‑syllabus tasks involve finding a distance or angle in a real‑world context, such as a ship’s bearing from a lighthouse or the height of a building using two angles of elevation measured from different points.

拓展课程中常见的任务是在实际情境中求距离或角度，例如求船舶相对于灯塔的方位角，或通过两点测得的仰角求建筑高度。

Check the reasonableness of your answer — a side length in a triangle cannot be negative, and an angle in a triangle must lie between 0° and 180°. Mastering these key points will give you confidence and a strong foundation for the trigonometry section of your IGCSE CIE Mathematics exam.

检验答案的合理性——三角形的边长不能为负，三角形的内角必须在 0° 到 180° 之间。掌握这些关键考点将使你在 IGCSE CIE 数学考试的三角函数部分充满信心、基础扎实。

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IGCSE CIE Mathematics: Trigonometry Key Points | IGCSE CIE 数学：三角函数 考点精讲