📚 IGCSE OCR Maths: Trigonometry Key Points | IGCSE OCR 数学：三角函数 考点精讲
Trigonometry is a branch of mathematics that studies the relationships between the angles and side lengths of triangles. In IGCSE OCR Maths, trigonometry is a core topic that appears in both calculator and non‑calculator papers. You need to understand trigonometric ratios, exact values of special angles, graph properties, and how to apply the sine and cosine rules to non‑right‑angled triangles. This article covers every key point you must master for the exam, with clear explanations and examples in both English and Chinese.
三角函数是数学中研究三角形角度与边长关系的一个分支。在 IGCSE OCR 数学中，三角函数是一个核心考点，会出现在可使用计算器与不可使用计算器的试卷中。你需要掌握三角比、特殊角的精确值、图像性质，以及如何将正弦定理和余弦定理应用到非直角三角形中。本文以中英双语讲解每个你必须掌握的考点，并配有清晰的解释和例题。
1. Introduction to Trigonometry | 三角函数简介
Trigonometry literally means ‘measuring triangles’. In IGCSE, we work with both right‑angled triangles and any triangle. The three basic trigonometric functions — sine, cosine and tangent — are defined as ratios of sides in a right‑angled triangle relative to a given acute angle. These ratios stay constant for a fixed angle, no matter how large the triangle is. Understanding this foundation is essential before moving on to harder applications like the sine rule or 3D problems.
三角函数的字面意思是“测量三角形”。在 IGCSE 中，我们会接触直角三角形和任意三角形。三个基本三角函数——正弦、余弦和正切——是根据直角三角形中一个给定锐角的边长比定义的。对于固定角度，无论三角形有多大，这些比值都保持不变。在进入正弦定理或三维问题等更难的考点之前，理解这个基础至关重要。
2. Trigonometric Ratios: SOH CAH TOA | 三角函数比：SOH CAH TOA
The mnemonic SOH CAH TOA helps you remember the three basic ratios for a right‑angled triangle:
SOH: sin θ = Opposite / Hypotenuse
CAH: cos θ = Adjacent / Hypotenuse
TOA: tan θ = Opposite / Adjacent
Always identify the hypotenuse (longest side, opposite the right angle), the opposite side (facing angle θ) and the adjacent side (next to θ) before substituting into the formula.
记忆口诀 SOH CAH TOA 能帮你记住直角三角形的三个基本比：
SOH：sin θ = 对边 / 斜边
CAH：cos θ = 邻边 / 斜边
TOA：tan θ = 对边 / 邻边
在代入公式之前，一定要先找出斜边（最长边，直角对边）、对边（面对角 θ）和邻边（紧靠 θ 的边）。
sin θ = Opposite / Hypotenuse cos θ = Adjacent / Hypotenuse tan θ = Opposite / Adjacent
3. Exact Values for Special Angles | 特殊角的精确值
In non‑calculator papers, you must know the exact trigonometric values for 0°, 30°, 45°, 60° and 90°. These can be derived from an equilateral triangle and a square. The key values are:
在不可使用计算器的试卷中，你必须记住 0°、30°、45°、60° 和 90° 的精确三角值。这些值可以从一个等边三角形和一个正方形推导出来。关键数值如下：
| Angle 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 |
Write the values in simplest surd form. For example, tan 30° = 1/√3 can be rationalised to √3/3. Do not give decimal approximations unless the question specifically asks for them.
请将数值写成最简根式。例如，tan 30° = 1/√3 可以有理化为 √3/3。除非题目明确要求，否则不要给出小数近似值。
4. The Sine, Cosine and Tangent Graphs | 正弦、余弦和正切图像
The graphs of y = sin x, y = cos x and y = tan x have distinct shapes that you must be able to sketch and interpret. The sine and cosine graphs are waves with amplitude 1 and period 360°. The sine graph starts at (0,0), rises to a maximum of 1 at 90°, returns to 0 at 180°, reaches a minimum of −1 at 270° and ends at 0 after 360°. The cosine graph starts at (0,1) and has the same amplitude and period. The tangent graph has period 180° and vertical asymptotes at x = ±90°, ±270°, where it is undefined. Knowing these graphs helps in solving trigonometric equations and understanding transformations.
y = sin x、y = cos x 和 y = tan x 的图像各具特色，你需要能够画出示意图并会读图。正弦和余弦图像是振幅为 1、周期为 360° 的波形。正弦图像从 (0,0) 出发，在 90° 达到最大值 1，在 180° 回到 0，在 270° 达到最小值 −1，360° 后回到 0。余弦图像从 (0,1) 出发，振幅和周期相同。正切图像周期为 180°，在 x = ±90°、±270° 等处有垂直渐近线，该处函数无定义。掌握这些图像有助于解三角方程和理解图像变换。
5. Solving Right‑Angled Triangles | 解直角三角形
To find an unknown side, choose the ratio that links the known angle with the known side and the unknown side. Set up the equation and solve. To find an unknown angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹). Always check that your answer makes sense: the longest side is the hypotenuse, and acute angles must be less than 90°. Labelling sides first is a good habit.
求未知边时，要选择能关联已知角、已知边和未知边的三角比，列出方程并求解。求未知角时，使用反三角函数（sin⁻¹、cos⁻¹、tan⁻¹）。务必检查你的答案是否合理：最长的边是斜边，锐角必须小于 90°。养成先标注各边的习惯会很有帮助。
Example: In a right‑angled triangle, hypotenuse = 12 cm, θ = 40°, find the opposite side.
sin 40° = Opposite / 12 → Opposite = 12 × sin 40°
6. Angles of Elevation and Depression | 仰角和俯角
The angle of elevation is the angle measured upwards from the horizontal to a line of sight. The angle of depression is measured downwards from the horizontal. Both angles are always made with the horizontal and are equal when the line of sight is parallel to the ground. These situations often involve right‑angled triangles where the vertical height and horizontal distance are two sides. Draw a clear diagram with the horizontal line, then apply SOH CAH TOA.
仰角是从水平线向上测量的视线夹角。俯角是从水平线向下测量的夹角。这两个角始终与水平线构成，且当视线与地面平行时，仰角和俯角相等。这类情境往往构成直角三角形，其中垂直高度和水平距离是两条边。先画出带有水平线的清晰示意图，再套用 SOH CAH TOA。
7. Bearings | 方位角
Bearings are a way of describing direction, measured clockwise from North, and always given as three figures (e.g. 045°, 120°, 325°). Trigonometry problems with bearings often require you to identify right‑angled triangles or apply the sine/cosine rules. Always draw a North line at each point and mark the angle carefully. Translate the bearing into an acute angle inside a triangle before using trigonometric ratios.
方位角是一种描述方向的方法，从正北顺时针测量，并总是以三位数字表示（如 045°、120°、325°）。含方位角的三角问题通常需要你先识别出直角三角形，或应用正弦定理和余弦定理。务必在每个点上画出指北线，并仔细标注角度。在套用三角比之前，先将方位角转化为三角形内部的锐角。
8. The Sine Rule | 正弦定理
For any non‑right‑angled triangle, the sine rule connects sides and angles:
对于任意非直角三角形，正弦定理将边长与角度关联起来：
a / sin A = b / sin B = c / sin C
Or equivalently: sin A / a = sin B / b = sin C / c. Use the sine rule when you know two angles and one side (AAS) or two sides and a non‑included angle (SSA). Be careful with the SSA case — there can be two possible solutions (the ambiguous case) if the given angle is acute and the side opposite it is shorter than the other given side. Always check whether the problem expects one or two triangles.
或等价形式：sin A / a = sin B / b = sin C / c。当已知两个角和一个边（AAS），或已知两边和一个非夹角（SSA）时，可应用正弦定理。SSA 情形要格外小心——若给定角为锐角且其对照边比另一给定边短，可能出现两个解（歧义情况）。务必确认题目要求的是一个还是两个三角形。
9. The Cosine Rule | 余弦定理
The cosine rule is used when the sine rule is not applicable — typically when you know two sides and the included angle (SAS) or all three sides (SSS). The formula has two common forms:
当正弦定理不适用时，就需要余弦定理——通常是已知两边及其夹角（SAS），或已知三边（SSS）。公式有两种常见形式：
a² = b² + c² − 2bc cos A
cos A = (b² + c² − a²) / (2bc)
The first form finds a side; the second finds an angle. Always label the side and angle opposite each other with the same letter. Work step by step: substitute values carefully, then use square roots or inverse cosine appropriately. The cosine rule can also handle obtuse angles seamlessly, as cos of an obtuse angle is negative.
第一个形式用来求边，第二个用来求角。始终给相对的边和角标上相同字母。按步骤计算：仔细代入数值，然后相应地求平方根或反余弦。余弦定理也能无缝处理钝角，因为钝角的余弦值为负。
10. Area of a Triangle: ½ ab sin C | 三角形面积公式：½ ab sin C
When the perpendicular height of a triangle is not known, the area can be found using two sides and the included angle:
当三角形的高未知时，可用两边及其夹角来求面积：
Area = ½ × a × b × sin C
Here a and b are any two sides, and C is the angle between them. This formula works for any triangle, not just right‑angled. In exam questions, you may need to combine this with the sine rule or cosine rule to find a missing angle or side first.
这里 a 和 b 是任意两边，C 是它们的夹角。这个公式对任意三角形都适用，不仅是直角三角形。在考题中，你可能需要先结合正弦或余弦定理求出一个未知角或边，再计算面积。
11. Trigonometry in 3D | 三维三角函数
3D trigonometry problems involve pyramids, cuboids, prisms, or wedges where you need to find lengths or angles in different planes. The key technique is to identify the relevant right‑angled triangles within the 3D shape. For example, the angle between a line and a plane is the angle between the line and its projection onto the plane. The angle between two planes is found by drawing a line perpendicular to their common edge and measuring the angle in a cross‑section. Break the problem into 2D slices, label all given lengths, and use Pythagoras’ theorem alongside trigonometric ratios.
三维三角问题涉及棱锥、长方体、棱柱或楔体，需要求不同平面上的长度或角度。关键技巧是在三维图形中识别出相关的直角三角形。例如，直线与平面之间的夹角就是该直线与其在平面上投影线之间的角。两个平面之间的角通过画出垂直于其公共棱的直线，并在截面中测量该角来求。要把问题拆分成二维切片来思考，标出所有给定长度，并使用勾股定理和三角比。
12. Summary and Exam Tips | 总结与应试技巧
Mastering trigonometry for IGCSE OCR Maths means practising with precision. Memorise SOH CAH TOA and exact values. Know when to use the sine rule versus the cosine rule. Always draw a labelled diagram, even if the question provides one. Check that your angle answers are within a sensible range (0° to 180° for triangles, 0° to 90° for acute angles). In non‑calculator questions, leave answers in exact surd form. For bearings, give three‑digit answers. And in 3D, sketch the right‑angled triangle separately. These habits will save you marks and build confidence.
要在 IGCSE OCR 数学中掌握三角函数，就必须精准练习。牢记 SOH CAH TOA 和特殊角的精确值。清楚何时用正弦定理，何时用余弦定理。即使题目给出了图，你也一定要自己画一个标注好的示意图。检查你求出的角度是否在合理范围内（三角形中角在 0° 到 180° 之间，锐角在 0° 到 90° 之间）。在不可使用计算器的题目中，答案要保留根式精确值。方位角要给出三位数答案。在三维题中，单独画出直角三角形草图。这些好习惯会帮你减少失分，增强信心。
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