📚 IGCSE Maths: Trigonometry in Right-Angled Triangles | IGCSE 数学：直角三角形中的三角学
Trigonometry in right-angled triangles is one of the most practical and heavily examined topics in IGCSE Mathematics. It connects angle measures with side lengths through the sine, cosine and tangent ratios, and it builds directly on Pythagoras’ theorem. In this revision guide, we will work through the key methods, common problem types and exam strategies you need to use these ideas confidently.
直角三角形中的三角学是 IGCSE 数学中非常实用且高频考查的主题之一。它通过正弦、余弦和正切比将角度与边长联系起来，并直接建立在勾股定理的基础上。在本复习指南中，我们将梳理关键方法、常见题型和考试策略，帮助你自信运用这些知识。
1. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies only to right-angled triangles. If c is the hypotenuse, the side opposite the right angle, then the square of the hypotenuse equals the sum of the squares of the other two sides.
勾股定理只适用于直角三角形。如果 c 是斜边，也就是直角所对的边，那么斜边的平方等于另外两条边的平方和。
c² = a² + b²
c = √(a² + b²)
Before using trigonometric ratios, you should always be able to identify the hypotenuse, the opposite side and the adjacent side relative to a given angle. This labelling is the foundation for everything that follows.
在使用三角比之前，你必须能够根据给定的角识别出斜边、对边和邻边。这种边的标记是后续所有内容的基础。
2. Trigonometric Ratios: Sine, Cosine and Tangent | 正弦、余弦与正切
For a right-angled triangle, the three basic trigonometric ratios are defined as follows:
对于直角三角形，三个基本三角比定义如下：
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
These ratios depend only on the angle θ, not on the size of the triangle. This is why trigonometric tables and calculator functions give consistent values for a given angle.
这些比值只取决于角 θ，而与三角形的大小无关。这就是为什么三角函数表和计算器对同一个角会给出相同数值的原因。
3. Choosing the Correct Ratio | 选择正确的三角比
To decide which ratio to use, first label the sides relative to the angle you are interested in. Then look at the two sides involved in the problem: if you know or need the opposite and hypotenuse, use sine; if adjacent and hypotenuse, use cosine; if opposite and adjacent, use tangent.
要决定使用哪个比值，首先根据你关注的角标记各边。然后看问题中涉及的两条边：如果知道或要求对边和斜边，就用正弦；如果是邻边和斜边，就用余弦；如果是对边和邻边，就用正切。
A useful memory aid is SOH CAH TOA, where SOH stands for Sine equals Opposite over Hypotenuse, CAH for Cosine equals Adjacent over Hypotenuse, and TOA for Tangent equals Opposite over Adjacent.
一个实用的记忆方法是 SOH CAH TOA：SOH 表示正弦等于对边比斜边，CAH 表示余弦等于邻边比斜边，TOA 表示正切等于对边比邻边。
4. Finding an Unknown Side | 求未知边
When finding an unknown side, choose the ratio that links the given side and the required side. Substitute the known values into the formula, then solve the resulting equation algebraically.
求未知边时，选择能联系已知边和所求边的三角比。将已知数值代入公式，然后通过代数运算求解。
For example, if a right-angled triangle has an angle of 35° and a hypotenuse of 12 cm, the side opposite the angle is found using sin 35° = x ÷ 12. Multiplying both sides by 12 gives x = 12 × sin 35°.
例如，一个直角三角形有一个角为 35°，斜边为 12 cm，那么这个角的对边可以用 sin 35° = x ÷ 12 来求。两边同时乘以 12，得到 x = 12 × sin 35°。
5. Finding an Unknown Angle | 求未知角
If you know two sides of a right-angled triangle and need to find an angle, use the inverse trigonometric functions. On most calculators these are labelled sin⁻¹, cos⁻¹ and tan⁻¹.
如果知道直角三角形的两条边并需要求某个角，可以使用反三角函数。在大多数计算器上，它们标记为 sin⁻¹、cos⁻¹ 和 tan⁻¹。
For instance, if the opposite side is 7 cm and the adjacent side is 10 cm, then tan θ = 7 ÷ 10. Therefore θ = tan⁻¹(0.7), which is approximately 35.0°.
例如，如果对边为 7 cm，邻边为 10 cm，那么 tan θ = 7 ÷ 10。因此 θ = tan⁻¹(0.7)，结果约为 35.0°。
Always check that your calculator is in degree mode because IGCSE trigonometry questions usually require answers in degrees.
一定要检查计算器是否处于角度模式，因为 IGCSE 三角学题目通常要求以度为单位给出答案。
6. Angles of Elevation and Depression | 仰角与俯角
An angle of elevation is the angle measured upwards from the horizontal, while an angle of depression is the angle measured downwards from the horizontal. Both angles are commonly used in practical trigonometry problems involving buildings, aeroplanes and ladders.
仰角是从水平线向上测量的角，俯角是从水平线向下测量的角。这两种角常见于涉及建筑物、飞机和梯子等实际三角学问题中。
When solving such problems, draw a clear diagram and translate the written description into a right-angled triangle. The angle of depression from one point is equal to the angle of elevation from the corresponding point because they are alternate angles on parallel lines.
解决这类问题时，要画出清晰的图，把文字描述转化为直角三角形。从某点测得的俯角与对应点测得的仰角相等，因为它们是平行线上的内错角。
7. Bearings and Trigonometry | 方位角与三角学
Bearings are measured clockwise from north and are always given as three digits. Trigonometry is often used with bearings to find distances or to determine unknown bearings after a journey.
方位角是从正北方向顺时针测量的角，并且总是用三位数字表示。三角学常与方位角结合使用，用来求距离或确定一段行程后的未知方位角。
To convert a bearing into an angle inside a triangle, draw a north-south line at the point of reference and find the angle between the path and this line. Then use the sine rule, cosine rule or right-angled triangle ratios as appropriate.
要把方位角转化为三角形内部的角，可以在参考点画出南北方向线，然后找出路径与这条线之间的夹角。接着根据情况使用正弦定理、余弦定理或直角三角形中的三角比。
8. Exact Values for 30°, 45° and 60° | 特殊角的精确值
IGCSE examination papers often ask for exact answers rather than calculator approximations. You must know the exact values of sine, cosine and tangent for 30°, 45° and 60°.
IGCSE 考试经常要求给出精确值，而不是计算器的近似值。你必须记住 30°、45° 和 60° 的正弦、余弦和正切的精确值。
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
These exact values come from two special triangles: the equilateral triangle divided into two 30°-60°-90° triangles, and the right-angled isosceles triangle with angles 45°-45°-90°.
这些精确值来自两个特殊三角形：等边三角形分成的两个 30°-60°-90° 三角形，以及角为 45°-45°-90° 的等腰直角三角形。
9. Multi-Step Problems and Problem Solving | 多步问题与解题策略
Many IGCSE trigonometry questions require more than one step. You may need to find a side using Pythagoras’ theorem first, then use that side to find an angle, or combine two different right-angled triangles that share a common side.
许多 IGCSE 三角学题目需要多个步骤。你可能需要先用勾股定理求出某条边，再用这条边去求一个角，或者结合两个共用一条边的直角三角形来解题。
A strong strategy is to label all given information on the diagram, identify which triangle you are working in at each stage, and write down the formula before substituting values. Keep intermediate results in the calculator memory or write them to at least four significant figures to avoid rounding errors.
一个有效的策略是在图上标出所有已知信息，确定每一步在处理哪个三角形，并在代入数值前写下公式。将中间结果保留在计算器内存中，或至少写出四位有效数字，以避免舍入误差。
10. Common Errors and How to Avoid Them | 常见错误及避免方法
One common mistake is mixing up the opposite and adjacent sides, especially when the triangle is rotated. Always imagine standing at the angle you are using and ask which side is directly opposite it.
一个常见错误是混淆对边和邻边，尤其是当三角形旋转后更是如此。始终想象自己站在所使用的角上，并判断哪条边正对着它。
Another error is forgetting to use the inverse trig function when finding an angle. If you have tan θ = 0.7, you must use tan⁻¹ on your calculator, not tan. Also make sure the calculator is in degree mode, not radian mode.
另一个错误是在求角时忘记使用反三角函数。如果 tan θ = 0.7，你必须在计算器上使用 tan⁻¹，而不是 tan。还要确保计算器处于角度模式，而不是弧度模式。
Rounding too early can change the final answer by more than one mark. Only round the final answer to the required degree of accuracy, usually three significant figures unless the question states otherwise.
过早舍入可能会使最终答案偏差超过一分。通常只在最后一步将答案舍入到要求的精度，一般为三位有效数字，除非题目另有说明。
11. Applications in 3D and Real-Life Contexts | 三维与实际生活中的应用
Trigonometry in right-angled triangles also appears in 3D problems, such as finding the length of a diagonal inside a cuboid or the angle between a line and a plane. The key is to identify the correct right-angled triangle within the 3D figure.
直角三角形中的三角学也出现在三维问题中，例如求长方体内部对角线的长度，或求一条直线与一个平面之间的夹角。关键是在三维图形中识别出正确的直角三角形。
Real-life contexts include ramps, roof slopes, navigation and surveying. In these questions, write down the model you are using, such as ‘the ladder makes a right angle with the ground after assuming the ground is horizontal’.
实际生活情境包括坡道、屋顶坡度、导航和测量。在这些问题中，写下你所使用的模型，例如“在假设地面是水平的前提下，梯子与地面构成直角”。
12. Exam Tips and Practice | 考试技巧与练习
In the exam, always show your working because method marks are awarded for the correct use of a trig ratio even if the final answer is wrong. Draw diagrams when one is not provided and label sides clearly.
考试时一定要写出解题过程，因为即使最终答案错误，正确使用三角比也能获得步骤分。没有图时要自己画图，并清晰地标出各边。
Practise past paper questions under timed conditions and mark them using the official mark scheme. Focus on questions that combine Pythagoras, trigonometry, bearings and angles of elevation because these appear frequently in IGCSE papers.
在计时条件下练习历年真题，并用官方评分标准进行批改。重点关注结合勾股定理、三角学、方位角和仰角的问题，因为它们在 IGCSE 试卷中频繁出现。
If you master labelling sides, selecting the correct ratio and using exact values, you will be able to tackle most right-angled triangle questions with confidence.
如果你能熟练掌握边的标记、正确选择三角比以及使用精确值，你就能自信地解答大多数直角三角形相关问题。
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