📚 Mastering Pythagoras and Right-Angled Trigonometry for IGCSE | 掌握 IGCSE 勾股定理与直角三角形三角学
In IGCSE Mathematics, Pythagoras' theorem and right-angled trigonometry are essential tools for solving problems involving side lengths, angles, distances and real-world measurements. These topics appear frequently in both Core and Extended papers, often combined with bearings, areas and three-dimensional shapes. This article reviews the key concepts, worked examples and common exam techniques you need to master this topic.
在 IGCSE 数学中，勾股定理和直角三角形三角比是解决边长、角度、距离及实际测量问题的基础工具。这些内容在核心和扩展试卷中频繁出现，经常与方位角、面积和三维图形综合考查。本文复习关键概念、典型例题和常见考试技巧，帮助你掌握该主题。
1. The Pythagoras Theorem: Statement and Conditions | 勾股定理：表述与适用条件
The Pythagoras theorem only applies to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. If c is the hypotenuse and a and b are the two shorter sides, then:
勾股定理只适用于直角三角形。它表明斜边的平方等于另外两条直角边的平方之和。如果 c 是斜边，a 和 b 是两条直角边，则有：
c² = a² + b²
The hypotenuse is always the longest side and is located opposite the right angle. You must identify the right angle first and label the sides correctly before applying the formula.
斜边总是最长边，位于直角对面。在应用公式之前，必须先确定直角并正确标注各边。
2. Finding the Hypotenuse | 求斜边
When both shorter sides are known, you can find the hypotenuse by squaring each shorter side, adding the results and taking the square root. For example, a right-angled triangle has legs of 6 cm and 8 cm. The hypotenuse c satisfies:
当两条直角边已知时，可以通过平方各直角边、相加并取平方根来求斜边。例如，一个直角三角形两直角边为 6 cm 和 8 cm，斜边 c 满足：
c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm
Always check that your final answer is longer than both shorter sides. If it is not, you have probably labelled the hypotenuse incorrectly.
始终检查最终答案是否比两条直角边长。若不是，你可能错误地标注了斜边。
3. Finding a Shorter Side | 求直角边
If the hypotenuse and one shorter side are known, rearrange the theorem to find the missing shorter side. For example, if the hypotenuse is 13 cm and one leg is 5 cm, the missing side a is given by:
如果已知斜边和一条直角边，可以重新整理定理来求缺失的直角边。例如，斜边为 13 cm，一条直角边为 5 cm，则缺失边 a 满足：
a² = c² - b² = 13² - 5² = 169 - 25 = 144, so a = √144 = 12 cm
Notice that subtraction is used, not addition. Many students lose marks by adding when they should subtract. Think carefully about which side is being found.
注意这里使用减法而非加法。许多学生因该用减法时用了加法而丢分。要仔细判断要求的是哪条边。
4. Pythagorean Triples and Applications | 勾股数与应用
Some common whole-number triples make calculations quicker. You should recognise these sets and their multiples:
一些常见的整数勾股数可以使计算更快。你应识别这些数组及其倍数：
- 3-4-5 (and 6-8-10, 9-12-15) | 3-4-5（以及 6-8-10、9-12-15）
- 5-12-13 (and 10-24-26) | 5-12-13（以及 10-24-26）
- 7-24-25 | 7-24-25
- 8-15-17 | 8-15-17
Pythagoras' theorem is also used to solve problems with rectangles, isosceles triangles, circles and coordinate geometry. For example, the diagonal of a rectangle with length 9 cm and width 12 cm is √9² + 12² = 15 cm by the 3-4-5 triple scaled by 3.
勾股定理也用于解决矩形、等腰三角形、圆和坐标几何问题。例如，长 9 cm、宽 12 cm 的矩形对角线为 √9² + 12² = 15 cm，这是 3-4-5 勾股数扩大 3 倍。
5. Introduction to Trigonometric Ratios | 三角比引入
The three basic trigonometric ratios relate the angles and side lengths of a right-angled triangle. For an acute angle θ, the sides are labelled as the opposite side, the adjacent side and the hypotenuse.
三个基本三角比将直角三角形的角度与边长联系起来。对于锐角 θ，各边分别标记为对边、邻边和斜边。
sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent
These ratios are constant for a given angle, regardless of the size of the triangle. This allows you to find missing lengths and angles when at least one side and one extra piece of information are known.
这些比值对于给定的角是恒定的，与三角形大小无关。这使你在已知至少一条边和另一个条件时可以求缺失的边长或角度。
6. SOH-CAH-TOA and Choosing the Ratio | SOH-CAH-TOA 与选择三角比
The acronym SOH-CAH-TOA helps you remember which sides are used in each ratio: Sin is Opposite over Hypotenuse, Cos is Adjacent over Hypotenuse, Tan is Opposite over Adjacent.
助记词 SOH-CAH-TOA 帮助你记住每个比值使用哪些边：sin 是对边比斜边，cos 是邻边比斜边，tan 是对边比邻边。
| Ratio | 比值 | Formula | 公式 | Used when | 使用场景 |
|---|---|---|
| Sine | sin θ = O / H | Opposite and hypotenuse involved |
| Cosine | cos θ = A / H | Adjacent and hypotenuse involved |
| Tangent | tan θ = O / A | Opposite and adjacent involved |
Choose the ratio that connects the side you know, the side you want and the angle you are using. If you know two sides and need an angle, use the inverse function.
选择能连接已知边、所求边和所使用角的比值。如果你知道两条边并需求角度，则使用反函数。
7. Calculating Missing Sides | 计算缺失边长
To find a missing side, substitute the known angle and side into the appropriate ratio, then rearrange. For example, a right-angled triangle has an angle of 35° and the adjacent side is 7 cm. To find the opposite side x, use:
要求缺失边，将已知角和边代入适当的比值，然后重新整理。例如，一个直角三角形有一个 35° 的角，邻边为 7 cm。求对边 x，使用：
tan 35° = x ÷ 7, so x = 7 × tan 35° ≈ 4.9 cm
Make sure your calculator is in degree mode, not radian mode. In IGCSE examinations, angles are usually given in degrees unless stated otherwise.
确保计算器处于度数模式，而不是弧度模式。在 IGCSE 考试中，除非特别说明，角度通常以度为单位。
8. Calculating Missing Angles | 计算缺失角度
verse trigonometric functions sin⁻¹, cos⁻¹ or tan⁻¹. For example, if sin θ = 5 ÷ 13, then:
如果你知道两条边并需求角度，使用反三角函数 sin⁻¹、cos⁻¹ 或 tan⁻¹。例如，如果 sin θ = 5 ÷ 13，则：
θ = sin⁻¹(5 ÷ 13) ≈ 22.6°
Round your answer to one decimal place unless the question asks for a different accuracy. Always show the ratio before using the inverse function so the examiner can follow your working.
除非题目要求不同精度，通常将答案保留一位小数。始终在应用反函数之前展示比值，以便阅卷人能看懂你的步骤。
9. Angles of Elevation and Depression | 仰角与俯角
Angles of elevation and depression are measured from the horizontal. The angle of elevation is the angle above the horizontal when looking up at an object. The angle of depression is the angle below the horizontal when looking down at an object.
仰角和俯角是从水平线量起的角。仰角是向上看物体时水平线上方的角。俯角是向下看物体时水平线下方的角。
These angles are often used in word problems involving cliffs, buildings, planes and ships. Drawing a clear right-angled triangle and labelling the horizontal and vertical sides will help you set up the correct trigonometric ratio.
这些角常用于涉及悬崖、建筑物、飞机和船只的应用题。画一个清晰的直角三角形并标注水平边和垂直边，有助于建立正确的三角比。
10. Problem-Solving Strategy and Common Pitfalls | 问题解决策略与常见错误
Follow a consistent four-step approach for every right-angled triangle problem: identify the right angle, label the sides, choose the ratio or theorem, and solve. Write down the formula before substituting values.
对每个直角三角形问题遵循一致的四步法：确定直角，标注各边，选择比值或定理，然后求解。在代入数值前先写出公式。
Common pitfalls include using Pythagoras when the triangle is not right-angled, mixing up adjacent and opposite sides, using the wrong mode on a calculator and forgetting to square root at the end. Avoid these by checking each step carefully.
常见错误包括在非直角三角形上使用勾股定理、混淆邻边和对边、计算器模式错误、最后忘记开平方。仔细检查每一步可以避免这些错误。
11. Past Paper Style Questions | 真题风格练习
Try this exam-style question: A ladder 5 m long leans against a vertical wall. The foot of the ladder is 2 m from the wall. How high up the wall does the ladder reach? Give your answer to two decimal places.
尝试这道考试风格题目：一架 5 m 长的梯子斜靠在竖直墙上。梯脚距墙 2 m。梯子到达墙上的高度是多少？答案保留两位小数。
Using Pythagoras, the height h satisfies h² = 5² - 2² = 25 - 4 = 21, so h = √21 ≈ 4.58 m.
使用勾股定理，高度 h 满足 h² = 5² - 2² = 25 - 4 = 21，所以 h = √21 ≈ 4.58 m。
12. Summary and Exam Tips | 总结与考试技巧
Pythagoras' theorem finds missing side lengths in right-angled triangles, while trigonometric ratios find missing sides and angles. Learn SOH-CAH-TOA, practise with calculator work and always draw a labelled diagram.
勾股定理求直角三角形中的缺失边长，而三角比可以求缺失的边和角。掌握 SOH-CAH-TOA，练习计算器操作，并始终画带标注的图形。
In the exam, show all working, check your calculator mode, and make sure your answer is sensible. A missing hypotenuse should be longer than either leg, and a missing angle should be between 0° and 90° in right-angled trigonometry.
在考试中，展示所有步骤，检查计算器模式，并确保答案合理。缺失的斜边应比任何直角边长，直角三角形中的缺失角应在 0° 到 90° 之间。
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