📚 Pythagoras' Theorem and Right-Angled Trigonometry | 勾股定理与直角三角形三角比
Right-angled triangles appear in every IGCSE Mathematics exam, and they link two of the most powerful tools in geometry: Pythagoras' theorem and the trigonometric ratios. This revision guide walks through the key formulas, typical question types and the exact steps examiners expect when you find a missing side or angle.
直角三角形在每套 IGCSE 数学试卷中都会出现，它把几何中两个最强大的工具——勾股定理和三角比——连接在一起。本复习指南将讲解关键公式、常见题型以及求未知边长或角度时考官期望的完整步骤。
1. Why This Topic Matters | 为什么这个主题重要
In IGCSE, questions on Pythagoras and trigonometry often carry 3–6 marks, either as standalone problems or embedded in bearing, circle and 3D applications. Understanding the difference between the two methods is crucial: Pythagoras links all three sides of a right triangle, while trigonometric ratios link angles to side pairs.
在 IGCSE 中，勾股定理与三角比题目通常占 3–6 分，可能是独立题，也可能嵌在方位角、圆和三维应用题中。理解两种方法的区别非常重要：勾股定理联系直角三角形的三条边，而三角比把角度与两条边的比值联系起来。
Strong exam technique includes labelling sides, choosing the correct formula and showing every step of working. Even a small labelling error can lead to the wrong ratio and a lost answer.
扎实的考试技巧包括标注边长、选择正确公式以及展示完整的计算步骤。即使只是标错边长，也可能导致选错三角比并丢失答案分。
2. Pythagoras' Theorem: The Core Rule | 勾股定理：核心法则
For any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the hypotenuse is c and the shorter sides are a and b, then the rule is:
对于任意直角三角形，斜边的平方等于另外两条直角边的平方和。如果用 c 表示斜边，用 a 和 b 表示两条直角边，则公式为：
a² + b² = c²
The hypotenuse is always the longest side and lies opposite the right angle. In a diagram, it is the side that does not touch the right angle. It is very common for an exam question to label the right angle and ask you to identify the hypotenuse before calculating.
斜边永远是最长的边，并且位于直角的对面。在图中，它是不接触直角的那条边。考试题通常会标出直角，并要求你在计算前先找出斜边。
3. Finding a Missing Hypotenuse | 求斜边
, simplify the sum, then take the square root. Do not round until the final answer unless the question says otherwise.
当两条直角边已知时，将数值代入 a² + b² = c²，先求平方和，再开平方。除非题目另有说明，否则不要提前四舍五入。
Example: A right-angled triangle has legs 6 cm and 8 cm. Find the hypotenuse.
例题：一个直角三角形两条直角边分别为 6 cm 和 8 cm，求斜边。
c² = 6² + 8² = 36 + 64 = 100 ⇒ c = √100 = 10 cm
Always state the final answer with correct units. In IGCSE, missing units on a length can lose accuracy marks, so write cm, m or mm exactly as given in the question.
最终答案务必带上正确单位。在 IGCSE 中，长度答案缺少单位可能丢失准确度分，因此要按题目给出的单位书写 cm、m 或 mm。
4. Finding a Missing Short Side | 求直角边
If the hypotenuse and one leg are known, rearrange the formula before you calculate. Since the hypotenuse is the largest side, the missing short side is found by subtracting the known square from the hypotenuse square.
如果已知斜边和一条直角边，需要先把公式变形，再计算。因为斜边是最长的边，所以求未知直角边时要用斜边的平方减去已知直角边的平方。
a² = c² − b²
Example: A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.
例题：一个直角三角形斜边为 13 cm，一条直角边为 5 cm，求另一条直角边。
a² = 13² − 5² = 169 − 25 = 144 ⇒ a = √144 = 12 cm
Subtracting in the correct order is essential. If you subtract the hypotenuse square from the short side square, you will get a negative number and cannot take a real square root.
减去时必须注意顺序。如果用短边的平方减去斜边的平方，会得到负数，无法在实数范围内开平方。
5. Trigonometric Ratios: SOH CAH TOA | 三角比：SOH CAH TOA
In a right-angled triangle, the sine, cosine and tangent of an angle θ are defined by three side ratios. The memory aid SOH CAH TOA helps you recall them quickly.
在直角三角形中，角 θ 的正弦、余弦和正切由三个边比定义。助记口诀 SOH CAH TOA 可以帮助你快速记住它们。
| Ratio | Formula | Memory aid |
| Sine | sin θ = opposite / hypotenuse | SOH |
| Cosine | cos θ = adjacent / hypotenuse | CAH |
| Tangent | tan θ = opposite / adjacent | TOA |
The opposite side is directly across from the angle θ. The adjacent side is next to θ but is not the hypotenuse. These labels change if you move to the other acute angle, so always label from the angle you are using.
对边是角 θ 正对面的边。邻边紧挨着 θ 但不是斜边。这些标注会随另一个锐角而变化，因此必须根据你正在使用的角来重新标注。
6. Choosing the Correct Ratio | 选择正确的三角比
Label the three sides first: opposite, adjacent and hypotenuse. Then decide which ratio uses the two sides you know or the side you need and the side you know.
首先标注三条边：对边、邻边和斜边。然后判断哪个三角比能用到你已知的边以及你需要求出的那条边。
If you know the angle and the adjacent side and need the hypotenuse, use cosine. If you know the angle and the opposite side and need the adjacent side, use tangent.
如果已知角度和邻边，要求斜边，则使用余弦。如果已知角度和对边，要求邻边，则使用正切。
Drawing a quick sketch and writing SOH CAH TOA beside it can prevent many common errors. In word problems, the words 'opposite' and 'adjacent' may not appear, so you must interpret the diagram correctly.
快速画一个草图，并在旁边写出 SOH CAH TOA，可以避免许多常见错误。在应用题中，题目不一定会直接出现“对边”和“邻边”这些词，因此你必须正确理解图形。
7. Finding an Unknown Side | 求未知边长
Set up an equation with the chosen ratio, substitute the known angle and side, then solve the equation. Make sure your calculator is in degree mode, because IGCSE angles are usually measured in degrees unless stated otherwise.
用选定的三角比建立方程，代入已知角度和边长，然后解方程。请确保计算器处于角度制模式，因为 IGCSE 中除非特别说明，角度通常以度为单位。
Example: If sin 30° = x / 12, multiply both sides by 12:
例题：如果 sin 30° = x / 12，两边同时乘以 12：
x = 12 sin 30° = 12 × 0.5 = 6
may need an extra rearrangement step. For example, cos 40° = 8 / x gives x = 8 / cos 40°.
当未知数在分母上时，可能需要多一步变形。例如 cos 40° = 8 / x，变形后得到 x = 8 / cos 40°。
Show every step clearly. Even if your final answer is incorrect, IGCSE examiners can award method marks for substituting into the correct ratio and rearranging properly.
每一步都要写得清楚。即使最终答案错误，只要正确代入三角比并合理变形，IGCSE 考官仍会给予方法分。
8. Finding an Unknown Angle | 求未知角度
angle, use the inverse trigonometric function on your calculator. If sin θ = 5 / 13, then:
当已知两条边要求角度时，使用计算器上的反三角函数。若 sin θ
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