📚 Pythagoras’ Theorem & Right-Angled Trigonometry | 勾股定理与直角三角形三角学
Pythagoras’ theorem and right-angled trigonometry are two of the most frequently examined topics in IGCSE Mathematics. They allow you to calculate missing side lengths and angles in right-angled triangles, and they form the basis for many real-world problems in navigation, construction and physics. This article explains the key formulas, common problem types and exam strategies step by step.
勾股定理和直角三角学是IGCSE数学中最常考查的主题之一。它们可以帮助你计算直角三角形中缺失的边长和角度，也是许多现实问题（如导航、建筑和物理）的基础。本文逐步讲解关键公式、常见题型和考试策略。
1. Identifying Right-Angled Triangles | 识别直角三角形
A right-angled triangle has one interior angle exactly equal to 90°. The side opposite the right angle is called the hypotenuse; it is always the longest side. The other two sides are usually labelled as ‘opposite’ and ‘adjacent’ relative to a given acute angle.
直角三角形有一个内角恰好等于90°。直角所对的边称为斜边，它总是最长的边。另外两条边通常相对于某个给定的锐角标记为“对边”和“邻边”。
- The hypotenuse is always opposite the 90° angle.
- 斜边总是正对着90°角。
- The opposite and adjacent sides depend on which acute angle you choose.
- 对边和邻边取决于你选择哪个锐角。
2. Pythagoras’ Theorem Statement | 勾股定理的表述
In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If the hypotenuse is labelled c and the other sides are a and b, then:
在任何直角三角形中，斜边的平方等于另外两条边的平方之和。如果斜边标记为c，另外两条边标记为a和b，那么：
a² + b² = c²
This relationship only works for right-angled triangles. It is used to find an unknown side when the other two sides are known.
这个关系只适用于直角三角形。当已知另外两条边时，可以用它来求未知边。
3. Finding the Hypotenuse | 求斜边
If you know the two shorter sides, substitute them into the formula and solve for c. For example, if a = 3 cm and b = 4 cm:
如果你知道两条较短的边，就把它们代入公式并解出c。例如，如果a = 3厘米，b = 4厘米：
3² + 4² = c²
9 + 16 = c²
c² = 25
c = √25 = 5 cm
Always take the positive square root because length cannot be negative.
始终取正的平方根，因为长度不可能为负数。
4. Finding a Shorter Side | 求一条较短的边
When the unknown side is not the hypotenuse, rearrange the formula first. For example, if b is missing, write b² = c² − a². Suppose c = 13 m and a = 5 m:
当未知边不是斜边时，需要先重新整理公式。例如，如果缺失的是b，则写作b² = c² − a²。假设c = 13米，a = 5米：
b² = 13² − 5² = 169 − 25 = 144
b = √144 = 12 m
Check that your answer is smaller than the hypotenuse; if not, you have likely made a rearrangement error.
检查你的答案是否小于斜边；如果不是，你很可能在移项时出错了。
5. The Converse of Pythagoras’ Theorem | 勾股定理的逆定理
The converse states that if a triangle has sides of lengths a, b and c, and a² + b² = c², then the triangle is right-angled. This is useful for proving whether a given triangle contains a right angle.
逆定理指出，如果一个三角形的三边长度分别为a、b和c，并且a² + b² = c²，那么这个三角形就是直角三角形。这可用于证明一个给定三角形是否包含直角。
For example, a triangle with sides 6 cm, 8 cm and 10 cm satisfies 6² + 8² = 10², so the angle between the two shorter sides is 90°.
例如，边长为6厘米、8厘米和10厘米的三角形满足6² + 8² = 10²，因此两条较短边之间的夹角是90°。
6. Trigonometric Ratios: Sin, Cos and Tan | 三角函数比：正弦、余弦和正切
In a right-angled triangle, the three trigonometric ratios are defined for an acute angle θ:
在直角三角形中，对于锐角θ，三个三角函数比定义如下：
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
Some students remember this using the phrase “SOH CAH TOA”. You must label the triangle carefully before choosing the correct ratio.
有些学生用短语“SOH CAH TOA”来记忆。在选择正确的比值之前，你必须仔细标记三角形。
7. Using Trigonometry to Find a Missing Side | 用三角函数求缺失的边
If you know one acute angle and one side in a right-angled triangle, you can find another side by choosing the ratio that links the known and unknown sides. For example, if θ = 30° and the hypotenuse is 12 cm, and you need the opposite side:
如果你知道直角三角形中的一个锐角和一条边，就可以选择一个能联系已知边与未知边的比值，从而求出另一条边。例如，如果θ = 30°，斜边为12厘米，并且你需要求对边：
sin 30° = opposite ÷ 12
opposite = 12 × sin 30° = 12 × 0.5 = 6 cm
Make sure your calculator is in degree mode when the angle is given in degrees.
当角度以度为单位时，请确保计算器设置为角度模式。
8. Finding a Missing Angle Using Inverse Trigonometry | 用反三角函数求缺失的角度
If you know two sides and need an acute angle, use the inverse trigonometric functions sin⁻¹, cos⁻¹ or tan⁻¹. First identify the ratio, then apply the inverse function. For example, if opposite = 8 and adjacent = 15:
如果你知道两条边而需要求一个锐角，就使用反三角函数sin⁻¹、cos⁻¹或tan⁻¹。首先确定比值，然后应用反函数。例如，如果对边 = 8，邻边 = 15：
tan θ = 8 ÷ 15
θ = tan⁻¹(8 ÷ 15) ≈ 28.1°
Round angles to one decimal place unless the question says otherwise. Always show the substitution step before the calculator step.
除非题目另有要求，角度通常保留一位小数。在计算器计算之前，务必写出代入步骤。
9. Angles of Elevation and Depression | 仰角与俯角
Angles of elevation and depression appear frequently in IGCSE word problems. The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward from the horizontal. These angles are equal when formed between parallel horizontal lines.
仰角和俯角在IGCSE应用题中经常出现。仰角是从水平线向上测量的角度；俯角是从水平线向下测量的角度。当它们位于两条平行的水平线之间时，这两个角相等。
- Draw a clear diagram and place the angle in the correct triangle.
- 绘制清晰的示意图，并把角度放在正确的三角形中。
- Use the horizontal line as the adjacent side or as a reference for the hypotenuse.
- 把水平线作为邻边，或作为斜边的参照线。
10. Exact Trigonometric Values | 特殊角的精确三角函数值
IGCSE exams often expect you to know the exact values for 30°, 45° and 60° without using a calculator. These values come from two special right-angled triangles: the 45°-45°-90° triangle and the 30°-60°-90° triangle.
IGCSE考试通常要求你无需计算器就能掌握30°、45°和60°的精确值。这些值来自两种特殊的直角三角形：45°-45°-90°三角形和30°-60°-90°三角形。
| θ | 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 |
Rationalising the denominator is not always required, but you should be able to recognise equivalent forms such as 1/√2 = √2/2.
分母有理化并不总是必需的，但你应该能识别等价形式，例如1/√2 = √2/2。
11. Multi-Step Problems and Combined Strategies | 多步骤问题与综合策略
Many IGCSE questions combine Pythagoras’ theorem and trigonometry in the same diagram. You may need to find a side first using one method, then find an angle using another. Split the diagram into smaller right-angled triangles where possible.
许多IGCSE题目会在同一个图形中综合考查勾股定理和三角函数。你可能需要先用一种方法求出一条边，再用另一种方法求出一个角。尽可能将图形拆分成较小的直角三角形。
Always write down each stage of working. Even if your final answer is wrong, clear method marks can still be gained. Do not round intermediate values until the final step.
始终写出每一步过程。即使最终答案错误，清晰的方法步骤仍然可以获得分数。在最后一步之前不要对中间值进行四舍五入。
12. Common Mistakes and Exam Tips | 常见错误与考试提示
- Using Pythagoras on a non-right-angled triangle. Check the angle is 90° first.
- 在非直角三角形上使用勾股定理。首先检查夹角是否为90°。
- Labelling the opposite and adjacent sides incorrectly when the angle changes.
- 当角度改变时，错误标记对边和邻边。
- Forgetting to take the square root or taking the negative square root.
- 忘记开平方，或者取了负的平方根。
- Having the calculator in radian mode instead of degree mode.
- 计算器设置为弧度模式而不是角度模式。
Before the exam, practise at least five mixed questions that require you to choose between Pythagoras and trigonometry. Draw a labelled triangle for every problem.
考试前，至少练习五道需要你在勾股定理和三角函数之间进行选择的综合题。为每一道题画出带标记的三角形。
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