📚 Pythagoras' Theorem and Trigonometric Ratios | 勾股定理与三角函数比
In IGCSE Mathematics, right-angled triangle problems appear frequently across both Foundation and Higher tiers. Pythagoras' theorem and the three trigonometric ratios -- sine, cosine and tangent -- provide a complete toolkit for finding unknown sides and angles. This article explains the core ideas, worked examples and common exam pitfalls in a clear, bilingual format.
在 IGCSE 数学中，直角三角形问题在基础卷和进阶卷中都经常出现。勾股定理以及正弦、余弦、正切这三个三角函数比，为解决未知边和未知角问题提供了完整工具。本文以清晰的中英双语形式讲解核心概念、例题和常见考试易错点。
1. The Pythagorean Theorem | 勾股定理
Pythagoras' theorem applies only 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.
勾股定理只适用于直角三角形。它指出：斜边的平方等于另外两条直角边的平方之和。
a² + b² = c²
Here, c represents the hypotenuse, which is the longest side and always opposite the right angle. The sides a and b are the two shorter legs that form the 90° angle.
其中，c 表示斜边，即最长边，并且总是与直角相对。边 a 和 b 是构成 90° 角的两条较短直角边。
2. Identifying the Hypotenuse | 识别斜边
Before applying the theorem, you must be able to identify the hypotenuse correctly. It is always the side opposite the right angle and is never labelled as a or b when using the theorem in its standard form.
在应用定理之前，你必须能够正确识别斜边。斜边总是直角的对边，在标准形式的定理中，它不会被标记为 a 或 b。
- Find the right angle first, usually shown with a small square symbol.
- 首次先找到直角，通常用一个小的方形符号标出。
- The side directly across from this angle is the hypotenuse, c.
- 与这个角直接相对的边就是斜边 c。
- The remaining two sides are the legs a and b.
- 剩下的两条边就是直角边 a 和 b。
3. Worked Example: Finding a Missing Hypotenuse | 求斜边的例题
Question: A right-angled triangle has legs of length 6 cm and 8 cm. Find the length of the hypotenuse.
题目：一个直角三角形的两条直角边分别为 6 厘米和 8 厘米。求斜边的长度。
Using a² + b² = c², substitute a = 6 and b = 8.
使用公式 a² + b² = c²，代入 a = 6 和 b = 8。
6² + 8² = c²
36 + 64 = c²
100 = c²
Take the square root of both sides to find c.
两边开平方，求出 c。
c = √100 = 10 cm
The hypotenuse is 10 cm long. This 6-8-10 triangle is one of the well-known Pythagorean triples.
斜边长度为 10 厘米。这个 6-8-10 三角形是著名的勾股数组合之一。
4. Worked Example: Finding a Missing Leg | 求直角边的例题
Question: A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the length of the other leg.
题目：一个直角三角形的斜边为 13 厘米，一条直角边为 5 厘米。求另一条直角边的长度。
Write the theorem as a² + b² = c². Here c = 13 and one leg, say a = 5, is known.
先写出定理 a² + b² = c²。这里 c = 13，已知一条直角边 a = 5。
5² + b² = 13²
25 + b² = 169
Subtract 25 from both sides.
两边同时减去 25。
b² = 169 - 25 = 144
b = √144 = 12 cm
The missing leg is 12 cm. Again, 5-12-13 is a Pythagorean triple, so recognising these can save time in the exam.
缺失的直角边为 12 厘米。同样，5-12-13 也是一组勾股数，因此考试中识别这些组合可以节省时间。
5. Pythagorean Triples | 勾股数
egers that satisfy a² + b² = c². Some common triples appear very often in IGCSE questions.
勾股数是满足 a² + b² = c² 的三个正整数。一些常见的勾股数经常出现在 IGCSE 考题中。
| Smaller leg a | 较小直角边 a | Larger leg b | 较大直角边 b | Hypotenuse c | 斜边 c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
Any multiple of a triple is also valid. For example, 6-8-10 is just 3-4-5 multiplied by 2.
任何勾股数的倍数也成立。例如，6-8-10 就是 3-4-5 的每项乘以 2。
6. Trigonometric Ratios: Sine, Cosine and Tangent | 三角函数比：正弦、余弦与正切
When you know one acute angle and one side of a right-angled triangle, trigonometric ratios allow you to find the other sides. The three ratios are defined using the labels opposite, adjacent and hypotenuse.
当你知道直角三角形的一个锐角和一条边时，三角函数比可以帮助你求出其他边。这三个比的定义使用了“对边”“邻边”和“斜边”的标记。
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
The phrase SOH CAH TOA is a useful memory aid for these relationships.
短语 SOH CAH TOA 是记忆这些关系的有用口诀。
- Sine = Opposite over Hypotenuse
- 正弦 = 对边 ÷ 斜边
- Cosine = Adjacent over Hypotenuse
- 余弦 = 邻边 ÷ 斜边
- Tangent = Opposite over Adjacent
- 正切 = 对边 ÷ 邻边
7. Choosing the Correct Ratio | 选择正确的三角函数比
To choose the correct ratio, first label the triangle relative to the angle you are using or trying to find. The side opposite the angle is labelled O, the side next to the angle but not the hypotenuse is labelled A, and the longest side is H.
要选择正确的比，首先相对于你正在使用或要求的角度标记三角形。与角相对的边标记为 O，与角相邻但不是斜边的边标记为 A，最长边标记为 H。
- If you have or need O and H, use sine.
- 如果已知或要求 O 和 H，使用正弦。
- If you have or need A and H, use cosine.
- 如果已知或要求 A 和 H，使用余弦。
- If you have or need O and A, use tangent.
- 如果已知或要求 O 和 A，使用正切。
8. Worked Example: Finding a Missing Side | 求未知边的例题
a right-angled triangle, angle θ = 35°. The hypotenuse is 12 cm. Find the side opposite θ.
题目：在一个直角三角形中，θ = 35°，斜边为 12 厘米。求 θ 的对边长度。
We need the opposite side and we know the hypotenuse, so use the sine ratio.
我们需要求对边，已知斜边，所以使用正弦比。
sin 35° = opposite ÷ 12
Multiply both sides by 12.
两边同时乘以 12。
opposite = 12 × sin 35° ≈ 6.88 cm
The opposite side is approximately 6.88 cm. Make sure your calculator is in degree mode.
对边长度约为 6.88 厘米。请确保计算器处于角度制模式。
9. Worked Example: Finding an Angle | 求未知角的例题
Question: A right-angled triangle has an opposite side of 7 cm and an adjacent side of 10 cm. Find the angle θ.
题目：一个直角三角形的对边为 7 厘米，邻边为 10 厘米。求角 θ。
We know O and A, so use the tangent ratio.
我们已知 O 和 A，所以使用正切比。
tan θ = 7 ÷ 10 = 0.7
Take the inverse tangent of both sides.
两边取反正切。
θ = tan⁻¹(0.7) ≈ 34.99° ≈ 35.0°
The angle is approximately 35.0°. The inverse trigonometric functions are often written as sin⁻¹, cos⁻¹ and tan⁻¹.
该角约为 35.0°。反三角函数通常写作 sin⁻¹、cos⁻¹ 和 tan⁻¹。
10. Angles of Elevation and Depression | 仰角与俯角
Exam questions often apply trigonometry to real-life contexts involving angles of elevation and depression. The angle of elevation is measured upwards from a horizontal line, while the angle of depression is measured downwards from a horizontal line.
考试题经常将三角函数应用于涉及仰角和俯角的实际情境中。仰角是从水平线向上测量的角，而俯角是从水平线向下测量的角。
- Draw a clear right-angled triangle from the problem.
- 根据题目画出清晰的直角三角形。
- Label the given angle, side and the side you need to find.
- 标出已知角、已知边和要求边。
- Choose the ratio that connects the labelled sides.
- 选择能够连接已标记边的三角函数比。
11. Common Mistakes | 常见错误
Students often lose marks by using the wrong side labels or by mixing up Pythagoras and trigonometry. These errors are avoidable with careful labelling.
学生常常因为错误标记边，或者混淆勾股定理和三角函数而失分。只要仔细标记，这些错误是可以避免的。
- Always check that the triangle is right-angled before applying these rules.
- 在应用这些规则之前，一定要先确认三角形是直角三角形。
- Do not use the hypotenuse as the adjacent side unless the angle is between them.
- 不要将斜边当作邻边使用，除非该角确实位于它们之间。
- Remember that inverse trig functions find angles, not sides.
- 记住，反三角函数用于求角度，而不是求边长。
- Ensure your calculator is in degree mode for IGCSE work.
- 确保计算器在 IGCSE 考试中处于角度制模式。
12. Exam Tips | 考试提示
In the IGCSE exam, method marks are awarded for correct substitution and working. Even if your final answer is slightly wrong, showing clear steps can earn most of the marks.
在 IGCSE 考试中，正确的代入和步骤可以拿到方法分。即使最终答案略有错误，展示清晰步骤也能获得大部分分数。
- Write down the formula or ratio before substituting numbers.
- 在代入数字之前先写出公式或三角函数比。
- Round only at the final answer, not during intermediate steps.
- 只在最终答案处四舍五入，不要在中间步骤中提前取近似值。
- Label your triangle with O, A and H in every question.
- 在每一道题中都标出 O、A 和 H。
- Check whether the answer is reasonable, for example the hypotenuse must be the longest side.
- 检查答案是否合理，例如斜边必须是最长边。
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