📚 IGCSE Mathematics: Trigonometry for Right-Angled Triangles | IGCSE 数学：直角三角形三角学
Trigonometry is a branch of mathematics that studies the relationships between angles and sides of triangles. In the IGCSE Mathematics syllabus, the first and most fundamental topic is trigonometry in right-angled triangles. Mastering this topic builds the foundation for later studies of sine and cosine rules, graphs of trigonometric functions, and even calculus in higher grades.
三角学是数学的一个分支，研究三角形中角度与边之间的关系。在 IGCSE 数学课程中，第一个最基础的内容就是直角三角形中的三角学。掌握这一主题可为后续学习正弦定理、余弦定理、三角函数图像乃至更高级的微积分打下坚实基础。
1. Key Terms and the Right-Angled Triangle | 关键术语与直角三角形
Every right-angled triangle has a right angle (90°). When we work with trigonometry, we always choose one acute angle θ. Relative to that angle, the sides are given special names.
每个直角三角形都有一个直角（90°）。当我们使用三角学方法时，总是选定一个锐角 θ。相对于这个角，三条边有特殊名称。
- Hypotenuse: the longest side, always opposite the right angle.
- Opposite: the side directly opposite the chosen angle θ.
- Adjacent: the side next to θ that is not the hypotenuse.
- 斜边：最长的边，始终对着直角。
- 对边：与选定的角 θ 正对的边。
- 邻边：与 θ 相邻但不是斜边的边。
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边
You must identify these sides carefully in any diagram before applying a ratio.
在应用比例前，你必须先在图中仔细确认这些边的位置。
2. The Three Trigonometric Ratios | 三个三角比
The core of right-angled trigonometry is the three ratios: sine, cosine and tangent. Each ratio compares two side lengths of the triangle.
直角三角形三角学的核心是三个比例：正弦、余弦和正切。每个比例比较三角形两条边的长度。
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
For example, if θ = 30° and hypotenuse = 10 cm, then sin 30° = 0.5, so opposite = 10 × 0.5 = 5 cm.
例如，若 θ = 30°，斜边 = 10 cm，则 sin 30° = 0.5，所以对边 = 10 × 0.5 = 5 cm。
3. SOHCAHTOA – The Easy Mnemonic | SOHCAHTOA——简便口诀
To memorise the three formulas, students use the mnemonic SOHCAHTOA.
为了记住这三个公式，学生常用口诀 SOHCAHTOA。
| Part | Meaning | Chinese |
|---|---|---|
| SOH | sin θ = Opposite/Hypotenuse | sin θ = 对边/斜边 |
| CAH | cos θ = Adjacent/Hypotenuse | cos θ = 邻边/斜边 |
| TOA | tan θ = Opposite/Adjacent | tan θ = 对边/邻边 |
For a memory phrase in English, some students say “Some Old Horses Can Always Hear Their Owners Approaching.”
英语中，有些学生用句子 “Some Old Horses Can Always Hear Their Owners Approaching” 来记忆，中文可直接记住 SOH-CAH-TOA 的读音。
4. Finding a Missing Side | 求未知边长
When you know one acute angle and one side, you can use a trigonometric ratio to find another side. Follow these steps:
当你已知一个锐角和一条边时，可以用三角比求出另一条边。步骤如下：
- Identify the known angle and the sides involved.
- Choose the correct ratio (SOH, CAH or TOA).
- Write the equation and solve for the unknown side.
- 确定已知角以及涉及的边。
- 选择正确的三角比（SOH、CAH 或 TOA）。
- 写出方程并解出未知边。
Worked Example: In a right-angled triangle, angle θ = 40° and adjacent side = 6 cm. Find the hypotenuse.
示例：在直角三角形中，角 θ = 40°，邻边 = 6 cm。求斜边。
Here we have adjacent and need hypotenuse, so we use cos θ = adjacent/hypotenuse.
已知邻边，要求斜边，使用 cos θ = 邻边/斜边。
cos 40° = 6 / hypotenuse
hypotenuse = 6 / cos 40° ≈ 6 / 0.766 ≈ 7.83 cm
Remember to round your answer to a sensible degree of accuracy, usually 3 significant figures.
注意将答案四舍五入到合理的精度，通常保留 3 位有效数字。
5. Finding a Missing Angle | 求未知角度
If you know two sides, you can find the unknown acute angle using the inverse trigonometric functions: sin⁻¹, cos⁻¹ and tan⁻¹.
如果已知两条边，可以用反三角函数：sin⁻¹、cos⁻¹ 和 tan⁻¹ 求出未知锐角。
Worked Example: In a right triangle, opposite = 5 cm and hypotenuse = 13 cm. Find the angle θ.
示例：在直角三角形中，对边 = 5 cm，斜边 = 13 cm。求角 θ。
We use sin θ = opposite/hypotenuse = 5/13 ≈ 0.3846.
使用 sin θ = 对边/斜边 = 5/13 ≈ 0.3846。
θ = sin⁻¹(0.3846) ≈ 22.6°
When using your calculator, make sure it is in degree mode. This is one of the most common errors.
使用计算器时，确保它处于角度模式（DEG）。这是最常见的错误之一。
6. Exact Trigonometric Values | 特殊角的精确值
Certain angles appear frequently in IGCSE problems: 0°, 30°, 45°, 60° and 90°. You are expected to know their exact values without a calculator.
某些角度在 IGCSE 题目中频繁出现：0°、30°、45°、60° 和 90°。你需要不借助计算器而知道它们的精确值。
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
These exact values are useful when solving problems without a calculator or when simplifying answers.
这些精确值在不用计算器解题或化简答案时非常有用。
7. Angles of Elevation and Depression | 仰角与俯角
In real-world applications, angles are measured from a horizontal line. An angle of elevation is the angle between the horizontal and the line of sight when looking up at an object. An angle of depression is the angle between the horizontal and the line of sight when looking down.
在现实应用中，角度从水平线开始测量。仰角是当视线向上看物体时，视线与水平线之间的夹角。俯角是当视线向下看时，视线与水平线之间的夹角。
Example: From a point 20 m from the base of a tower, the angle of elevation to the top is 35°. Find the height of the tower.
示例：从塔基水平距离 20 m 处，测得塔顶的仰角为 35°。求塔高。
Here the opposite side is the height h, and the adjacent side is 20 m. Use tan 35° = h/20, so h = 20 tan 35° ≈ 14.0 m.
此处对边是高度 h，邻边是 20 m。使用 tan 35° = h/20，所以 h = 20 tan 35° ≈ 14.0 m。
Always draw a clear right-angled triangle to model the situation. The angle of depression is equal to the angle of elevation inside the triangle due to parallel lines.
务必画出清晰的直角三角形来表示情境。由于平行线性质，俯角等于三角形内部的仰角。
8. Bearings and Trigonometry | 方位角与三角学
A bearing is a direction measured clockwise from North, usually given as a three-digit angle such as 045° or 120°. Trigonometry can be used to solve problems involving bearings.
方位角是从正北方向顺时针测量的方向，通常用三位角度表示，如 045° 或 120°。三角学可用于解决涉及方位角的问题。
Example: A ship sails 50 km on a bearing of 060°. How far east and how far north has it travelled?
示例：一艘船沿方位角 060° 航行了 50 km。它向东和向北各行驶了多远？
The east displacement is 50 sin 60° = 50 × √3/2 ≈ 43.3 km. The north displacement is 50 cos 60° = 50 × ½ = 25 km.
向东的位移为 50 sin 60° = 50 × √3/2 ≈ 43.3 km。向北的位移为 50 cos 60° = 50 × ½ = 25 km。
In bearing problems, make sure you interpret the angle correctly relative to the vertical north direction.
在方位角问题中，要确保正确理解相对于正北方向的角。
9. Common Mistakes and How to Avoid Them | 常见错误及其避免方法
Many students lose marks in trigonometry due to small but repeated mistakes. Here are the most common ones:
许多学生在三角学中因为细小但重复的错误而失分。以下是最常见的几类错误：
- Using the wrong ratio: Always check which sides are given.
- Calculator in radian mode: For degree problems, set the calculator to DEG.
- Misidentifying sides: Mark the angle θ first, then label opposite and adjacent.
- Rounding too early: Keep full accuracy until the final answer.
- 用错比例：先检查给了哪些边。
- 计算器处于弧度模式：角度问题要将计算器设为 DEG。
- 认错边：先标出角 θ，再标对边和邻边。
- 过早四舍五入：在最终答案前保持完整精度。
Additionally, do not forget that the hypotenuse is always the largest side. Tan θ can also be written as sin θ / cos θ.
此外，不要忘记斜边总是最长边。tan θ 也可写成 sin θ / cos θ。
10. Worked Exam-Style Problem | 考试风格例题
Let us combine several skills in one complete problem.
让我们在一个完整题目中综合运用多项技能。
Question: ABC is a right-angled triangle at B. AB = 8 cm and BC = 6 cm. Calculate the length AC and the angle ACB.
题目：△ABC 中，∠B = 90°，AB = 8 cm，BC = 6 cm。求 AC 的长度和角 ACB。
Solution: Using Pythagoras: AC² = 8² + 6² = 64 + 36 = 100, so AC = 10 cm. For angle ACB, the opposite side to C is AB = 8, and the adjacent side is BC = 6. Thus tan C = 8/6 = 4/3, so C = tan⁻¹(4/3) ≈ 53.1°.
解答：先用勾股定理：AC² = 8² + 6² = 64 + 36 = 100，所以 AC = 10 cm。对于角 ACB，C 角的对边是 AB = 8，邻边是 BC = 6。因此 tan C = 8/6 = 4/3，所以 C = tan⁻¹(4/3) ≈ 53.1°。
This shows how Pythagoras and trigonometry often appear together.
这说明勾股定理和三角学经常一起出现。
11. Practice Questions | 练习题目
Try these questions by yourself before checking the answers.
请先自行尝试以下题目，再对照答案。
- In a right triangle, sin θ = 0.6 and hypotenuse = 15. Find the opposite side.
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