📚 IGCSE Maths Topic 5: Trigonometry and Pythagoras | IGCSE 数学第 5 讲：三角学与勾股定理
Welcome to this IGCSE Mathematics revision guide on Topic 5: Trigonometry and Pythagoras' theorem. In this article, you will learn how to use Pythagoras' theorem to find missing sides in right-angled triangles, apply the sine, cosine and tangent ratios to solve for unknown sides and angles, and extend these ideas to non-right-angled triangles using the sine rule, cosine rule and area formula.
欢迎阅读本篇 IGCSE 数学复习指南，主题为第 5 讲：三角学与勾股定理。在本文中，你将学习如何使用勾股定理求直角三角形的未知边，应用正弦、余弦和正切比求解未知边和角，并将这些方法拓展到非直角三角形，利用正弦定理、余弦定理和面积公式。
1. Pythagoras' Theorem: The Foundation | 勾股定理：基础
Pythagoras' theorem states that 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 has length c and the other two sides have lengths a and b, then the relationship is:
勾股定理指出，在任意直角三角形中，斜边的平方等于另外两条边的平方和。若斜边长为 c，另外两条边长分别为 a 和 b，则关系式为：
a² + b² = c²
The hypotenuse is always the longest side and is opposite the right angle. The two shorter sides are often called the legs or the perpendicular sides. You can use this theorem to find a missing side when the other two sides are known.
斜边始终是最长边，并且正对直角。两条较短的边通常称为直角边或垂边。当已知另外两条边时，可以使用该定理求未知边。
To find the hypotenuse, add the squares of the legs and then take the square root. To find a leg, subtract the square of the known leg from the square of the hypotenuse and take the square root.
求斜边时，将两条直角边的平方相加，再开平方。求直角边时，用斜边的平方减去已知直角边的平方，再开平方。
- Missing hypotenuse: c = √a² + b²
- Missing leg: a = √c² - b²
For example, if a = 3 and b = 4, then c = √3² + 4² = √9 + 16 = √25 = 5. This is the well-known 3-4-5 right-angled triangle.
例如，若 a = 3 且 b = 4，则 c = √3² + 4² = √9 + 16 = √25 = 5。这就是著名的 3-4-5 直角三角形。
2. Pythagorean Triples and Real-Life Applications | 勾股数与实际应用
Some sets of whole numbers satisfy Pythagoras' theorem exactly. These are called Pythagorean triples. Common examples include 3-4-5, 5-12-13, 7-24-25 and 8-15-17. Any multiple of a triple also works, such as 6-8-10 or 9-12-15.
一些整数集合恰好满足勾股定理，这些数称为勾股数。常见的例子包括 3-4-5、5-12-13、7-24-25 和 8-15-17。勾股数的任意倍数也成立，例如 6-8-10 或 9-12-15。
In real life, Pythagoras' theorem can be used to find the length of a ladder leaning against a wall, the diagonal of a rectangle, or the shortest distance between two points on a map when the path forms a right angle.
在实际生活中，勾股定理可用于求斜靠在墙上的梯子长度、矩形的对角线长度，或当地图中路线构成直角时求两点之间的最短距离。
Worked example: a ladder rests against a vertical wall. The foot of the ladder is 5 m from the wall, and the top reaches 12 m high. Find the length of the ladder.
例题：一架梯子靠在竖直的墙上。梯子底端距墙 5 m，顶端高 12 m。求梯子的长度。
Using c² = 5² + 12² = 25 + 144 = 169, so c = √169 = 13 m.
使用 c² = 5² + 12² = 25 + 144 = 169，得到 c = √169 = 13 m。
3. Introducing Sine, Cosine and Tangent | 引入正弦、余弦与正切
In a right-angled triangle, the three basic trigonometric ratios relate an acute angle to the lengths of two sides. Before using them, you must label the sides correctly relative to the angle θ. The side opposite θ is called the opposite side, the side next to θ but not the hypotenuse is called the adjacent side, and the longest side is the hypotenuse.
在直角三角形中，三个基本三角比将一个锐角与两条边的长度联系起来。使用它们之前，必须根据角 θ 正确标出各边。θ 正对的边称为对边，与 θ 相邻但不是斜边的边称为邻边，最长的边是斜边。
The three ratios are defined as:
三个三角比定义如下：
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
A useful memory trick is SOH CAH TOA: Sin equals Opposite over Hypotenuse, Cos equals Adjacent over Hypotenuse, Tan equals Opposite over Adjacent.
一个有用的记忆口诀是 SOH CAH TOA：正弦等于对边比斜边，余弦等于邻边比斜边，正切等于对边比邻边。
| Ratio | Formula | Mnemonic |
|---|---|---|
| sine | sin θ = opposite / hypotenuse | SOH |
| cosine | cos θ = adjacent / hypotenuse | CAH |
| tangent | tan θ = opposite / adjacent | TOA |
4. Using Trigonometric Ratios to Find a Missing Side | 使用三角比求未知边
When finding a missing side, first label the sides opposite, adjacent and hypotenuse relative to the given angle. Then choose the ratio that connects the side you know and the side you want to find. Write an equation, substitute the known values, and solve for the unknown length.
求未知边时，首先根据给定角标出对边、邻边和斜边。然后选择同时连接已知边和未知边的三角比。列出方程，代入已知值，再求出未知长度。
Worked example: in a right-angled triangle, the angle is 35° and the hypotenuse is 12 cm. Find the length of the side opposite the 35° angle.
例题：在一个直角三角形中，角为 35°，斜边为 12 cm。求 35° 角对边的长度。
sine: sin 35° = opposite ÷ 12. Therefore opposite = 12 × sin 35° ≈ 12 × 0.5736 = 6.88 cm.
我们已知斜边，要求对边，因此使用正弦：sin 35° = 对边 ÷ 12。因此对边 = 12 × sin 35° ≈ 12 × 0.5736 = 6.88 cm。
Always check that your calculator is in degree mode. If it is set to radians, your answers for angles in degrees will be incorrect.
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