G-3 Student Book Topic 100: Pythagoras and Trigonometry | G-3 学生用书专题 100：勾股定理与三角比

📚 G-3 Student Book Topic 100: Pythagoras and Trigonometry | G-3 学生用书专题 100：勾股定理与三角比

In IGCSE Mathematics, right-angled triangles appear in many contexts, from finding the shortest distance across a field to calculating the height of a building using one angle. This article reviews the two core tools you need: Pythagoras’ theorem and the three trigonometric ratios sine, cosine and tangent. You will see how to identify the correct sides, set up equations and apply the ideas to exam-style problems.

在 IGCSE 数学中，直角三角形出现在许多情境里，例如求一块场地上的最短距离，或利用一个角度计算建筑物的高度。本文复习你需要掌握的两个核心工具：勾股定理以及正弦、余弦、正切三个三角比。你将学会如何识别正确的边、建立方程，并把它们应用到考试题型中。

1. Right-Angled Triangles at a Glance | 直角三角形要点

Every right-angled triangle has one 90° angle, a longest side called the hypotenuse, and two shorter sides. Relative to a chosen acute angle, the sides are labelled as the opposite side, the adjacent side and the hypotenuse.

每个直角三角形都有一个 90° 角，最长的边称为斜边，另外两条边为直角边。相对于选定的锐角，边被标记为对边、邻边和斜边。

Always start by drawing a small diagram and labelling the sides. This habit prevents many errors, especially when the angle used in trigonometry changes.

始终先画一个简图并标出边。这个习惯可以避免很多错误，尤其是当三角比中使用的角发生变化时。

2. The Pythagorean Theorem | 勾股定理

Pythagoras’ theorem only works for 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边的平方和。

a² + b² = c²

Here, c is the hypotenuse, while a and b are the two shorter sides. The order of a and b does not matter because addition is commutative.

这里 c 是斜边，a 和 b 是两条直角边。因为加法满足交换律，所以 a 和 b 的顺序无关紧要。

3. Finding the Hypotenuse | 求斜边

If the two shorter sides are known, substitute them into a² + b² = c² and then take the positive square root.

如果已知两条直角边，将它们代入 a² + b² = c²，然后取正平方根即可。

Example: Find the hypotenuse when the shorter sides are 5 cm and 12 cm.

示例：当两条直角边分别是 5 cm 和 12 cm 时，求斜边。

5² + 12² = c² → 25 + 144 = c² → c = √169 = 13

The hypotenuse is 13 cm. Remember that length is always positive, so reject any negative square root.

斜边为 13 cm。记住长度始终为正，因此负平方根应舍去。

4. Finding a Shorter Side | 求直角边

When the hypotenuse and one shorter side are given, rearrange the theorem to isolate the unknown side: b² = c² – a².

当已知斜边和一条直角边时，重新整理定理以隔离未知边：b² = c² – a²。

Example: The hypotenuse is 13 cm and one shorter side is 5 cm. Find the other side.

示例：斜边为 13 cm，一条直角边为 5 cm，求另一条直角边。

b² = 13² – 5² = 169 – 25 = 144 → b = 12

The missing side is 12 cm. Always check whether you should be adding or subtracting before you calculate.

缺失的边为 12 cm。在计算之前务必检查应该用加法还是减法。

5. Pythagorean Triples | 勾股数组

A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². Knowing common triples can speed up your work.

勾股数组是满足 a² + b² =

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