Pythagoras' Theorem and Trigonometry in Right-Angled Triangles | 勾股定理与直角三角形三角学

📚 Pythagoras' Theorem and Trigonometry in Right-Angled Triangles | 勾股定理与直角三角形三角学

Pythagoras' theorem and the sine, cosine and tangent ratios are essential tools in IGCSE Mathematics. They allow you to find unknown sides and angles in right-angled triangles, and they appear frequently in both calculator and non-calculator papers. This article explains the key ideas, worked examples and common exam pitfalls.

勾股定理以及正弦、余弦和正切比是 IGCSE 数学的重要工具。它们可以帮助你求解直角三角形的未知边和角，在计算器和非计算器试卷中都经常出现。本文将讲解核心概念、典型例题和常见考试易错点。

1. Pythagoras' Theorem: The Foundation | 勾股定理：基础

In any right-angled triangle, the longest side is called the hypotenuse. It is always opposite the right angle. The other two sides are often labelled as the legs or shorter sides.

在任何直角三角形中，最长的边称为斜边。它总是正对直角。另外两条边通常称为直角边或较短的边。

Pythagoras' theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. If the hypotenuse is c and the other sides are a and b, then:

勾股定理指出，斜边的平方等于另外两条边的平方和。如果斜边为 c，另外两条边为 a 和 b，则：

a² + b² = c²

This formula only works for right-angled triangles. Always check that the angle between a and b is 90° before using it.

该公式只适用于直角三角形。在使用前，务必确认 a 和 b 之间的夹角为 90°。

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2. Finding the Hypotenuse | 求斜边

o a² + b² = c², then take the square root. Example: if a = 6 cm and b = 8 cm, calculate c.

求斜边时，将两条较短边代入 a² + b² = c²，然后开平方。例如：若 a = 6 cm，b = 8 cm，求 c。

c = √6² + 8² = √36 + 64 = √100 = 10 cm

Remember that c is a length, so it must be positive. The negative square root is not a valid answer in geometry.

请记住 c 是长度，因此必须为正。负的平方根在几何中不是有效答案。

For a non-calculator question, you can often simplify the square root before evaluating. For example, √50 = √25 × 2 = 5√2.

对于非计算器题，通常可以先化简平方根再求值。例如，√50 = √25 × 2 = 5√2。

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3. Finding a Shorter Side | 求较短的直角边

To find a shorter side, rearrange the formula. If you know the hypotenuse c and one shorter side a, then:

求较短边时，需要对公式进行移项。如果已知斜边 c 和一条较短边 a，则：

b² = c² − a²

For example, a right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the missing side b.

例如，一个直角三角形的斜边为 13 cm，一条直角边为 5 cm，求另一条直角边 b。

b = √13² − 5² = √169 − 25 = √144 = 12 cm

Always label the hypotenuse first. This reduces mistakes when rearranging the equation.

务必先标出斜边。这样可以减少移项时的错误。

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4. Pythagorean Triples | 勾股数

ntegers that satisfy a² + b² = c². The most common triples are 3-4-5, 5-12-13 and 7-24-25.

勾股数是指满足 a² + b² = c² 的三个正整数。最常见的勾股数有 3-4-5、5-12-13 和 7-24-25。

Multiples of these triples also work. For example, 6-8-10 and 9-12-15 are multiples of 3-4-5.

这些勾股数的倍数也成立。例如 6-8-10 和 9-12-15 都是 3-4-5 的倍数。

Recognising triples can save time in non-calculator questions. However, you must check which side is the hypotenuse before matching a triple.

识别勾股数可以节省非计算器题目的时间。但在匹配勾股数之前，必须先判断哪条边是斜边。

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5. Introduction to Sine, Cosine and Tangent | 正弦、余弦和正切简介

In a right-angled triangle, the trigonometric ratios relate an acute angle to two sides. For an angle θ, label the sides as follows: the hypotenuse is opposite the right angle; the opposite side is opposite θ; the adjacent side is next to θ and is not the hypotenuse.

在直角三角形中，三角比将一个锐角与两条边联系起来。对于角 θ，按以下方式标记各边：斜边正对直角；对边正对 θ；邻边紧邻 θ 且不是斜边。

The three ratios are summarised below.

三个比总结如下。

Ratio name	Formula	中文解释
sine	sin θ = opposite / hypotenuse	正弦：对边比斜边
cosine	cos θ = adjacent / hypotenuse	余弦：邻边比斜边
tangent	tan θ = opposite / adjacent	正切：对边比邻边

A useful memory aid is SOHCAHTO

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