Pythagoras’ Theorem & Trigonometric Ratios | 勾股定理与三角函数

📚 Pythagoras’ Theorem & Trigonometric Ratios | 勾股定理与三角函数

In the IGCSE Mathematics curriculum, Pythagoras’ Theorem and trigonometric ratios form the foundational toolkit for solving problems involving right-angled triangles. These concepts appear repeatedly across exam papers, from straightforward calculation questions to multi-step problem-solving and applications in real-world contexts.

在 IGCSE 数学课程中，勾股定理和三角函数是解决直角三角形问题的基础工具。这些概念在试卷中反复出现，从直接的计算题到多步骤的问题解决，以及现实情境中的应用。

1. What is Pythagoras’ Theorem? | 什么是勾股定理？

Pythagoras’ Theorem describes the relationship between the three sides of a right-angled triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle, which is the longest side) is equal to the sum of the squares of the lengths of the other two sides.

勾股定理描述直角三角形三条边之间的关系。它指出：斜边（即直角所对的边，也是三角形中最长的边）的平方等于另外两条直角边的平方之和。

a² + b² = c²

Here, c represents the length of the hypotenuse, while a and b represent the lengths of the other two sides. This theorem only applies to right-angled triangles, so be sure to confirm the triangle contains a right angle before applying it.

其中，c 表示斜边的长度，a 和 b 表示另外两条边的长度。该定理仅适用于直角三角形，因此在应用之前务必确认三角形中含有直角。

2. Finding the Hypotenuse | 求斜边

When you know the lengths of both shorter sides and want to find the hypotenuse, you add the squares of the two known sides and then take the square root.

当已知两条直角边的长度而需要求斜边时，将两条已知边的平方相加，然后再开平方根。

Example | 例题： In a right-angled triangle, the two shorter sides are 3 cm and 4 cm. Find the length of the hypotenuse.

例题： 在一个直角三角形中，两条直角边分别为 3 厘米和 4 厘米。求斜边的长度。

c² = 3² + 4² = 9 + 16 = 25 → c = √25 = 5 cm

Therefore, the hypotenuse is 5 cm. This is the well-known 3-4-5 Pythagorean triple, a special set of integers that satisfy the theorem. Other common triples include 5-12-13 and 8-15-17. Recognising these triples can save you valuable time in exams.

因此，斜边长度为 5 厘米。这就是著名的 3-4-5 勾股数，即一组满足勾股定理的特殊整数。其他常见的勾股数包括 5-12-13 和 8-15-17。在考试中识别这些勾股数可以节省宝贵的时间。

3. Finding a Shorter Side | 求直角边

When the hypotenuse and one shorter side are known, you must subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.

当已知斜边和一条直角边时，需要用斜边的平方减去已知直角边的平方，然后再开平方根。

Example | 例题： A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the remaining side.

例题： 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。

b² = 13² − 5² = 169 − 25 = 144 → b = √144 = 12 cm

It is essential to set up the correct equation. A common mistake is to add the squares of the two given sides even when one is the hypotenuse. Always identify which side is the hypotenuse first: it is the longest side and is opposite the right angle.

列出正确的方程至关重要。一个常见错误是即使已知的一边是斜边，仍然把两边的平方相加。务必先判断哪条边是斜边：斜边是最长的边，且正对直角。

4. Introduction to Trigonometric Ratios | 三角函数入门

Trigonometry extends the study of right-angled triangles to include angles. For a given acute angle θ in a right-angled triangle, three primary ratios are defined:

三角函数将直角三角形的研究扩展到角度。对于直角三角形中的给定锐角 θ，定义了三个基本比率：

In these definitions, “opposite” refers to the side directly across from angle θ, “adjacent” is the side next to angle θ (but not the hypotenuse), and “hypotenuse” is the longest side opposite the right angle.

在这些定义中，”对边” 指正对着角 θ 的边，”邻边” 是紧邻角 θ 的边（但不是斜边），”斜边” 是直角所对的最长的边。

To remember these ratios, use the

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