📚 Right-Angled Triangles: Pythagoras and Trigonometry | 直角三角形：勾股定理与三角函数
Right-angled triangles appear throughout IGCSE mathematics, from practical measurements to higher-level coordinate geometry and 3D problems. Two essential tools for solving these triangles are Pythagoras' theorem and the trigonometric ratios sine, cosine and tangent. This revision article builds each skill step by step, includes exact special angles, and shows how exam questions commonly combine the ideas.
直角三角形贯穿 IGCSE 数学的众多内容，从实际测量到较高层次的坐标几何与三维问题。求解直角三角形的两个核心工具是勾股定理以及正弦、余弦和正切这三个三角比。本文循序渐进地讲解各项技能，包含特殊角精确值，并展示考题常见的综合方式。
1. Labelling a Right-Angled Triangle | 直角三角形的边标注
Before applying any formula, always label the sides relative to the angle you are using. The hypotenuse is the longest side and is opposite the right angle. The opposite side faces the given acute angle, and the adjacent side is the one next to that angle but not the hypotenuse.
在使用任何公式之前，一定要根据所选的角标注各边。斜边是最长边，并且正对直角。对边正对已知锐角，邻边与该锐角相邻但不是斜边。
2. Pythagoras' Theorem | 勾股定理
ypotenuse, add the squares of the two shorter sides and take the positive square root: c = √a² + b².
对于任意直角三角形，设两条直角边为 a 和 b，斜边为 c，则有 a² + b² = c²。求斜边时，将两条较短边的平方相加，再取正平方根：c = √a² + b²。
a² + b² = c²
This formula only applies to right-angled triangles, so identify the right angle and the hypotenuse before substituting values.
该公式只适用于直角三角形，因此在代入数值之前，必须先确定直角与斜边。
3. Finding a Shorter Side | 求直角边
. Be careful: the missing side is usually shorter than the hypotenuse, so your answer must be less than c.
当已知斜边和一条直角边时，移项求出未知边。若求 b，则使用 b² = c² − a²，然后开平方。注意：未知直角边通常比斜边短，因此所得结果必须小于 c。
b = √c² − a²
Always check that the value under the square root is positive; a negative value usually means you have labelled the hypotenuse incorrectly.
务必检查平方根内的值是否为正；若为负，通常说明你把斜边标错了。
4. Pythagorean Triples and the Converse | 勾股数及其逆定理
triangle satisfy a² + b² = c², then the triangle is right-angled with c as the hypotenuse.
识别 3-4-5、5-12-13、7-24-25 等简单勾股数及其倍数可以加快解题。逆定理指出：若三角形三边满足 a² + b² = c²，则该三角形是直角三角形，且 c 为斜边。
This is particularly useful when a question asks you to prove that a triangle is right-angled without showing a diagram.
当题目要求在不给出图形的情况下证明一个三角形是直角三角形时，这一逆定理尤其有用。
5. The Trigonometric Ratios | 三角比
cent / hypotenuse, tan θ = opposite / adjacent. θ is the acute angle, not the right angle.
三个基本三角比将锐角与直角三角形的两条边联系起来。使用 SOH CAH TOA：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。这里的 θ 是锐角，不是直角。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Label the sides from the angle you know or want to find; never use the 90° angle inside SOH CAH TOA.
从已知角或所求角出发标注边；切勿在 SOH CAH TOA 中使用 90° 角。
6. Finding Unknown Sides with Trigonometry | 用三角比求未知边
ample, if the hypotenuse is 10 cm and θ = 30°, the opposite side is x = 10 sin 30° = 5 cm.
选择包含已知边与所求边的三角比。代入已知数值，然后解方程。例如，斜边为 10 cm，θ = 30°，则对边 x = 10 sin 30° = 5 cm。
x = 10 sin 30° = 5 cm
Always write the substitution line in your working, even if you then use a calculator; it often earns method marks.
即使使用计算器，也要在解题过程中写出代入步骤，这通常能获得方法分。
7. Finding Unknown Angles | 求未知角
an⁻¹(opposite/adjacent). Always check that your result is an acute angle between 0° and 90° for a right-angled triangle.
求角时，在计算器上使用反三角函数：θ = sin⁻¹(对边/斜边)，θ = cos⁻¹(邻边/斜边)，θ = tan⁻¹(对边/邻边)。务必检查结果是否为 0° 到 90° 之间的锐角。
θ = sin⁻¹(opposite / hypotenuse)
Make sure your calculator is in degree mode, not radian mode, before answering IGCSE angle questions.
在解答 IGCSE 角度问题之前，确保计算器处于角度模式，而不是弧度模式。
8. Exact
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