📚 Right-Angled Triangle Trigonometry: Pythagoras, Sine, Cosine and Tangent | 直角三角形三角学：勾股定理、正弦、余弦与正切
Right-angled triangle trigonometry is one of the most practical and frequently examined topics in IGCSE Mathematics. It combines Pythagoras' theorem with the sine, cosine and tangent ratios, allowing you to find unknown sides and angles in two-dimensional and three-dimensional problems. Mastering this topic is essential for success in both Core and Extended papers, and it provides a foundation for further study in physics, engineering and design.
直角三角形三角学是 IGCSE 数学中最实用、最常考查的主题之一。它将勾股定理与正弦、余弦和正切比结合起来，使你能在二维和三维问题中求解未知边和角。掌握这一主题对于在核心课程和扩展课程试卷中取得好成绩至关重要，也为物理、工程和设计等后续学习打下基础。
1. Labelling a Right-Angled Triangle | 直角三角形的标记
Before using Pythagoras' theorem or any trigonometric ratio, you must label the triangle correctly. The side opposite the right angle is always the hypotenuse, and it is the longest side. The side opposite a given acute angle θ is called the opposite side, while the side next to θ (but not the hypotenuse) is called the adjacent side.
在使用勾股定理或任何三角比之前，你必须正确标记三角形。直角所对的边始终是斜边，也是最长的边。给定锐角 θ 所对的边称为对边，而紧邻 θ（但不是斜边）的边称为邻边。
The labels 'opposite' and 'adjacent' depend on which acute angle you are using. If you switch from angle θ to the other acute angle, the opposite and adjacent sides swap roles, but the hypotenuse does not change.
“对边”和“邻边”的标记取决于你使用哪一个锐角。如果从角 θ 切换到另一个锐角，对边和邻边会互换角色，但斜边不变。
A common exam technique is to draw a small diagram and write O, A and H next to the three sides before starting any calculation. This reduces the chance of substituting values into the wrong formula.
常见的考试技巧是画一个小图，在三条边旁标出 O、A 和 H，然后再开始计算。这可以减少把数值代入错误公式的可能性。
2. Pythagoras' Theorem | 勾股定理
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. In symbols:
勾股定理指出，在任何直角三角形中，斜边的平方等于另外两条边的平方和。用符号表示为：
a² + b² = c²
where c is the hypotenuse and a and b are the two shorter sides. This theorem only works for right-angled triangles, so you must first check that the triangle contains a 90° angle.
其中 c 是斜边，a 和 b 是两条较短的直角边。该定理只适用于直角三角形，因此你必须先确认三角形含有一个 90° 角。
For example, if the two shorter sides are 6 cm and 8 cm, the hypotenuse is √6² + 8² = √36 + 64 = √100 = 10 cm. If the
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