📚 T-5-1017 IGCSE Maths: Mastering Right-Angled Trigonometry | IGCSE 数学 T-5-1017：直角三角形三角比精讲
Right-angled trigonometry is one of the most practical and heavily examined topics in IGCSE Mathematics. It connects an acute angle in a right-angled triangle with the ratios of two side lengths, allowing you to find unknown sides or angles in real-world problems such as ramps, ladders, bearings and navigation. A clear understanding of sine, cosine and tangent is essential for both Core and Extended IGCSE papers.
直角三角形三角比是 IGCSE 数学中非常实用且常考的核心内容。它把直角三角形中的一个锐角与两条边长之比联系起来，使你能够求解斜道、梯子、方位角与导航等实际问题中的未知边或未知角。无论 Core 还是 Extended 试卷，清晰理解正弦、余弦和正切都至关重要。
This revision guide covers the key definitions, the SOH CAH TOA selection method, finding unknown sides and angles, angles of elevation and depression, and common exam pitfalls. Work through each section carefully and always show your method when solving past-paper questions.
本复习指南涵盖关键定义、SOH CAH TOA 选择方法、求未知边和未知角、仰角与俯角以及常见考试陷阱。请仔细学习每一节，并在做真题时始终写出完整步骤。
1. Introduction to Trigonometric Ratios | 三角比简介
The three basic trigonometric ratios are sine, cosine and tangent. Each ratio is defined from the perspective of one acute angle in a right-angled triangle. They are abbreviated as sin, cos and tan, and they do not represent lengths themselves but rather the quotient of two sides. Because similar right-angled triangles have the same side ratios, these trigonometric values depend only on the angle θ, not on the size of the triangle.
三个基本三角比是正弦、余弦和正切。每个三角比都是从一个锐角的角度来定义的。它们分别简写为 sin、cos 和 tan，它们本身不代表长度，而是两条边长的比值。由于相似的直角三角形具有相同的边长比例，这些三角比值只取决于角 θ，而与三角形的大小无关。
In IGCSE Mathematics, you will mainly use trigonometry to solve right-angled triangles. A right-angled triangle has one 90° angle and two acute angles. The side relationships are fixed for a given acute angle, which is why scientific calculators can instantly give sin 30°, cos 60° or tan 45°.
在 IGCSE 数学中，你主要使用三角比来解直角三角形。直角三角形有一个 90° 角与两个锐角。对于一个给定的锐角，边长关系是固定的，这就是为什么科学计算器可以立即给出 sin 30°、cos 60° 或 tan 45° 的值。
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
2. Labelling the Right-Angled Triangle | 直角三角形的边标记
Before using any trigonometric ratio, you must label the three sides correctly in relation to the angle θ. The hypotenuse is always the longest side and lies opposite the right angle. The opposite side faces the angle θ, while the adjacent side is the one next to θ but is not the hypotenuse.
在使用任何三角比之前，你必须根据角 θ 正确标记三条边。斜边总是最长，并且位于直角对面。对边正对角 θ，邻边是与 θ 相邻但不是斜边的那条边。
Labelling errors are one of the main reasons students lose marks in trigonometry questions. Always write the labels on your diagram before substituting any values. If you move to the other acute angle in the same triangle, the opposite and adjacent labels must swap, but the hypotenuse stays the same.
标记错误是学生在三角比题目中失分的主要原因之一。代入数值之前，一定要先在图形上写出各边标记。如果换到同一个三角形中的另一个锐角，对边和邻边的标记必须互换，但斜边保持不变。
- Hypotenuse: opposite the 90° angle; 斜边：90° 角的对边。
- Opposite: the side directly facing θ; 对边：正对角 θ 的边。
- Adjacent: the side next to θ and not the hypotenuse; 邻边：与 θ 相邻且不是斜边的边。
3. The Sine Ratio | 正弦比
5 cm and the hypotenuse is 10 cm, then sin θ = 5 ÷ 10 = 0.5, which corresponds to θ = 30°.
锐角 θ 的正弦是对边与斜边长度的比值。对于锐角，这个比值始终在 0 和 1 之间。例如，如果对边为 5 cm，斜边为 10 cm，则 sin θ = 5 ÷ 10 = 0.5，对应的角 θ = 30°。
The sine ratio is often used when the question involves the hypotenuse and the side opposite the given angle. In word problems, the 'height' or 'vertical distance' is frequently the opposite side, especially when a ramp or ladder is modelled using horizontal and vertical lines.
当题目涉及斜边和给定角的对边时，通常使用正弦比。在文字题中，“高度”或“竖直距离”常常是对边，尤其是在用水平线和竖直线建模斜道或梯子时。
sin θ = opposite ÷ hypotenuse
4. The Cosine Ratio | 余弦比
and 1. If the adjacent side is 8 m and the hypotenuse is 10 m, then cos θ = 8 ÷ 10 = 0.8.
锐角 θ 的余弦是邻边与斜边长度的比值。和正弦一样，锐角的余弦也在 0 和 1 之间。如果邻边为 8 m，斜边为 10 m，则 cos θ = 8 ÷ 10 = 0.8。
Use the cosine ratio when the known or required side is next to the angle and the hypotenuse is also involved. In diagrams involving a horizontal distance and a slanted length, the horizontal side is usually the adjacent side, so cos is the natural choice.
当已知或要求的边与角相邻并且也涉及斜边时，使用余弦比。在涉及水平距离和倾斜长度的图形中，水平边通常是邻边，因此余弦是自然的选择。
cos θ = adjacent ÷ hypotenuse
5. The Tangent Ratio | 正切比
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