📚 Right-Angled Trigonometry: Sine, Cosine and Tangent | 直角三角函数：正弦、余弦与正切
In IGCSE Mathematics, trigonometric ratios allow you to find unknown sides and angles in right-angled triangles. The three main ratios are sine, cosine and tangent, often remembered by the mnemonic SOH CAH TOA. This article guides you through labelling the triangle, selecting the correct ratio, and applying it in straightforward and real-life problems.
在 IGCSE 数学中，三角比可以帮助你求解直角三角形中未知的边和角。三个主要比是正弦、余弦和正切，通常用助记口诀 SOH CAH TOA 来记忆。本文将引导你完成三角形的边标注、选择正确的比，并将其应用于简单问题和实际问题中。
1. What Is Trigonometry? | 什么是三角学？
Trigonometry is the branch of mathematics that studies the relationships between the angles and side lengths of triangles. For IGCSE, the focus is on right-angled triangles, where one angle is exactly 90°. The key idea is that for a given acute angle, the ratio of two particular sides remains constant no matter how large or small the triangle is.
三角学是研究三角形中角与边长之间关系的数学分支。在 IGCSE 中，重点是直角三角形，其中一个角恰好为 90°。关键思想是：对于一个给定的锐角，两条特定边的比保持不变，无论三角形是大是小。
The three ratios are defined as follows:
三个比的定义如下：
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
Here θ (theta) represents any acute angle in the right-angled triangle.
这里的 θ（西塔）表示直角三角形中任意一个锐角。
2. Labelling the Sides Correctly | 正确标注边
Before using any formula, you must label the three sides of the right-angled triangle relative to the angle you are interested in. The hypotenuse is always the longest side and is opposite the right angle. The opposite side is directly across from the chosen angle θ. The adjacent side is next to θ and is not the hypotenuse.
在使用任何公式之前，你必须相对于所关注的角来标注直角三角形的三条边。斜边始终是最长的边，并且与直角相对。对边是正对着所选角 θ 的边。邻边是与 θ 相邻且不是斜边的边。
For example, if θ is at the bottom left of a triangle, the vertical side on the right is the opposite, the horizontal side along the bottom is the adjacent, and the longest side is the hypotenuse.
例如，如果 θ 在三角形的左下角，那么右侧的竖直边是对边，底部的水平边是邻边，最长的边是斜边。
A very common mistake is to label the opposite and adjacent sides without first stating which angle is being used. Always write 'opposite to θ' and 'adjacent to θ' to avoid confusion.
一个非常常见的错误是在没有首先说明使用的是哪个角的情况下，就标注对边和邻边。为了避免混淆，请始终写出 'θ 的对边' 和 'θ 的邻边'。
3. The Sine Ratio | 正弦比
The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. It is written as:
一个角的正弦是对边长度与斜边长度之比。它写作：
sin θ = opposite ÷ hypotenuse
nstance, if θ = 30° and the hypotenuse is 10 cm, the opposite side is 10 × sin 30° = 5 cm.
当你已知或需要求对边和斜边时，正弦非常有用。例如，如果 θ = 30° 且斜边为 10 厘米，那么对边为 10 × sin 30° = 5 厘米。
Remember that sin θ is always less than or equal to 1 for acute angles, because the hypotenuse is the longest side.
请记住，对于锐角，sin θ 始终小于或等于 1，因为斜边是最长的边。
4. The Cosine Ratio | 余弦比
The cosine of an angle is the ratio of the adjacent side to the hypotenuse. It is written as:
一个角的余弦是邻边长度与斜边长度之比。它写作：
cos θ = adjacent ÷ hypotenuse
θ = 60° and the adjacent side is 4 cm, the hypotenuse can be found by 4 ÷ cos 60° = 8 cm.
当涉及邻边和斜边时，使用余弦。例如，如果 θ = 60° 且邻边为 4 厘米，那么斜边可以通过 4 ÷ cos 60° = 8 厘米来求得。
Cosine values for acute angles also lie between 0 and 1. This property is helpful for checking whether an answer is reasonable.
锐角的余弦值也介于 0 和 1 之间。这一性质有助于检查答案是否合理。
5. The Tangent Ratio | 正切比
The tangent of an angle is the ratio of the opposite side to the adjacent side. It is written as:
一个角的正切是对
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