📚 Mastering Right-Angled Triangle Trigonometry | 掌握直角三角形三角函数
In IGCSE Mathematics, trigonometry is a powerful tool for solving problems involving right-angled triangles. The three basic trigonometric ratios - sine, cosine and tangent - connect the angles of a triangle to the ratios of its side lengths. Understanding these ratios allows you to find unknown sides, calculate missing angles, and model real-world situations such as heights, distances and slopes. This revision guide walks you through the key concepts, step-by-step methods and exam tips you need to master right-angled triangle trigonometry.
在IGCSE数学中，三角函数是解决直角三角形问题的有力工具。正弦、余弦和正切这三个基本三角比将三角形的角与其边长的比值联系起来。理解这些比可以帮助你求解未知边、计算缺失角，并对高度、距离和坡度等实际问题进行建模。本复习指南将带你掌握直角三角形三角函数的核心概念、解题步骤和考试技巧。
1. Trigonometric Ratios: The Core Idea | 三角比的核心概念
Trigonometry means 'measurement of triangles'. In a right-angled triangle, one angle is always 90°, and the other two angles are acute, meaning they are each less than 90°. For any acute angle θ, the three side ratios remain constant no matter how large or small the triangle is, as long as the angle stays the same. This property is the foundation of all trigonometric ratios.
三角学的意思是“三角形的测量”。在直角三角形中，一个角始终为90°，另外两个角为锐角，即它们都小于90°。对于任意锐角θ，只要角度保持不变，无论三角形是大是小，三个边的比值都保持不变。这一性质是所有三角比的基础。
For an acute angle θ in a right-angled triangle, we define three ratios:
对于直角三角形中的锐角θ，我们定义三个比：
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
These ratios are usually abbreviated as sin, cos and tan. They are the building blocks of right-angled triangle trigonometry and appear throughout the IGCSE syllabus.
这些比通常缩写为sin、cos和tan。它们是直角三角形三角函数的基础，并在整个IGCSE大纲中频繁出现。
2. Labelling the Sides Correctly | 正确标记边
Before applying any trigonometric ratio, you must label the three sides of the triangle relative to the angle you are working with. The hypotenuse is always the side opposite the right angle and is the longest side. The opposite side is directly across from the angle θ. The adjacent side is next to the angle θ and is not the hypotenuse.
在应用任何三角比之前，你必须根据所处理的角来标记三角形的三条边。斜边始终是直角对面的边，也是最长边。对边是角θ正对的那条边。邻边是角θ旁边、但不是斜边的那条边。
For example, if θ is the angle at the bottom left of a triangle, the vertical side is the opposite, the horizontal side is the adjacent, and the sloping side is the hypotenuse. If you choose the other acute angle instead, the labels for opposite and adjacent swap places, but the hypotenuse stays the same.
例如，如果θ是三角形左下角的角，则竖直边是对边，水平边是邻边，倾斜边是斜边。如果你选择另一个锐角，对边和邻边的标记会交换位置，但斜边保持不变。
- Always label sides from the point of view of the angle θ, not from the right angle.
- 始终从角θ的角度来标记边，而不是从直角的角度。
- If you choose the other acute angle, the opposite and adjacent labels swap.
- 如果你选择另一个锐角，对边和邻边的标记会互换。
3. The Sine Ratio (sin θ) | 正弦比（sin θ）
length of the opposite side to the length of the hypotenuse. It is often remembered by the phrase 'SOH' e. To use this ratio, identify the opposite side and the hypotenuse, then form the fraction.
角的正弦是对边长度与斜边长度之比。人们常用“SOH”来记忆，意思是正弦 = 对边 ÷ 斜边。使用这个比时，先确定对边和斜边，然后写出分数。
sin θ = opposite ÷ hypotenuse
o is useful when the hypotenuse is known and you need to find the side opposite the angle.
例如，如果对边是6厘米，斜边是10厘米，那么sin θ = 6 ÷ 10 = 0.6。当已知斜边且需要求角对面的边时，正弦比非常有用。
4. The Cosine Ratio (cos θ) | 余弦比（cos θ）
the length of the adjacent side to the length of the hypotenuse. It is often remembered by 'CAH' use. This ratio is used when the side next to the angle is known and the hypotenuse is involved.
角的余弦是邻边长度与斜边长度之比。人们常用“CAH”来记忆，意思是余弦 = 邻边 ÷ 斜边。当已知角旁边的边并且涉及斜边时，使用这个比。
cos θ = adjacent ÷ hypotenuse
you know the side next to the angle and need to find the hypotenuse, or when the hypotenuse is known and you need the adjacent side.
例如，如果邻边是8厘米，斜边是10厘米，那么cos θ = 8 ÷ 10 = 0.8。当你已知角旁边的边而需要求斜边，或已知斜边而需要求邻边时，余弦非常有用。
5. The Tangent Ratio (tan θ) | 正切比（tan θ）
of the length of the opposite side to the length of the adjacent side. It is remembered by 'TOA' nt. Tangent is particularly useful when the hypotenuse is not given or not required in the problem.
角的正切是对边长度与邻边长度之比。人们用“TOA”来记忆，意思是正切 = 对边 ÷ 邻边。当问题中没有给出或不需要求斜边时，正切特别有用。
tan θ = opposite ÷ adjacent
ed in problems involving heights and horizontal distances, such as finding the height of a building from its shadow length.
例如，如果对边是6厘米，邻边是8厘米，那么tan θ = 6 ÷ 8 = 0.75。正切比常用于涉及高度和水平距离的问题，例如根据影子长度求建筑物的高度。
6. Choosing the Right Ratio | 选择正确的比
Deciding which trigonometric ratio to use depends on which sides you know and which side or angle you need to find. A common mnemonic is SOH-CAH-TOA, which summarises all three definitions. Use the following guide to select the correct ratio.
选择使用哪个三角比取决于你已知哪些边以及你需要求哪条边或哪个角。常见的记忆口诀是SOH-CAH-TOA，它概括了这三个定义。使用以下指南来选择正确的比。
- If you know or need the opposite side and the hypotenuse, use sine.
- 如果你知道或需要求对边和斜边，使用正弦。
- If you know or need the adjacent side and the hypotenuse, use cosine.
- 如果你知道或需要求邻边和斜边，使用余弦。
- If you know or need the opposite and the adjacent sides, use tangent.
- 如果你知道或需要求对边和邻边，使用正切。
If two sides are known and you need the third side, you may prefer to use Pythagoras' theorem first, then use trigonometry for angles. If only one side and one acute angle are known, you must use a trigonometric ratio to find another side.
如果已知两条边而需要求第三条边，你可以先用毕达哥拉斯定理，再用三角函数求角。如果只知道一条边和一个锐角，你必须使用三角比来求另一条边。
7. Finding a Missing Side |
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