📚 IGCSE Maths Topic 4: Trigonometry in Right-Angled Triangles | IGCSE 数学主题 4：直角三角形中的三角学
Trigonometry is one of the most important toolkits in the IGCSE Mathematics syllabus. It begins with right-angled triangles, where the relationships between angles and side lengths lead directly to Pythagoras' theorem and the sine, cosine and tangent ratios. This article reviews the core ideas, worked applications and common exam techniques you need for Topic 4, including exact values, bearings and angles of elevation.
三角学是 IGCSE 数学大纲中最重要的工具之一。它从直角三角形开始，角度和边长之间的关系直接引出了毕达哥拉斯定理以及正弦、余弦和正切比。本文回顾主题 4 的核心概念、实际应用和常见考试技巧，包括精确值、方位角和仰角/俯角。
1. Pythagoras' Theorem: The Foundation | 毕达哥拉斯定理：基础
c². This is the first tool to use when you know two sides and need to find the third side.
在任何一个直角三角形中，斜边的平方等于另外两条边的平方和。如果斜边标记为 c，另外两条边标记为 a 和 b，那么该定理可以写成 a² + b² = c²。当你知道两条边并需要求第三条边时，这是第一个要使用的工具。
For example, if a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 quation: a² = c² − b². Always check that the side you are solving for is placed correctly.
例如，如果 a = 3 cm，b = 4 cm，那么 c² = 3² + 4² = 9 + 16 = 25，所以 c = √25 = 5 cm。同样，如果你知道斜边和一条较短的边，你可以重新整理方程：a² = c² − b²。务必检查你要求解的边是否放在了正确的位置。
a² + b² = c² and a² = c² − b²
2. Naming the Sides | 边长的命名
Before using trigonometry, you must label the sides of a right-angled triangle correctly relative to a given acute angle θ. The side opposite the right angle is always the hypotenuse. The side opposite the angle θ is the opposite side, and the side next to θ but not the hypotenuse is the adjacent side.
在使用三角学之前，你必须相对于给定的锐角 θ 正确标记直角三角形的边。直角所对的边始终是斜边。与角 θ 相对的边是对边，与角 θ 相邻但不是斜边的边是邻边。
This labelling changes if you move from one acute angle to the other. For example, in a triangle with angles of 30° and 60°, the side that is opposite to 30° is adjacent to 60°. This is why every trig calculation must begin by clearly identifying the reference angle.
如果你从一个锐角换到另一个锐角，这种标记会发生变化。例如，在一个含有 30° 和 60° 的三角形中，30° 的对边是 60° 的邻边。这就是为什么每个三角函数计算都必须先清楚地确定参考角。
3. The Three Trigonometric Ratios: SOHCAHTOA | 三种三角函数比：SOHCAHTOA
For any acute angle θ in a right-angled triangle, the three basic trigonometric ratios are defined as follows. Sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.
对于直角三角形中的任意锐角 θ，三个基本的三角函数比定义如下。正弦是对边与斜边的比，余弦是邻边与斜边的比，正切是对边与邻边的比。
tenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent
SE exam questions simply require you to choose the correct ratio based on which two sides are involved.
记忆口诀 SOHCAHTOA 非常有用：正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。许多 IGCSE 考试题只需要你根据涉及的两条边选择正确的比例。
4. Choosing the Right Ratio | 选择正确的比例
To decide which ratio to use, first label the sides relative to the given angle, then look at the two sides you know or need. If the problem involves the opposite side and the hypotenuse, use sine. If it involves the adjacent side and the hypotenuse, use cosine. If it involves the opposite and adjacent sides, use tangent.
要决定使用哪个比例，首先要相对于已知角标记各边，然后观察你知道或需要求的两条边。如果问题涉及对边和斜边，使用正弦。如果涉及邻边和斜边，使用余弦。如果涉及对边和邻边，使用正切。
sides are opposite and hypotenuse, so you set up sin 35° = x / 12. Then x = 12 × sin 35°.
例如，假设你知道一个 35° 的角，斜边是 12 cm，需要求 35° 的对边。相关的两条边是对边和斜边，因此你可以设立 sin 35° = x / 12。然后 x = 12 × sin 35°。
5. Finding Missing Sides | 求缺失的边
ng side is the denominator, you may need to divide instead: hypotenuse = adjacent / cos θ.
求缺失的边时，如果已知边是比例中的分母，那么用已知边乘以相应的三角比。例如，如果 cos θ = 邻边 / 斜边，那么邻边 = 斜边 × cos θ。如果缺失的边是分母，你可能需要做除法：斜边 = 邻边 / cos θ。
s the hypotenuse, so use sin 65° = h / 6. Therefore h = 6 × sin 65° ≈ 6 × 0.9063 = 5.44 m.
例题：一架梯子靠在墙上。梯子长 6 m，与水平地面成 65° 角。求梯子达到的垂直高度。高度是 65° 的对边，梯子是斜边，所以使用 sin 65° = h / 6。因此 h = 6 × sin 65° ≈ 6 × 0.9063 = 5.44 m。
6. Finding Missing Angles | 求缺失的角
If you know two sides of a right-angled triangle and need an acute angle, use the inverse trigonometric functions. These are written as sin⁻¹, cos⁻¹ and tan⁻¹ or sometimes arcsin, arccos and arctan. Your calculator may require pressing the shift or second function key.
如果你知道直角三角形的两条边，需要求一个锐角，可以使用反三角函数。它们写作 sin⁻¹、cos⁻¹ 和 tan⁻¹，有时也写作 arcsin、arccos 和 arctan。你的计算器可能需要按 shift 键或第二功能键。
25) ≈ 32.0°. Always state angles to the nearest degree unless the question asks otherwise.
例如，如果对边是 5 cm，邻边是 8 cm，那么 tan θ = 5 / 8 = 0.625。要求 θ，使用 θ = tan⁻¹(0.625) ≈ 32.0°。除非题目另有要求，角度通常保留到最接近的度数。
7. Exact Values for 30°, 45° and 60° | 30°、45° 和 60° 的精确值
IGCSE candidates are expected to know the exact trigonometric values for 30°, 45° and 60° without using a calculator. These values can be derived from an equilateral triangle for 30° and 60°, and from a square or isosceles right triangle for 45°.
IGCSE 考生需要在不使用计算器的情况下知道 30°、45° 和 60° 的精确三角函数值。这些值可以从等边三角形推导 30° 和 60°，从正方形或等腰直角三角形推导 45°。
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
These exact values are commonly tested in non-calculator papers. For example, simplify sin 30° + cos 60°: both are 1/2, so the sum is 1. Knowing the exact forms also helps when answers need to be given in surd form.
这些精确值常在非计算器试卷中考查。例如，化简 sin 30° + cos 60°：两者都等于 1/2，所以和为 1。知道精确值形式也有助于答案需要以根式形式给出时。
8. Angles of Elevation and Depression | 仰角与俯角
An angle of elevation is the angle measured upwards from the horizontal when an observer looks at an object above them. An angle of depression is the angle measured downwards from the horizontal when an observer looks at an object below them.
仰角是观察者看向上方物体时，从水平线向上测量的角度。俯角是观察者看向下方物体时，从水平线向下测量的角度。
The key fact is that the angle of elevation from one point to another equals the angle of depression from the second point back to the first, provided the two horizontal lines are parallel. This alternate angle property often appears in IGCSE problem-solving contexts such as cliffs, towers and ships.
关键事实是，从一个点到另一个点的仰角等于从第二个点回到第一个点的俯角，条件是两条水平线平行。这个内错角性质经常出现在 IGCSE 的应用题中，如悬崖、塔和船只。
9. Bearings and Trigonometry | 方位角与三角学
Bearings are used to describe direction. A bearing is always measured clockwise from north, and it is normally written as a three-digit number, such as 075° or 240°. Trigonometry is often combined with bearings to calculate distances and angles between moving objects or fixed points.
方位角用于描述方向。方位角始终从正北方向顺时针测量，通常写成三位数，例如 075° 或 240°。三角学经常与方位角结合，用于计算运动物体或固定点之间的距离和角度。
orth component is 20 × cos 60° = 10 km, and the east component is 20 × sin 60° + 15 = 20 × √3/2 + 15 ≈ 32.32 km.
例如，一艘船沿方位角 060° 航行 20 km，然后向正东航行 15 km。南北方向和东西方向的分量可以用正弦和余弦求出：北方向分量为 20 × cos 60° = 10 km，东方向分量为 20 × sin 60° + 15 = 20 × √3/2 + 15 ≈ 32.32 km。
10. Multi-Step Problems and Exam Tips | 多步问题与考试技巧
Many high-mark IGCSE questions combine Pythagoras' theorem and trigonometric ratios in the same problem. You may first need to find a missing side using Pythagoras, then use that side in a sine, cosine or tangent ratio to find an angle or another side.
许多高分 IGCSE 题目将毕达哥拉斯定理和三角函数比结合在同一个问题中。你可能首先需要用毕达哥拉斯定理求出一条缺失边，然后将该边用于正弦、余弦或正切比例中来求一个角或另一条边。
Always draw and label a clear diagram, identify the right angle, write down the formula you are using, and round only at the final answer to avoid rounding errors. If the question asks for an angle, remember to use the inverse trig function. If it asks for a side, use the normal trig ratio and solve the equation carefully.
务必画出并标记清晰的图形，确定直角，写下你使用的公式，并且只在最终答案处进行四舍五入，以避免舍入误差。如果题目要求求角，记得使用反三角函数。如果要求求边，使用普通三角比并仔细解方程。
Finally, check that your answer is sensible. The hypotenuse must always be the longest side, sine and cosine values must be between 0 and 1 for acute angles, and angles should be less than 90° in right-angled triangle problems unless the question specifies otherwise.
最后，检查你的答案是否合理。斜边必须始终是最长的边，锐角的正弦和余弦值必须在 0 到 1 之间，直角三角形问题中的角应小于 90°，除非题目另有说明。
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