📚 Trigonometry in Right-Angled Triangles | 直角三角形中的三角学
Right-angled triangle trigonometry is one of the most practical topics in IGCSE Mathematics. It gives you a set of tools for finding unknown sides or angles when you know certain other measurements, and it appears in questions ranging from simple calculations to real-world problems involving heights and distances.
直角三角形中的三角学是 IGCSE 数学中最实用的主题之一。当你已知某些边长或角度时，它提供了一套求解未知边或角的工具；从简单计算到涉及高度与距离的实际问题，这个主题都会出现。
In this article, you will learn how to label a right-angled triangle correctly, use the sine, cosine and tangent ratios, solve for missing sides and angles, and apply these ideas to elevation and depression problems. Each section includes worked examples and common exam tips.
在本文中，你将学习如何正确标记直角三角形，使用正弦、余弦和正切比，求解未知边和角，并将这些方法应用于仰角与俯角问题。每一节都包含例题和常见考试技巧。
1. What Is Trigonometry? | 什么是三角学？
Trigonometry is the branch of mathematics that connects the angles of a triangle with the lengths of its sides. In IGCSE Mathematics, the focus is on right-angled triangles, where one angle is exactly 90°.
三角学是数学中把三角形的内角与边长联系起来的分支。在 IGCSE 数学中，重点是直角三角形，即其中一个角恰好为 90° 的三角形。
The word trigonometry comes from Greek words meaning 'triangle measure'. It is not just an abstract topic: surveyors, engineers, architects and pilots all use trigonometric ratios to calculate distances and heights that are difficult to measure directly.
“三角学”一词来自希腊语，意思是“三角形的测量”。它并不是一个抽象的课题：测量员、工程师、建筑师和飞行员都会使用三角比来计算难以直接测量的距离和高度。
There are three main trigonometric ratios you need to know: sine, cosine and tangent. These are usually shortened to sin, cos and tan.
你需要掌握三个主要的三角比：正弦、余弦和正切，通常简写为 sin、cos 和 tan。
The key idea is that for a fixed acute angle in a right-angled triangle, certain side ratios always remain the same, no matter how large or small the triangle is.
关键思想是：在直角三角形中，对于一个固定的锐角，某些边长之比始终保持不变，无论三角形的大小如何。
2. Labelling the Right-Angled Triangle | 标记直角三角形
Before using any trigonometric ratio, you must be able to label the three sides of a right-angled triangle correctly. The three labels are the hypotenuse, the opposite side and the adjacent side.
在使用任何三角比之前，你必须能够正确标记直角三角形的三条边。这三个标记分别是斜边、对边和邻边。
The hypotenuse is always the longest side. It is the side directly opposite the right angle and is never touched by the right angle itself.
斜边始终是最长的边。它正对直角，并且永远不会与直角相邻。
The opposite side is the side directly opposite the angle you are working with. The adjacent side is the side that forms the angle together with the hypotenuse, but it is not the hypotenuse.
对边是正对你所研究的角的边。邻边是与斜边一起构成该角的那条边，但它不是斜边。
For example, if the marked angle is at the bottom left of a triangle, then the vertical side is the opposite, the horizontal side is the adjacent, and the sloping side is the hypotenuse.
例如，如果所标记的角位于三角形的左下角，那么竖直边是对边，水平边是邻边，斜边是倾斜的那条边。
Always draw a small sketch and label the sides before choosing a ratio. Many mistakes in exams come from labelling the opposite and adjacent sides incorrectly.
在选择三角比之前，一定要画一个简图并标记各条边。考试中的许多错误都来自于对边和邻边标记错误。
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. In symbols, this is written as:
一个角的正弦是对边长度与斜边长度之比。用符号表示为：
sin θ = opposite ÷ hypotenuse
xample, if the opposite side is 3 cm and the hypotenuse is 5 cm, then sin θ = 3 ÷ 5 = 0.6.
这个比告诉你斜边中有多大比例是由该角的对边构成的。例如，如果对边为 3 cm，斜边为 5 cm，则 sin θ = 3 ÷ 5 = 0.6。
Because the opposite side is always shorter than the hypotenuse in a right-angled triangle, the value of sin θ is always less than 1 for an acute angle.
因为在直角三角形中，对边总是比斜边短，所以对于锐角，sin θ 的值始终小于 1。
The sine ratio is especially useful when the question gives you the hypotenuse and asks for the opposite side, or gives you the opposite side and asks for the hypotenuse.
当题目给出斜边并要求求对边，或给出对边并要求求斜边时，正弦比特别有用。
4. The Cosine Ratio | 余弦比
The cosine of an angle is the ratio of the length of the adjacent side to the length of 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
tenuse. If the adjacent side is 4 cm and the hypotenuse is 5 cm, then cos θ = 4 ÷ 5 = 0.8.
余弦比将角的邻边与斜边联系起来。如果邻边为 4 cm，斜边为 5 cm，则 cos θ = 4 ÷ 5 = 0.8。
Cosine is often confused with sine because both ratios involve the hypotenuse. The difference is that sine uses the opposite side, while cosine uses the adjacent side.
余弦经常与正弦混淆，因为两者都涉及斜边。区别在于正弦使用对边，而余弦使用邻边。
Use the cosine ratio when the question involves the hypotenuse and the side next to the marked angle. For acute angles, cos θ also always lies between 0 and 1, just like sin θ.
当题目涉及斜边和所标记角的邻边时，使用余弦比。对于锐角，cos θ 也始终在 0 到 1 之间，就像 sin θ 一样。
5. The Tangent Ratio | 正切比
The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. It is written as:
一个角的正切是对边长度与邻边长度之比。它写作：
tan θ = opposite ÷ adjacent
The tangent ratio is different from sine and cosine because it does not involve the hypotenuse. This makes it very useful when you know or need the two shorter sides only.
正切比不同于正弦和余弦，因为它不涉及斜边。这在你只知道或只需两条较短边时非常有用。
m, then tan θ = 3 ÷ 4 = 0.75. There is no need to know the hypotenuse in this calculation.
例如，如果对边为 3 cm，邻边为 4 cm，则 tan θ = 3 ÷ 4 = 0.75。在这个计算中不需要知道斜边。
However, unlike sine and cosine, the tangent ratio can be greater than 1. This happens when the opposite side is longer than the adjacent side.
然而，与正弦和余弦不同，正切比可以大于 1。当对边比邻边长时就会出现这种情况。
Always check which two sides are involved in the question. If neither side is the hypotenuse, tangent is likely the correct ratio to use.
一定要检查题目中涉及哪两条边。如果两条边都不是斜边，那么正切很可能是正确的三角比。
6. Choosing the Correct Ratio | 选择正确的三角比
To choose the correct trigonometric ratio, many students use the memory aid SOHCAHTOA. Each group of letters stands for one ratio:
为了选择正确的三角比，许多学生使用记忆口诀 SOHCAHTOA。每组字母代表一个比：
- Sine: Opposite ÷ Hypotenuse — 正弦：对边 ÷ 斜边
- Cosine: Adjacent ÷ Hypotenuse — 余弦：邻边 ÷ 斜边
- Tangent: Opposite ÷ Adjacent — 正切：对边 ÷ 邻边
Start by identifying the angle you are working with, then label the sides. Next, decide which side you already know and which side you need to find.
首先确定你所研究的角，然后标记各条边。接着确定你已知哪条边以及需要求哪条边。
If you know the hypotenuse and need the opposite, use sine. If you know the hypotenuse and need the adjacent, use cosine. If you know both shorter sides, use tangent.
如果你已知斜边并需要求对边，使用正弦。如果你已知斜边并需要求邻边，使用余弦。如果你已知两条较短边，使用正切。
The table below summarises the three ratios and the sides they connect:
下表总结了三个三角比及其所联系的边：
| Ratio | Formula | 中文 |
|---|---|---|
| Sine | sin θ = opposite ÷ hypotenuse | 正弦：对边 ÷ 斜边 |
| Cosine | cos θ = adjacent ÷ hypotenuse | 余弦：邻边 ÷ 斜边 |
| Tangent | tan θ = opposite ÷ adjacent | 正切：对边 ÷ 邻边 |
Taking a few seconds to label the triangle and choose the correct ratio will help you avoid unnecessary errors in working.
花几秒钟标记三角形并选择正确的三角比，将帮助你避免解题过程中的不必要错误。
7. Finding a Missing Side | 求未知边
When you are given one side and one acute angle, you can use a trigonometric ratio to find another side. Follow these steps: label the sides, choose the correct ratio, write the equation, substitute the known values, and solve for the unknown side.
当你已知一条边和一个锐角时，可以使用三角比来求另一条边。请遵循以下步骤：标记各边，选择正确的比，写出方程，代入已知值，然后求解未知边。
Worked example: A right-angled triangle has a hypotenuse of 12 cm and an angle of 35°. Find the length of the side opposite the 35° angle.
例题：一个直角三角形的斜边为 12 cm，一个角为 35°。求 35° 角的对边长度。
te side, use the sine ratio. Write sin 35° = x ÷ 12, where x is the unknown opposite side.
因为题目涉及斜边和对边，所以使用正弦比。写出 sin 35° = x ÷ 12，其中 x 是未知的对边。
To solve for x, multiply both sides by 12:
为了求解 x，将两边同时乘以 12：
x = 12 × sin 35° ≈ 12 × 0.5736 ≈ 6.88 cm
Always give your final answer to a sensible degree of accuracy, usually three significant figures unless the question states otherwise.
除非题目另有说明，否则最终答案通常保留三位有效数字，以合理的精度给出。
42°, you can find the opposite side using tangent: tan 42° = x ÷ 10, so x = 10 × tan 42°.
另一个例子：如果邻边为 10 m，角为 42°，你可以使用正切求对边：tan 42° = x ÷ 10，因此 x = 10 × tan 42°。
x = 10 × tan 42° ≈ 10 × 0.9004 ≈ 9.00 m
Remember to keep the unknown side on its own before using your calculator. Rearranging the formula first can prevent input mistakes.
在使用计算器之前，记得先让未知边单独在等号一边。先整理公式可以避免输入错误。
8. Finding a Missing Angle | 求未知角
If you know two sides of a right-angled triangle and need to find an acute angle, use the inverse trigonometric functions: sin⁻¹, cos⁻¹ and tan⁻¹.
如果你已知直角三角形的两条边并需要求一个锐角，使用反三角函数：sin⁻¹、cos⁻¹ 和 tan⁻¹。
The inverse functions answer the question: 'What angle gives this ratio?' For example, if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°.
反三角函数回答的问题是：“哪个角能得到这个比？”例如，如果 sin θ = 0.5，那么 θ = sin⁻¹(0.5) = 30°。
Worked example: A right-angled triangle has an opposite side of 5 cm and an adjacent side of 8 cm. Find the angle θ to the nearest degree.
例题：一个直角三角形的对边为 5 cm，邻边为 8 cm。求角 θ，精确到度。
Use the tangent ratio because the two known sides are the opposite and adjacent sides:
使用正切比，因为已知的两条边是对边和邻边：
tan θ = 5 ÷ 8
Apply the inverse tangent function to both sides:
对两边同时使用反正切函数：
θ = tan⁻¹(5 ÷ 8) ≈ tan⁻¹(0.625) ≈ 32.0°
Make sure your calculator is in degree mode, not radian mode. In IGCSE questions, angles are almost always measured in degrees.
确保你的计算器处于角度模式，而不是弧度模式。在 IGCSE 考试中，角度几乎总是以度为单位。
A quick check: if the opposite side is shorter than the adjacent side, the angle should be less than 45° when using tangent. This type of check helps you spot calculator errors.
快速检查：如果对边比邻边短，那么使用正切时角度应小于 45°。这种检查有助于发现计算器错误。
9. Angles of Elevation and Depression | 仰角与俯角
An angle of elevation is the angle measured upwards from the horizontal. An angle of depression is the angle measured downwards from the horizontal. Both are often used in trigonometry word problems.
仰角是从水平线向上测量的角。俯角是从水平线向下测量的角。两者经常出现在三角学应用题中。
In a typical problem, you might stand at a certain distance from a building and measure the angle of elevation to the top. The horizontal distance, the vertical height and the line of sight form a right-angled triangle.
在典型问题中，你可能站在离建筑物一定距离的地方，测量到建筑物顶端的仰角。水平距离、竖直高度和视线形成一个直角三角形。
Worked example: From a point 40 m from the base of a tower, the angle of elevation to the top is 28°. Find the height of the tower.
例题：从距离塔底 40 m 的一点测得塔顶的仰角为 28°。求塔的高度。
Let h be the height of the tower. The horizontal distance is the adjacent side, and the height is the opposite side, so use tangent:
设 h 为塔的高度。水平距离是邻边，高度是对边，因此使用正切：
tan 28° = h ÷ 40
Multiply both sides by 40:
将两边同时乘以 40：
h = 40 × tan 28° ≈ 40 × 0.5317 ≈ 21.3 m
Problems involving angle of depression are solved in the same way. The angle of depression from a cliff top to a boat is equal to the angle of elevation from the boat to the cliff top, because they are alternate angles.
涉及俯角的问题也可以用同样的方法解决。从悬崖顶部到船只的俯角等于从船只到悬崖顶部的仰角，因为它们是内错角。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
One common mistake is labelling the opposite and adjacent sides incorrectly. Always identify the marked angle first, then decide which side is opposite and which is adjacent.
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