📚 Trigonometry at KS3: Essential Revision Guide | KS3 数学：三角函数考点精讲
Trigonometry might sound intimidating, but it simply explores the powerful link between angles and side lengths in right-angled triangles. This guide breaks down every key concept you need to master at KS3, from labelling triangles to solving real-world problems.
三角函数听起来可能有点吓人，但它其实只是研究直角三角形中角度与边长之间强大的联系。这篇指南将逐一分解你在 KS3 阶段需要掌握的每一个关键概念，从标记三角形到解决实际问题。
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. At KS3, we focus exclusively on right-angled triangles, where one angle is exactly 90°. These relationships remain constant for any given acute angle, no matter how large the triangle is drawn.
三角学是数学中研究三角形角度与边长关系的一个分支。在 KS3 阶段，我们只讨论直角三角形，其中一个角恰好是 90°。无论你把三角形画得多大，对于一个给定的锐角，这些边长比例关系始终保持不变。
2. Labelling Right-Angled Triangles | 直角三角形的标记
Before using any formula, you must label the triangle’s sides correctly relative to a chosen acute angle (often denoted θ). The hypotenuse is always the longest side and lies opposite the right angle. The opposite side is directly across from the angle θ, and the adjacent side is the side next to θ that is not the hypotenuse.
在使用任何公式之前，你必须根据所选锐角（通常用 θ 表示）正确标记三角形的边。斜边永远是最长的边，并且对着直角。对边是正对着角度 θ 的边，邻边则是挨着 θ 且不是斜边的那条边。
3. The Three Trigonometric Ratios | 三个三角比
Sine, cosine and tangent are ratios of two specific sides. The famous mnemonic SOH CAH TOA helps you remember them: Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent. These ratios allow you to find unknown sides or angles.
正弦、余弦和正切是两条特定边的比值。著名的记忆口诀 SOH CAH TOA 能帮你记住它们：正弦 = 对边 / 斜边，余弦 = 邻边 / 斜边，正切 = 对边 / 邻边。这些比值能让你求出未知的边长或角度。
- SOH: sin θ = opposite / hypotenuse
- CAH: cos θ = adjacent / hypotenuse
- TOA: tan θ = opposite / adjacent
4. Finding a Missing Side | 求未知边长
When you know one acute angle and one side, you can calculate an unknown side. First, label the sides relative to the given angle, decide which ratio involves the known side and the side you want, set up an equation, and solve it using a calculator or exact values. Always round your final answer as instructed.
当你知道一个锐角和一条边长时，就能计算出未知边长。先根据已知角标记各边，确定涉及已知边和你要求边的三角比，设定方程，然后用计算器或精确值求解。最后务必按要求对结果进行四舍五入。
Example: Find x in sin 35° = x / 8 → x = 8 × sin 35°
示例：已知 sin 35° = x / 8 → x = 8 × sin 35°
5. Finding a Missing Angle | 求未知角
If you know two sides but need an acute angle, use the inverse trigonometric functions sin⁻¹, cos⁻¹ or tan⁻¹. Write the ratio from the known sides, then apply the inverse function to find the angle. Make sure your calculator is set to degree mode, not radians.
如果你知道两条边长而需要求锐角，就要使用反三角函数 sin⁻¹、cos⁻¹ 或 tan⁻¹。先用已知边写出比值，再使用反函数求出角度。请确保你的计算器设置在度数模式下，而不是弧度模式。
If cos θ = 5/13, then θ = cos⁻¹(5/13)
若 cos θ = 5/13，则 θ = cos⁻¹(5/13)
6. Using Trigonometry in Practical Problems | 三角函数实际应用
Trigonometry appears in problems involving angles of elevation (looking up) and depression (looking down). Draw a clear diagram, identify the right-angled triangle, and label the sides based on whether you are finding a height, a distance, or an angle. Always include units in your final answer.
三角函数常见于涉及仰角和俯角的问题中。画一幅清晰的示意图，找出直角三角形，并根据你是在求高度、距离还是角度来标记各边。最终答案一定要带上单位。
Imagine a ladder leaning against a wall: the ladder is the hypotenuse, the ground is the adjacent side to the angle the ladder makes with the floor, and the wall height is the opposite side.
想象一架梯子靠在墙上：梯子是斜边，地面是梯子与地面夹角的邻边，墙的高度则是对边。
7. Exact Values for Key Angles (30°, 45°, 60°) | 特殊角的精确值
KS3 often introduces the exact trigonometric values for 30°, 45° and 60° without a calculator. These values come from special triangles: an isosceles right-angled triangle gives values for 45°, and half an equilateral triangle gives values for 30° and 60°. Memorising them saves time.
KS3 阶段通常会介绍 30°、45° 和 60° 角在没有计算器时的精确三角值。这些值来自特殊三角形：等腰直角三角形给出 45° 的值，等边三角形的一半则给出 30° 和 60° 的值。记住它们可以节省大量时间。
| Angle θ | sin θ | cos θ | tan θ |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
8. Common Mistakes to Avoid | 常见错误
Labelling the opposite and adjacent sides incorrectly is the most frequent error; always re-check which angle you are using. Another mistake is forgetting to set the calculator to degrees, which gives wildly wrong answers. Using the wrong ratio (e.g., sine instead of tangent) also leads to errors.
最常见的错误就是把对边和邻边标错了；一定要反复确认你正在使用哪个角。另一个错误是忘记把计算器设为度数模式，这会导致完全错误的答案。此外，用错三角比也会导致错误。
When solving for a side, students sometimes divide when they should multiply. Rearrange the equation carefully: if sin θ = x / hypotenuse, then x = sin θ × hypotenuse.
求边长时，有些学生会在该乘的时候除了。要仔细变换公式：如果 sin θ = x / 斜边，那么 x = sin θ × 斜边。
9. Practice Questions and Tips | 练习题与技巧
Try this: In a right-angled triangle, angle A = 40° and the adjacent side is 12 cm. Find the opposite side. First, identify that you need tangent because opposite and adjacent are involved: tan 40° = opposite / 12, so opposite = 12 × tan 40°. Always sketch the triangle first.
试试这道题：在一个直角三角形中，角 A = 40°，邻边长为 12 厘米，求对边长。首先，确定你需要使用正切，因为涉及到对边和邻边：tan 40° = 对边 / 12，所以 对边 = 12 × tan 40°。解题时永远先画出三角形草图。
For angle problems, if sin θ = 0.7, then θ = sin⁻¹(0.7). Use the shift or 2nd function key on your calculator. Write down every step to avoid simple arithmetic slips.
对于求角度的问题，若 sin θ = 0.7，则 θ = sin⁻¹(0.7)。使用计算器上的 shift 或第二功能键。写出每一步的计算过程，以避免简单的算术错误。
10. Summary and Key Takeaways | 总结与关键点
Trigonometry at KS3 revolves around the three ratios sine, cosine and tangent in right-angled triangles. Master labelling the hypotenuse, opposite and adjacent sides relative to a chosen angle, and use SOH CAH TOA to choose the correct ratio. With regular practice, you will confidently find missing sides and angles in any right-angled triangle.
KS3 阶段的三角函数围绕直角三角形中的正弦、余弦和正切这三个比值展开。掌握根据所选角来标记斜边、对边和邻边，并用 SOH CAH TOA 选择正确的比值。通过持续练习，你将能自信地求出任何直角三角形中未知的边长和角度。
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