📚 Mastering Trigonometric Ratios: Sine, Cosine and Tangent | 掌握三角函数比：正弦、余弦与正切
Trigonometry in IGCSE mathematics begins with the three fundamental ratios in a right-angled triangle: sine, cosine and tangent. These ratios connect the sides of a triangle to its acute angles and appear in many geometry and real-world problems. Mastering them is essential for success in both Core and Extended papers.
IGCSE 数学中的三角学从直角三角形里的三个基本比开始：正弦、余弦和正切。这些比把三角形的边与锐角联系起来，出现在许多几何和实际问题中。掌握它们是打好核心卷和扩展卷基础的关键。
1. The Right-Angled Triangle Vocabulary | 直角三角形术语
In any right-angled triangle, the longest side is called the hypotenuse. It is always opposite the right angle. The other two sides are named relative to a chosen acute angle: the side facing that angle is the opposite, and the side next to it is the adjacent. The hypotenuse never changes label, but opposite and adjacent depend on which angle you are using.
在任何一个直角三角形中，最长的边称为斜边。它总是正对着直角。另外两条边根据所选的锐角来命名：正对着这个角的边称为对边，与这个角相邻的边称为邻边。斜边的名称始终不变，但对边和邻边取决于你使用哪个角。
2. Labelling Sides: Opposite, Adjacent and Hypotenuse | 标记边：对边、邻边与斜边
Suppose angle θ is one of the two acute angles. The side directly across from θ is the opposite. The side next to θ, but not the hypotenuse, is the adjacent. A common mistake is to label the adjacent as the remaining side without checking the angle. Always draw a small diagram and write O, A and H next to the sides before selecting a formula.
假设角 θ 是两个锐角之一。正对着 θ 的边是对边。与 θ 相邻但不是斜边的边是邻边。常见错误是不检查角度就把剩下的边标为邻边。在选择公式之前，一定要画出简图并在边上标出 O、A 和 H。
- Hypotenuse: opposite the right angle | 斜边：正对着直角
- Opposite: facing the chosen angle θ | 对边：正对着所选的角 θ
- Adjacent: next to θ, not the hypotenuse | 邻边：与 θ 相邻，不是斜边
3. Defining Sine, Cosine and Tangent | 定义正弦、余弦和正切
The three trigonometric ratios are defined as follows: sine of θ equals the opposite side divided by the hypotenuse, cosine of θ equals the adjacent side divided by the hypotenuse, and tangent of θ equals the opposite side divided by the adjacent side. The memory aid SOH-CAH-TOA helps recall these definitions. It is important to learn both the fraction forms and their word meanings.
三个三角比的定义如下：θ 的正弦等于对边除以斜边，θ 的余弦等于邻边除以斜边，θ 的正切等于对边除以邻边。记忆口诀 SOH-CAH-TOA 可以帮助记住这些定义。重要的是既要记住分数形式，也要理解其含义。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
4. Choosing the Correct Ratio | 选择正确的比值
To decide which ratio to use, first identify which side you know or want to find, and which other side is given. If the two sides involved are the opposite and hypotenuse, use sine. If they are adjacent and hypotenuse, use cosine. If they are opposite and adjacent, use tangent. Selecting the wrong ratio is one of the most frequent errors, so write down the two sides before choosing.
要决定使用哪个比，首先要确定你已知或想求的是哪条边，以及给出的另一条边是哪条边。如果涉及的两条边是对边和斜边，就使用正弦。如果是邻边和斜边，就使用余弦。如果是对边和邻边，就使用正切。选错比值是最常见的错误之一，因此在选择之前先把两条边写下来。
| Given Sides | Ratio to Use |
| Opposite + Hypotenuse | sin θ |
| Adjacent + Hypotenuse | cos θ |
| Opposite + Adjacent | tan θ |
5. Using a Calculator for Trigonometry | 使用计算器进行三角计算
IGCSE calculators can evaluate trigonometric functions directly. Make sure the calculator is in degree mode, not radian mode, unless the question uses radians. To find sin 30°, press sin then 30, or use the exact value if you are working without a calculator. For inverse operations, use the shift or second function key together with sin⁻¹, cos⁻¹ or tan⁻¹ to find an angle from a ratio.
IGCSE 计算器可以直接计算三角函数。确保计算器处于角度模式而不是弧度模式，除非题目中使用弧度。要求 sin 30° 时，按下 sin 再输入 30；如果不使用计算器，也可以直接用精确值。反运算则使用 shift 或第二功能键配合 sin⁻¹、cos⁻¹ 或 tan⁻¹，从一个比值求角度。
Remember that sin⁻¹ θ is not the same as 1/sin θ. The notation means the inverse sine function, which undoes the sine operation. | 记住 sin⁻¹ θ 不等于 1/sin θ。这个记号表示反正弦函数，它撤销正弦运算。
6. Finding a Missing Side | 求缺失的边长
involved, use sine: sin 35° = x / 12. Multiply both sides by 12 to obtain x = 12 sin 35°.
求缺失的边长时，选择包含已知边、未知边和已知角的比值。例如，如果斜边为 12 厘米、角为 35°，并且你需要求角的对边，设它为 x。因为涉及对边和斜边，使用正弦：sin 35° = x / 12。两边同乘以 12 得到 x = 12 sin 35°。
x = 12 sin 35° ≈ 6.88 cm
Always keep the unknown on its own first, then substitute the numbers. This reduces rearrangement errors and makes your working clear to the examiner. | 总是先让未知数单独在一边，再代入数字。这样可以减少移项错误，也能让阅卷老师清楚地看到你的解题过程。
7. Finding a Missing Angle | 求缺失的角度
potenuse is 8, then sin θ = 5/8. Therefore θ = sin⁻¹(5/8). The calculator gives θ ≈ 38.7°.
求角度时，先用已知边写出比值，然后使用反三角函数。例如，如果 θ 的对边是 5，斜边是 8，那么 sin θ = 5/8。因此 θ = sin⁻¹(5/8)。计算器给出 θ ≈ 38.7°。
Use the inverse ratio that matches the sides: opposite and adjacent give tan⁻¹, adjacent and hypotenuse give cos⁻¹. Then round your answer to the degree of accuracy requested in the question. | 使用与边匹配的反比值：对边和邻边用 tan⁻¹，邻边和斜边用 cos⁻¹。然后按题目要求的精确度四舍五入答案。
8. Angles of Elevation and Depression | 仰角与俯角
Angles of elevation and depression are measured from the horizontal. The angle of elevation is the angle above the horizontal when you look up at an object. The angle of depression is the angle below the horizontal when you look down from a higher point. These angles are equal when they are alternate angles formed by parallel horizontal lines.
仰角和俯角是从水平线开始测量的。仰角是你向上看物体时与水平线之间的角。俯角是你从较高处向下看时与水平线之间的角。当它们由两条平行水平线形成时，这两个角作为内错角是相等的。
In a diagram, draw the horizontal line, mark the angle clearly, and identify the right-angled triangle before applying SOH-CAH-TOA. Many questions ask you to find the height of a cliff, tower or tree from a given distance and angle. | 在图中，画出水平线，清楚地标出角度，并在应用 SOH-CAH-TOA 之前找出直角三角形。许多题目会要求你根据给定的距离和角度求出悬崖、塔或树的高度。
9. Applications: Bearings and Real-World Problems | 应用：方位角与实际问题
Bearings are measured clockwise from north and written as three digits, such as 045° or 210°. Trigonometry can find the distance travelled north, east, or the angle between a ship's path and the north direction. Break the journey into right-angled triangles, label the sides and use sine, cosine or tangent.
方位角是从北方顺时针测量并用三位数表示的角，例如 045° 或 210°。三角学可以用来求向北或向东移动的距离，或者船的航线与北方方向之间的角。把路线分解成直角三角形，标出各边，再使用正弦、余弦或正切。
For example, a plane flies 20 km on a bearing of 060°. The eastward distance is 20 sin 60° km, and the northward distance is 20 cos 60° km. Sketching the north and east components as perpendicular sides makes the right-angled triangle clear. | 例如，一架飞机沿着 060° 的方位角飞行 20 公里。向东的距离是 20 sin 60° 公里，向北的距离是 20 cos 60° 公里。把向北和向东的分量画成互相垂直的边，直角三角形就清晰了。
10. Exact Trigonometric Values | 特殊角的精确三角函数值
, 30°, 45°, 60° and 90°. For example, sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 45° = √2/2 and cos 30° = √3/2. These values come from special triangles: the isosceles right-angled triangle and the equilateral triangle split in half.
IGCSE 考试通常要求你知道 0°、30°、45°、60° 和 90° 的精确值。例如 sin 30° = 1/2、cos 60° = 1/2、tan 45° = 1、sin 45° = √2/2、cos 30° = √3/2。这些值来自特殊三角形：等腰直角三角形和等边三角形的一半。
| θ | sin θ | cos θ | tan θ |
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 or √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common mistakes include using the wrong mode on the calculator, labelling sides incorrectly, and confusing sine with cosine. Also remember that sin θ and sin⁻¹ θ are inverse operations, not reciprocals. Draw a diagram for every question, write the SOH-CAH-TOA rule, and round only at the final step.
常见错误包括计算器模式选择错误、边标记错误以及混淆正弦和余弦。还要记住 sin θ 与 sin⁻¹ θ 是逆运算，不是倒数。每道题都画出图形，写出 SOH-CAH-TOA 规则，并且只在最后一步才进行四舍五入。
- Check the angle is acute when using right-angled triangle trigonometry | 使用直角三角形三角学时确认角度是锐角
- Use exact values where the question asks for exact form | 题目要求精确形式时使用精确值
- Do not round intermediate answers; keep full calculator accuracy until the end | 不要对中间答案四舍五入；在最后一步之前保留计算器的完整精度
12. Summary and Checklist | 总结与清单
By the end of this topic, you should be able to label the hypotenuse, opposite and adjacent sides, write the three trigonometric ratios, find missing sides and angles, and apply trigonometry to elevation, depression and bearings. Practise past paper questions under timed conditions to build confidence.
在本主题结束时，你应该能够标出斜边、对边和邻边，写出三个三角比，求缺失的边和角，并能把三角学应用于仰角、俯角和方位角。限时练习历年真题，增强信心。
Final check: can you explain why tan θ = sin θ / cos θ for any acute angle? If yes, you are ready for the next level of trigonometric identities. | 最后检查：你能解释为什么对于任何锐角都有 tan θ = sin θ / cos θ 吗？如果能，你就为下一级的三角恒等式做好了准备。
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