📚 Trigonometry: Pythagoras, Sine and Cosine Rules | 三角函数：勾股定理、正弦与余弦定理
In IGCSE Mathematics, trigonometry moves beyond the right-angled triangle. You need to apply Pythagoras' theorem precisely, use sine, cosine and tangent ratios, and then extend these ideas to non-right-angled triangles with the sine rule, cosine rule and the sine area formula. This article revises the key formulae, shows the decision process, and highlights exam techniques.
在IGCSE数学中，三角学不局限于直角三角形。你需要准确应用勾股定理，使用正弦、余弦和正切比，然后通过正弦定理、余弦定理和正弦面积公式将这些思想推广到非直角三角形。本文复习关键公式，展示选择过程，并强调考试技巧。
1. From Right Angles to Any Triangle | 从直角三角形到任意三角形
Trigonometry begins with ratios in a right-angled triangle but becomes more powerful when applied to any triangle. In a right-angled triangle, one angle is fixed at 90°, so the three sides can be labelled as opposite, adjacent and hypotenuse relative to a chosen acute angle. For a general triangle, there is no fixed hypotenuse, so the sine rule and cosine rule replace the basic ratios.
三角学始于直角三角形中的比，但应用于任意三角形时更加强大。在直角三角形中，一个角固定为90°，因此三条边可以相对于选定的锐角标记为对边、邻边和斜边。对于一般三角形，没有固定的斜边，因此正弦定理和余弦定理取代了基本比。
A clear labelling system is essential: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This convention keeps the sine and cosine rules consistent and reduces careless mistakes.
清晰的标记系统至关重要：边 a 对角 A，边 b 对角 B，边 c 对角 C。这一约定使正弦定理和余弦定理保持一致，并减少粗心错误。
2. Pythagoras' Theorem Recap | 勾股定理回顾
heorem is used both to find a missing side and to test whether a triangle is right-angled.
在任何直角三角形中，斜边的平方等于另外两条边的平方和。如果 c 是斜边，a、b 是直角边，则 c² = a² + b²。该定理既用于求缺失边，也用于检验三角形是否为直角三角形。
c² = a² + b²
angle is right-angled with the right angle opposite side c. In exam questions, always identify the hypotenuse first because it is the longest side and is opposite the right angle.
其逆定理也很有用：如果三条边满足 a² + b² = c²，则该三角形是直角三角形，直角对边为 c。在考试题中，务必首先确定斜边，因为它是最长边，且直角对边。
3. Trigonometric Ratios: Sine, Cosine and Tangent | 三角函数：正弦、余弦与正切
e / adjacent. A common memory aid is SOH CAH TOA. These ratios allow you to connect angles and side lengths.
对于直角三角形中的锐角 θ，三个基本比是 sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。常用记忆法是 SOH CAH TOA。这些比将角度与边长联系起来。
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
You should memorise the exact values of these ratios for 0°, 30°, 45°, 60° and 90°. They appear frequently without a calculator, especially in exact-value questions.
你应该记住 0°、30°、45°、60° 和 90° 时这些比的精确值。它们经常在无计算器的题目中出现，尤其是在精确值问题中。
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Remember that tan θ is undefined at 90° because division by zero occurs. This is a common conceptual test point in IGCSE multiple-choice questions.
请记住，tan θ 在 90° 处没有定义，因为会出现除以零的情况。这是 IGCSE 选择题中常见的概念考点。
4. Finding a Side in a Right-Angled Triangle | 求直角三角形中的边长
ide is found by sin θ = opposite / hypotenuse. Then opposite = 10 sin 30° = 10 × ½ = 5 cm.
要求缺失边，选择使用已知边和未知边的比。例如，如果斜边为 10 cm，角 θ = 30°，对边通过 sin θ = 对边 / 斜边求得。于是对边 = 10 sin 30° = 10 × ½ = 5 cm。
opposite = 10 × sin 30° = 5 cm
Always write down the ratio first before substituting numbers. This shows your method clearly and allows partial credit if you make a numerical mistake later.
务必先写出比例式，再代入数值。这样可以清楚地展示你的方法，即使后面的数值计算有误，也能获得部分分数。
5. Finding an Angle with Inverse Ratios | 用反三角函数求角度
tan⁻¹(7/24) ≈ 16.26°. Always check that your calculator is in degree mode for IGCSE work.
当已知边的比但未知角度时，使用反三角函数 sin⁻¹、cos⁻¹ 和 tan⁻¹。例如，如果 tan θ = 7/24，则 θ = tan⁻¹(7/24) ≈ 16.26°。做IGCSE题目时务必检查计算器处于角度模式。
