IGCSE Maths: Sine and Cosine Rules | IGCSE 数学：正弦定理与余弦定理

📚 IGCSE Maths: Sine and Cosine Rules | IGCSE 数学：正弦定理与余弦定理

In IGCSE Mathematics, solving problems with non-right-angled triangles requires two powerful tools: the sine rule and the cosine rule. These rules allow you to find missing sides or angles in any triangle, not just right-angled ones. This article explains when and how to use each rule, covers the area formula involving sine, and includes common pitfalls and exam strategies.

在 IGCSE 数学中，解非直角三角形问题需要两个重要工具：正弦定理和余弦定理。这些定理能帮助你求解任意三角形中的未知边或角，而不仅限于直角三角形。本文将讲解何时以及如何使用每条定理，介绍含正弦的面积公式，并总结常见错误与考试策略。

1. Introduction to Non-Right-Angled Triangles | 非直角三角形简介

A triangle without a 90° angle is called a non-right-angled or oblique triangle. In IGCSE, labelling is standard: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

没有 90° 角的三角形称为非直角三角形或斜三角形。在 IGCSE 中，标注方式是标准的：边 a 对应对角 A，边 b 对应对角 B，边 c 对应对角 C。

Since SOH-CAH-TOA applies only to right-angled triangles, you need the sine rule and cosine rule to solve general triangles. Always label the given sides and angles clearly before choosing a rule.

因为 SOH-CAH-TOA 只适用于直角三角形，所以解一般三角形时需要使用正弦定理和余弦定理。在选择定理之前，一定要清楚地标出已知的边和角。

---

2. The Sine Rule: Formula and Use | 正弦定理：公式与使用

mbols, a / sin A = b / sin B = c / sin C. Equivalently, sin A / a = sin B / b = sin C / c.

正弦定理指出，边长与其对角正弦的比值是恒定的。用符号表示为：a / sin A = b / sin B = c / sin C。等价形式为：sin A / a = sin B / b = sin C / c。

a / sin A = b / sin B = c / sin C

b sin A / sin B. Use the inverted form when finding a missing angle: sin A = a sin B / b.

求未知边时使用形式：a = b sin A / sin B。求未知角时使用倒数形式：sin A = a sin B / b。

---

3. When to Use the Sine Rule | 何时使用正弦定理

e, if you know angle A, angle B, and side a, you can find side b with b = a sin B / sin A.

当已知两个角和一个边（AAS 或 ASA），或两条边和一个非夹角（SSA）时，可以使用正弦定理。例如，如果已知角 A、角 B 和边 a，则可通过 b = a sin B / sin A 求出边 b。

Pairing a known side with its opposite angle is the key to using the sine rule correctly. If that pair is missing, the sine rule is not the first choice.

使用正弦定理的关键是将已知边与其对角配对。如果没有这样的配对，正弦定理就不是首选方法。

---

4. The Cosine Rule: Formula and Use | 余弦定理：公式与使用

b² + c² - 2bc cos A. It can be rearranged to find an angle: cos A = (b² + c² - a²) / 2bc.

余弦定理将三角形的三条边与其中一个角联系起来。标准形式为：a² = b² + c² - 2bc cos A。它可以变形用于求角：cos A = (b² + c² - a²) / 2bc。

a² = b² + c² - 2bc cos A

This rule works for any triangle and becomes Pythagoras' A = 90° because cos 90° = 0. The side a must be opposite the angle A used in the formula.

该定理适用于任意三角形；当 A = 90° 时，因为 cos 90° = 0，它就变成勾股定理。公式中的边 a 必须与所用的角 A 相对。

---

广告 / Sponsored

5. When to Use the Cosine Rule | 何时使用余弦定理

e sides (SSS). If you need a missing side, use the standard form a² = b² + c² - 2bc cos A.

当已知两边及其夹角（SAS），或三条边（SSS）时，应使用余弦定理。如果求未知边，使用标准形式 a² = b² + c² - 2bc cos A。

. Do not use the sine rule in these cases unless you already have a matched side-angle pair.

如果求未知角，则使用变形形式 cos A = (b² + c² - a²) / 2bc。在这些情况下不要使用正弦定理，除非你已经有一组对应的边角对。

---

6. Area of a Triangle Using Sine | 用正弦求三角形面积

ides and the included angle: Area = ½ bc sin A. Similarly, Area = ½ ac sin B = ½ ab sin C.

如果已知两条边及其夹角，可以无需垂直高度就计算三角形面积：面积 = ½ bc sin A。类似地，面积 = ½ ac sin B = ½ ab sin C。

Area = ½ bc sin A

This formula is particularly useful when the perpendicular height is not given but two sides and their included angle are known. It is often used in IGCSE questions that combine area with sine or cosine rules.

当题目未给出垂直高度但已知两边及其夹角时，这个公式尤其有用。IGCSE 题目常会将面积公式与正弦或余弦定理结合起来考查。

---

7. The Ambiguous Case of the Sine Rule | 正弦定理的歧义情形

gle (SSA). Because sin θ = sin (180° - θ), there can be two possible values for the angle.

歧义情形发生在使用正弦定理求角，且已知两边和一个非夹角（SSA）时。由于 sin θ = sin (180° - θ)，角可能有两个可能的值。

d be 30° or 150°. Always check whether both solutions are valid in the triangle by testing if the angle sum is less than 180°.

例如，若 sin A = 0.5，则 A 可能是 30° 或 150°。务必检查两个解是否都能放入三角形，即角度和是否小于 180°。

If the unknown angle is opposite the longer given side, only the acute angle may be valid. Drawing a diagram helps you decide which answer is realistic.

如果未知角对的是较长的已知边，可能只有锐角解有效。画图有助于判断哪个答案更符合实际。

---

8. Bearings and Trigonometry | 方位角与三角学

Bearings are measured clockwise from north and written with three digits, such as 065° or 240°. Non-right-angled trigonometry is often used in navigation problems involving bearings.

方位角是从正北方向起顺时针测量的角度，并用三位数表示，例如 065° 或 240°。涉及方位角的导航问题常需要使用非直角三角学。

Draw a clear diagram, convert bearing directions into angles inside the triangle, and identify matched side-angle pairs before applying the sine or cosine rule.

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）