Sine and Cosine Rules in IGCSE Maths | IGCSE 数学中的正弦与余弦定理

📚 Sine and Cosine Rules in IGCSE Maths | IGCSE 数学中的正弦与余弦定理

In IGCSE Mathematics, non-right-angled triangles appear very often in geometry, bearings, and real-world modelling problems. The sine rule and cosine rule extend trigonometry beyond right-angled triangles, giving you a reliable way to find missing sides and angles, and to calculate areas without knowing a perpendicular height.

在 IGCSE 数学中，非直角三角形经常出现在几何、方位角和实际建模问题中。正弦定理与余弦定理将三角学从直角三角形推广到任意三角形，让你能够可靠地求未知边、未知角，并在不知道垂直高度的情况下计算面积。

1. Why the Sine and Cosine Rules Matter | 为什么正弦与余弦定理重要

In a right-angled triangle, you can always use Pythagoras’ theorem or the basic trigonometric ratios sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. However, many IGCSE problems involve triangles that do not contain a right angle. The sine and cosine rules are the standard tools for these non-right-angled triangles.

在直角三角形中，你总是可以使用毕达哥拉斯定理或基本三角比 sin θ = 对边/斜边、cos θ = 邻边/斜边、tan θ = 对边/邻边。然而，许多 IGCSE 题目涉及不含直角的三角形。正弦定理和余弦定理就是处理这些非直角三角形的标准工具。

They are essential for bearings problems, navigation, surveying, and for finding the area of a triangle when only two sides and the included angle are known.

它们在方位角问题、导航、测量以及仅已知两边及其夹角时求三角形面积等方面至关重要。

2. Standard Labelling Convention | 标准标记习惯

Before applying either rule, you must label the triangle consistently. IGCSE exam papers usually use capital letters A, B, C for the angles and lower-case letters a, b, c for the sides opposite those angles.

在应用任一规则之前，你必须一致地标记三角形。IGCSE 试卷通常用大写字母 A、B、C 表示角，用小写字母 a、b、c 表示这些角所对的边。

Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This convention is vital because both rules link an angle with its opposite side.

边 a 对着角 A，边 b 对着角 B，边 c 对着角 C。这一约定至关重要，因为两条规则都将角与其对边联系起来。

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

The sine rule states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of a triangle.

正弦定理指出，三角形任意一边的长度与其对角的正弦之比是一个常数。

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

You should use the sine rule when you know one complete pair of an angle and its opposite side, and one other piece of information: either another side or another angle.

当你已知一对完整的角及其对边，并且还知道另一条信息（另一条边或另一个角）时，应使用正弦定理。

The formula can also be turned upside down: sin A / a = sin B / b = sin C / c. Use whichever version makes your unknown the numerator.

该公式也可以倒过来使用：sin A / a = sin B / b = sin C / c。使用能让未知量位于分子的版本即可。

4. Worked Example: Finding a Missing Side | 例题：求未知边

In triangle ABC, angle A = 38°, angle B = 64° and side a = 7.2 cm. Find side b correct to 3 significant figures.

在三角形 ABC 中，角 A = 38°，角 B = 64°，边 a =

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