Mastering Trigonometry for Edexcel IGCSE | 掌握三角函数：Edexcel IGCSE 数学复习

📚 Mastering Trigonometry for Edexcel IGCSE | 掌握三角函数：Edexcel IGCSE 数学复习

Trigonometry is a core topic in the Edexcel IGCSE Mathematics syllabus. It appears in both Paper 1 and Paper 2, often worth multiple marks in a range of question styles. This article will guide you through every key skill, from basic right-angled triangle ratios to advanced problem solving.

三角学是 Edexcel IGCSE 数学大纲中的核心主题。它在 Paper 1 和 Paper 2 中都会出现，并在多种题型中占有不小分值。本文将引导你掌握从直角三角形基本比值到高级问题解决的所有关键技能。

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1. Right-Angled Triangles: SOH CAH TOA | 直角三角形：SOH CAH TOA

For any right-angled triangle, we can define three trigonometric ratios for an angle θ. First identify the sides: the hypotenuse is the longest side, opposite the right angle; the opposite side is opposite the angle θ; the adjacent side is next to the angle θ and is not the hypotenuse.

对于任何直角三角形，我们可以定义角 θ 的三个三角比。首先识别各边：斜边是最长边，对着直角；对边是角 θ 所对的边；邻边是角 θ 旁边且不是斜边的边。

The three ratios are remembered by the mnemonic SOH CAH TOA:

这三个比值可以用口诀 SOH CAH TOA 记忆：

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent

Example: In a right-angled triangle, θ = 30°, hypotenuse = 10 cm. Find the opposite side.

例子：在直角三角形中，θ = 30°，斜边 = 10 厘米。求对边。

sin 30° = opposite / 10, so opposite = 10 × sin 30° = 5 cm

To find a missing angle, use the inverse functions: if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°. Make sure your calculator is in degree mode.

求缺失角时使用反函数：若 sin θ = 0.5，则 θ = sin⁻¹(0.5) = 30°。确保你的计算器处于角度模式。

Common exact values you should know:

你应该记住的常见精确值：

Angle	sin	cos	tan
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3

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2. The Sine Rule | 正弦定理

For any triangle with sides a, b, c opposite angles A, B, C respectively, the sine rule states:

对于任意三角形，边 a, b, c 分别对角 A, B, C，正弦定理表明：

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

Use the sine rule when:

使用正弦定理的情形：

• two angles and one side are known – to find another side;
• 已知两角和一边 – 求另一边；
• two sides and a non-included angle are known – to find another angle.
• 已知两边和其中一边的对角 – 求另一角。

Example: A = 40°, B = 60°, a = 8 cm. Find side b.

例子：A = 40°，B = 60°，a = 8 厘米。求边 b。

b = a × sin B / sin A = 8 × sin 60° / sin 40° ≈ 10.77 cm

Always check that the side you are finding corresponds to its opposite angle.

求边长时，始终确认所求边与其对角对应。

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3. The Cosine Rule | 余弦定理

The cosine rule relates a side to the other two sides and the included angle:

余弦定理将一条边与其他两边及其夹角联系起来：

a² = b² + c² − 2bc cos A

It is used in two main situations:

它主要用于两种情况：

• two sides and the included angle are known – to find the third side;
• 已知两边及其夹角 – 求第三边；
• all three sides are known – to find an angle.
• 已知三边 – 求某个角。

To find an angle, rearrange the formula:

求角时可变形公式：

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