📚 T-K-1017: IGCSE Trigonometry – Pythagoras, Sine and Cosine Rules | IGCSE 三角函数：勾股定理、正弦与余弦定理
This revision guide covers the core IGCSE trigonometry toolkit: Pythagoras' theorem, the sine and cosine rules, exact values, bearings and 3D applications. Worked examples are kept simple so that you can apply each formula with confidence.
本复习指南涵盖 IGCSE 三角学的核心工具：勾股定理、正弦定理与余弦定理、特殊角精确值、方位角以及三维应用。示例保持简洁，帮助你自信地使用每一个公式。
1. Trigonometry in IGCSE: Why It Matters | IGCSE 三角函数的重要性
IGCSE trigonometry appears in both Core and Extended tiers. Core mainly uses right-angled triangles, Pythagoras' theorem and the basic sin, cos, tan ratios. Extended adds the sine rule, cosine rule and the sine area formula for non-right-angled triangles.
IGCSE 三角学在 Core 和 Extended 层级都会出现。Core 主要涉及直角三角形、勾股定理以及基本的 sin、cos、tan 比。Extended 增加了正弦定理、余弦定理和用于非直角三角形的正弦面积公式。
Questions often combine trigonometry with bearings, 3D shapes, graphs or algebra, so strong triangle skills are essential for problem solving.
试题经常将三角学与方位角、三维图形、图像或代数结合起来，因此扎实的三角形技巧对解题至关重要。
2. Right-Angled Triangles and Pythagoras' Theorem | 直角三角形与勾股定理
In a right-angled triangle, the longest side is the hypotenuse, opposite the right angle. If the hypotenuse is c and the other two sides are a and b, then the theorem is written as:
在直角三角形中，最长边是斜边，位于直角的对侧。若斜边为 c，另外两条边为 a 和 b，则该定理写作：
c² = a² + b²
This relation only works for right-angled triangles. It is often used to find a missing side when the other two sides are known.
这一关系只适用于直角三角形。它常用于在已知另外两边时求未知边。
rearrange the formula: a² = c² − b². This avoids sign errors when the hypotenuse is given.
求较短边时，可重排公式：a² = c² − b²。这样可以在已知斜边时避免符号错误。
Example: If a = 6 cm, b = 8 cm, then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
示例：如果 a = 6 cm，b = 8 cm，则 c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。
3. Trigonometric Ratios: sin, cos, tan | 三角比：sin、cos、tan
In a right-angled triangle, the three basic trigonometric ratios are defined using the opposite side, adjacent side and hypotenuse. For an angle θ, they are:
在直角三角形中，三个基本三角比利用对边、邻边和斜边来定义。对于角 θ，它们为：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
A useful memory aid is SOH CAH TOA: Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, Tangent equals Opposite over Adjacent.
一个有用的记忆方法是 SOH CAH TOA：正弦等于对边除以斜边，余弦等于邻边除以斜边，正切等于对边除以邻边。
st a wall, making a 40° angle with the ground. The height reached is 8 × sin 40° ≈ 5.14 m.
示例：一个 8 m 的梯子斜靠在墙上，与地面成 40° 角。到达的高度为 8 × sin 40° ≈ 5.14 m。
4. Exact Values for Special Angles | 特殊角的精确值
IGCSE non-calculator papers often require exact values for 30°, 45° and 60°. You should memorise these.
IGCSE 无计算器试卷常要求 30°、45° 和 60° 的精确值，需要记住。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Remember that sin 30° = cos 60°, sin 60° = cos 30°, and tan θ = sin θ / cos θ.
记住 sin 30° = cos 60°，sin 60° = cos 30°，并且 tan θ = sin θ / cos θ。
5. The Sine Rule: Non-Right-Angled Triangles | 正弦定理：非直角三角形
For any triangle ABC, with side a opposite angle A, side b opposite angle B and side c opposite angle C, the sine rule states:
对于任意三角形 ABC，边 a 对向角 A，边 b 对向角 B，边 c 对向角 C，正弦定理指出：
a / sin A = b / sin B = c / sin C
Use the sine rule when you know two angles and one side, or two sides and a non-included angle.
当已知两角和一边，或已知两边和一个非夹角时，使用正弦定理。
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