📚 Pythagoras Theorem and Right-Angled Trigonometry | 勾股定理与直角三角形三角学
In IGCSE Mathematics, Pythagoras' theorem and right-angled trigonometry are essential for solving problems involving missing sides and angles in right-angled triangles. This revision guide covers the formulas, worked examples, common pitfalls, and exam strategies.
在 IGCSE 数学中，勾股定理和直角三角形三角学是求解直角三角形中缺失边长与角度的基础工具。本复习指南涵盖公式、例题、常见错误和考试策略。
1. Pythagoras' Theorem | 勾股定理
wo sides. If the hypotenuse is c and the two shorter sides are a and b, then a² + b² = c².
对于任意直角三角形，斜边上的正方形面积等于两条直角边上的正方形面积之和。如果斜边为 c，两条直角边为 a 和 b，则 a² + b² = c²。
a² + b² = c²
The theorem only works when the angle between a and b is exactly 90°.
该定理仅在 a 与 b 之间的夹角为 90° 时成立。
2. Identifying Sides | 识别边
The hypotenuse is always opposite the right angle and is the longest side. The opposite and adjacent sides depend on which acute angle you choose.
斜边始终与直角相对，是最长的边。对边和邻边取决于你选择的锐角。
For angle θ, label the side facing θ as opposite, the side next to θ but not the hypotenuse as adjacent, and the side opposite 90° as hypotenuse.
对于角 θ，将正对 θ 的边标记为对边，将紧挨 θ 且不是斜边的边标记为邻边，将正对 90° 的边标记为斜边。
3. Worked Example: Finding the Hypotenuse | 求斜边的例题
A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse c.
一个直角三角形的两条直角边分别为 6 cm 和 8 cm。求斜边 c。
c = √6² + 8² = √36 + 64 = √100 = 10 cm
So the hypotenuse is 10 cm.
因此斜边为 10 cm。
4. Worked Example: Finding a Shorter Side | 求直角边的例题
A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the missing side a.
一个直角三角形斜边为 13 cm，一条直角边为 5 cm。求缺失的直角边 a。
a = √13² − 5² = √169 − 25 = √144 = 12 cm
Rearrange the formula carefully: a² = c² − b².
请仔细移项：a² = c² − b²。
5. Trigonometric Ratios | 三角比
For an acute angle θ in a right-angled triangle, the three basic trigonometric ratios are defined as follows:
对于直角三角形中的锐角 θ，三个基本三角比定义如下：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
ne = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.
可以使用助记口诀 SOH CAH TOA：正弦 = 对边 / 斜边，余弦 = 邻边 / 斜边，正切 = 对边 / 邻边。
6. Choosing the Correct Ratio | 选择正确的三角比
First label the sides relative to the angle you are using. Then identify which two sides are known or needed.
首先根据所用角标记各边。然后找出已知或需要的两条边。
If you are working with opposite and hypotenuse, use sine. If you have adjacent and hypotenuse, use cosine. If you have opposite and adjacent, use tangent.
如果涉及对边和斜边，使用正弦；涉及邻边和斜边，使用余弦；涉及对边和邻边，使用正切。
7. Using Trigonometry to Find a Side | 用三角函数求边长
Example: θ = 35°, hypotenuse = 12 cm, find the opposite side x.
例题：θ = 35°，斜边 = 12 cm，求对边 x。
sin 35° = x / 12 → x = 12 sin 35° ≈ 6.88 cm
Make sure your calculator is in degree mode, not radian mode.
请确保计算器处于角度模式，而不是弧度模式。
8. Using Inverse Trigonometry to Find an Angle | 用反三角函数求角度
find the angle. For example, if opposite = 5 cm and hypotenuse = 13 cm, then sin θ = 5/13.
当已知两条边时，使用反函数求角度。例如，如果对边 = 5 cm，斜边 = 13 cm，则 sin θ = 5/13。
θ = sin⁻¹(5/13) ≈ 22.6°
On a calculator, use the sin⁻¹, cos⁻¹, or tan⁻¹ button after entering the ratio.
在计算器上，输入比值后使用 sin⁻¹、cos⁻¹ 或 tan⁻¹ 键。
9. Angles of Elevation and Depression | 仰角与俯角
The angle of elevation is the angle measured upwards from a horizontal line. The angle of depression is measured downwards from a horizontal line.
仰角是从水平线向上测量的角；俯角是从水平线向下测量的角。
These angles are equal when an observer looks at an object at a different height, because they are alternate angles on parallel horizontal lines.
当观察者看向不同高度的物体时，这两个角相等，因为它们是平行水平线上的内错角。
Use sine, cosine, or tangent to link vertical height, horizontal distance, and the line of sight.
使用正弦、余弦或正切将垂直高度、水平距离和视线联系起来。
10. Special Angles | 特殊角
Certain angles (30°, 45°, 60°) appear very often, and their exact trigonometric values should be memorised to save time.
某些角（30°、45°、60°）经常出现，应熟记它们的精确三角函数值以节省时间。
| 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 |
These exact values can be derived from an equilateral triangle and a square, which helps if you forget them in the exam.
这些精确值可由等边三角形和正方形推导出来，如果考试时忘记，也可以通过这种方式帮助回忆。
11. Common Mistakes | 常见错误
Do not use Pythagoras' theorem on non-right-angled triangles; for non-right-angled cases, use the sine rule or cosine rule.
不要在非直角三角形上使用勾股定理；非直角三角形应使用正弦定理或余弦定理。
Confusing opposite and adjacent sides is a frequent error. Always label the sides for the specific angle first.
混淆对边和邻边是常见错误。请始终先针对具体角标记各边。
Avoid rounding intermediate values too early; keep full calculator precision until the final answer.
避免过早舍入中间值；在最终答案之前保持计算器的完整精度。
Also check that your calculator is in degree mode for IGCSE angle questions.
还要检查计算器在 IGCSE 角度题中是否处于角度模式。
12. Summary and Exam Tips | 总结与考试技巧
Draw and label the triangle, list the known and unknown sides or angles, decide between Pythagoras and trigonometry, then solve step by step.
先画图并标记三角形，列出已知和未知的边或角，决定使用勾股定理还是三角函数，然后逐步求解。
Pythagoras is used when no angle other than the right angle is involved; trigonometry is used when an acute angle and at least one side are known or required.
当只涉及直角而不涉及其他角时使用勾股定理；当已知或求解锐角且至少已知一条边时使用三角函数。Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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