📚 IGCSE Mathematics 066: Pythagoras and Trigonometry | IGCSE数学066：勾股定理与三角函数
Pythagoras’ theorem and trigonometry are two of the most frequently tested topics in Edexcel IGCSE Mathematics. They form the foundation for solving problems involving right-angled triangles, vectors, bearings, and even 3D geometry. This revision guide will walk you through every key skill you need, step by step.
勾股定理和三角函数是Edexcel IGCSE数学中最高频考查的两大主题。它们构成了解决直角三角形、向量、方位角乃至三维几何问题的基础。本复习指南将带你逐步掌握每一个关键技能。
1. The Core Pythagorean Theorem | 勾股定理核心
For any right-angled triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. If the sides are \(a\), \(b\), and the hypotenuse is \(c\), we write:
对于任意直角三角形，斜边（最长边）的平方等于另外两边的平方之和。若两直角边为 \(a\)、\(b\)，斜边为 \(c\)，则写作：
c² = a² + b²
This equation works only for right-angled triangles. In an exam, always check for the little square symbol marking the right angle.
该等式只适用于直角三角形。在考试中，务必留意表示直角的小方块符号。
The theorem can be rearranged to find any side if two others are known. It is also the basis for the distance formula between two points on a coordinate grid.
该定理可以变形，在已知两边时求出第三边。它也是坐标系中两点间距离公式的基础。
2. Finding the Hypotenuse | 求斜边
When the two shorter sides of a right-angled triangle are given, you can find the hypotenuse by adding their squares and taking the square root.
当已知直角三角形的两条直角边时，可通过将两边的平方相加后开平方根来求斜边。
- English: Example: A triangle has sides 5 cm and 12 cm. Find the hypotenuse. Solution: c² = 5² + 12² = 25 + 144 = 169, so c = √169 = 13 cm.
- 中文：例如：一个三角形的两条直角边为5 cm和12 cm，求斜边。解：c² = 5² + 12² = 25 + 144 = 169，所以 c = √169 = 13 cm。
Always write the formula first, then substitute values. This earns method marks even if the final calculation is wrong.
务必先写出公式再代入数值。这样即使最终计算有误，也能获得方法分。
3. Finding a Shorter Side | 求直角边
To find a missing shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
求未知直角边时，用斜边的平方减去已知直角边的平方，再开平方根。
- English: Example: The hypotenuse is 17 cm and one leg is 8 cm. The other leg b satisfies b² = 17² − 8² = 289 − 64 = 225, so b = 15 cm.
- 中文：例如：斜边为17 cm，一条直角边为8 cm。另一条直角边 b 满足 b² = 17² − 8² = 289 − 64 = 225，所以 b = 15 cm。
Be careful with the order—do not subtract the hypotenuse square from the leg square.
注意顺序——不要将斜边的平方减去直角边的平方。
4. Pythagorean Triples | 勾股数
A Pythagorean triple is a set of three integers that satisfy the theorem. Recognising common triples can save time in non-calculator papers.
勾股数是一组满足定理的三个整数。识别常见勾股数可以在不使用计算器的试卷中节省时间。
| (3, 4, 5) | (5, 12, 13) | (7, 24, 25) |
| (8, 15, 17) | (9, 40, 41) | (12, 35, 37) |
Any multiple of a triple is also a triple. For example, doubling (3, 4, 5) gives (6, 8, 10).
任何一组勾股数的倍数仍是勾股数。例如，(3, 4, 5) 加倍得到 (6, 8, 10)。
5. Introducing Sin, Cos and Tan | 引入正弦、余弦和正切
Trigonometry links the angles of a right-angled triangle to the ratios of its sides. The three primary ratios are sine (sin), cosine (cos) and tangent (tan).
三角函数将直角三角形的角与其边之比联系起来。三种基本比率为正弦（sin）、余弦（cos）和正切（tan）。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Here, ‘opposite’ and ‘adjacent’ are relative to the angle θ, not the right angle.
这里的“对边”和“邻边”是相对于角 θ 而言的，而不是直角。
6. SOHCAHTOA – The Memory Aid | SOHCAHTOA 记忆口诀
SOHCAHTOA is a mnemonic to remember which sides go with which ratio:
SOHCAHTOA 是一个帮助你记住各比率对应哪条边的记忆口诀：
- SOH: Sin = Opposite / Hypotenuse
- CAH: Cos = Adjacent / Hypotenuse
- TOA: Tan = Opposite / Adjacent
- 中文： SOH：正弦 = 对边 / 斜边；CAH：余弦 = 邻边 / 斜边；TOA：正切 = 对边 / 邻边
Label the triangle first: mark the hypotenuse, then the side opposite the angle, then the remaining side adjacent to the angle.
先标注三角形：标出斜边，然后标出角的对边，最后标出角的邻边。
7. Finding an Angle | 求角度
When two sides are known, you can find an acute angle by using the inverse trigonometric functions: sin⁻¹, cos⁻¹, tan⁻¹ (also written arcsin, arccos, arctan).
当已知两边时，可通过反三角函数 sin⁻¹、cos⁻¹、tan⁻¹（也可写作 arcsin、arccos、arctan）求出锐角。
- English: Example: In a right triangle, the opposite side is 4 and the hypotenuse is 5. Then sin θ = 4/5 = 0.8, so θ = sin⁻¹(0.8) ≈ 53.13°.
- 中文：例如：在直角三角形中，对边为4，斜边为5。则 sin θ = 4/5 = 0.8，所以 θ = sin⁻¹(0.8) ≈ 53.13°。
Make sure your calculator is in degree mode. In Edexcel IGCSE, angles are usually in degrees unless stated otherwise.
确保计算器处于角度制模式。在Edexcel IGCSE中，除非特别说明，角度通常用度表示。
8. Finding a Side Length | 求边长
If one angle and one side are known, use the appropriate trigonometric ratio to calculate a missing side.
当已知一个角和一条边时，使用适当的三角比来计算未知边长。
- English: Example: A right triangle has angle 35° and adjacent side 10 cm. To find the opposite side x, use tan 35° = x / 10, so x = 10 × tan 35° ≈ 7.00 cm.
- 中文：例如：直角三角形中有角35°，邻边为10 cm。求对边 x，使用 tan 35° = x / 10，所以 x = 10 × tan 35° ≈ 7.00 cm。
Always decide which ratio to use by identifying the sides you have and the side you need.
通过确定你已知的边和你需要求的边来决定使用哪个比率。
9. Trigonometry in Bearings | 方位角中的三角函数
Bearings are three-figure angles measured clockwise from north. Trigonometry can be used to find distances and positions when solving bearing problems.
方位角是从正北方向顺时针测量的三位角度。在解决方位角问题时，可使用三角函数求距离和位置。
For example, if a ship travels 20 km on a bearing of 060°, its eastward displacement is 20 × sin 60° ≈ 17.32 km and its northward displacement is 20 × cos 60° = 10 km.
例如，一艘船沿方位角060°行驶20 km，其向东位移为 20 × sin 60° ≈ 17.32 km，向北位移为 20 × cos 60° = 10 km。
- English: The tangent ratio is also useful for calculating the bearing of a return journey after resolving a position into north and east components.
- 中文：正切比率也可用于将位置分解为南北和东西分量后计算返程方位角。
10. Exam-Style Problem: Combining Skills | 综合考查题：技能组合
Many high-scoring questions require you to use Pythagoras and trigonometry together, sometimes within the same triangle.
许多高分题要求你在同一三角形中综合使用勾股定理和三角函数。
- English: In triangle ABC, angle B = 90°, AB = 9 cm, and angle C = 40°. Find AC first: tan 40° = AB / BC, so BC = 9 / tan 40° ≈ 10.73 cm. Then use Pythagoras: AC² = 9² + 10.73² ≈ 196.0, so AC ≈ 14.0 cm.
- 中文：在三角形ABC中，角B = 90°，AB = 9 cm，角C = 40°。先求AC：tan 40° = AB / BC，所以 BC = 9 / tan 40° ≈ 10.73 cm。再用勾股定理：AC² = 9² + 10.73² ≈ 196.0，所以 AC ≈ 14.0 cm。
Always draw a clear diagram and label all given information. Check your answer for reasonableness.
务必画出清晰的图表并标注所有已知信息。检查答案是否合理。
Mastering these ten areas will give you a solid command of Pythagoras and trigonometry in the IGCSE Mathematics syllabus. Practice past-paper questions daily to build speed and accuracy.
掌握以上十个部分，你将能扎实驾驭IGCSE数学大纲中的勾股定理与三角函数。每天练习真题以提升速度与准确度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导