📚 Pythagoras' Theorem and Right-Angled Trigonometry | 勾股定理与直角三角形三角比
In IGCSE Mathematics, two foundational tools for working with right‑angled triangles are Pythagoras' theorem and the trigonometric ratios sine, cosine and tangent. These methods allow you to calculate unknown side lengths and angles in a huge range of geometric problems, from simple triangles to real‑world bearings and 3D applications. Mastering these skills is essential for success on the Edexcel exam.
在 IGCSE 数学中，处理直角三角形的两个基础工具是勾股定理以及三角函数正弦、余弦和正切。这些方法可以帮助你计算各种几何问题中的未知边长和角度，从简单的三角形到现实中的方位角和三维应用。掌握这些技能对于 Edexcel 考试至关重要。
1. Right‑angled triangles and labelling | 直角三角形与边的命名
Every right‑angled triangle has a hypotenuse – the side opposite the right angle, always the longest side. The other two sides are called legs. When working with a specific acute angle, we label the sides relative to that angle: opposite is the side facing the angle, and adjacent is the side next to the angle (not the hypotenuse). Correct labelling is the first step to avoiding mistakes.
每个直角三角形都有一条斜边——即直角对边，总是最长的一边。另外两条边称为直角边。当针对某个锐角进行计算时，我们相对于该角来命名各边：对边是正对该角的边，邻边是紧挨着该角的边（不是斜边）。正确命名是避免错误的第一步。
2. Pythagoras' theorem statement and proof | 勾股定理的表述与证明
Pythagoras' theorem states that in any right‑ se. Edexcel often expects you to recall a simple proof, such as using four identical right‑angled triangles arranged inside a square to show that (a+b)² − 2ab = c².
勾股定理指出，在任何直角三角形中，斜边的平方等于另外两条边的平方和：a² + b² = c²，其中 c 是斜边。Edexcel 经常要求学生回忆起一个简单的证明，比如用四个相同的直角三角形在一个大正方形内拼出一个较小的正方形，从而得出 (a+b)² − 2ab = c²。
3. Using Pythagoras to find missing sides | 用勾股定理求未知边长
To find the hypotenuse given the two shorter sides, use c = √a² + b². To find a shorter side, rearrange to a = √c² − b². For example, if a triangle has legs of 6 cm and 8 cm, the hypotenuse is √6² + 8² = √36+64 = √100 = 10 cm. If the hypotenuse is 13 cm and one leg is 5 cm, the missing leg is √13² − 5² = √169−25 = √144 = 12 cm.
已知两条直角边求斜边，用 c = √a² + b²。求直角边则变形为 a = √c² − b²。例如，若直角边为 6 cm 和 8 cm，斜边为 √6² + 8² = √36+64 = √100 = 10 cm。若斜边为 13 cm，一条直角边为 5 cm，则未知直角边为 √13² − 5² = √169−25 = √144 = 12 cm。
4. Trigonometric ratios: sine, cosine, tangent | 三角函数：正弦、余弦、正切
In any right‑ djacent. To remember them, use the mnemonic SOH CAH TOA. These ratios are constant for a given acute angle regardless of the size of the triangle.
在任何直角三角形中，三个主要的三角比定义为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记忆口诀是 SOH CAH TOA。这些比值对于给定的锐角是恒定的，与三角形的大小无关。
5. Finding unknown sides using trigonometry | 用三角函数求未知边长
tance, given angle 35° and adjacent side 8 cm, the opposite side is 8 × tan 35° ≈ 5.60 cm.
已知一个锐角和一条边，可以求任何其他边。选择包含已知边和所求边的三角比，列出方程：比值 = 分数，求解未知数。例如，已知角 35° 和邻边 8 cm，对边 = 8 × tan 35° ≈ 5.60 cm。
6. Finding angles using inverse trigonometry | 用反三角函数求角度
. When using inverse functions, always check your calculator is in degree mode (DEG) for IGCSE questions.
已知两条边，可以用计算器上的反三角函数求未知锐角：θ = sin⁻¹(对边/斜边)，θ = cos⁻¹(邻边/斜边) 或 θ = tan⁻¹(对边/邻边)。使用反函数时，务必确认计算器处于角度模式 (DEG)，以应对 IGCSE 题目要求。
7. Solving problems with angles of elevation and depression | 仰角与俯角问题
Angles of elevation are measured upwards from a horizontal line; angles of depression are measured downwards from a horizontal line. These problems almost always involve creating a right‑angled triangle where the horizontal line is one leg and the vertical distance is another. The angle of depression from the top of a cliff to a boat equals the angle of elevation from the boat to the cliff (alternate interior angles).
仰角是从水平线向上测量；俯角是从水平线向下测量。这类问题几乎总是构建一个直角三角形，水平线为一条直角边，垂直距离为另一条直角边。悬崖顶部到船的俯角等于船到悬崖的仰角（内错角相等）。
8. Bearings and right‑angled trigonometry | 方位角与直角三角比
A bearing is a direction measured clockwise from north, written as a three‑digit angle (e.g. 045°). Many bearing problems can be simplified by drawing a north‑south line and identifying a right‑angled triangle with known angles. Use trigonometry to find the distance travelled east/west or north/south. Always start by placing a right angle where the path meets a cardinal direction.
方位角是从正北顺时针方向测量的角度，用三位数字表示（如 045°）。许多方位角问题可以通过绘制南北线并确定含有已知角的直角三角形来简化。用三角函数求向东/向西或向北/向南移动的距离。始终从路径与基本方向相交处放置直角开始。
9. 3D Pythagoras and trigonometry | 立体几何中的勾股定理与三角比
Three‑dimensional problems involve finding lengths inside cuboids, prisms or square‑based pyramids. First identify a right‑angled triangle within the 3D shape (usually on a face or a cross‑section). If the required side is not directly visible, you may need to use Pythagoras twice – once to find a diagonal of a face, then again to find the space diagonal. For angles, the same approach works using trigonometric ratios.
三维问题包括求长方体、棱柱或正四棱锥内部的长度。先找出三维图形内的一个直角三角形（通常在一个面或截面上）。若所需边不直接可见，可能需要两次使用勾股定理——一次求面对角线，再一次求体对角线。求角度时同理，用三角比进行计算。
10. Exact trigonometric values for key angles | 特殊角的精确三角函数值
Edexcel IGCSE requires you to know exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°. These values can be derived from an equilateral triangle of side 2 (for 30° and 60°) and an isosceles right‑angled triangle of legs 1 (for 45°). They often appear in non‑calculator papers, so memorising them is crucial.
Edexcel IGCSE 要求掌握 0°, 30°, 45°, 60° 和 90° 的精确三角函数值。这些值可以由边长为 2 的等边三角形得到（用于 30° 和 60°），以及直角边长为 1 的等腰直角三角形得到（用于 45°）。它们常出现在非计算器试卷中，因此记住这些值至关重要。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 or √3/3 |
| 45° | 1/√2 or √2/2 | 1/√2 or √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
These exact values help simplify expressions without a calculator and are frequently tested alongside Pythagoras and trigonometric ratios.
这些精确值有助于在没有计算器的情况下简化表达式，并且经常与勾股定理和三角比一同考查。
11. Choosing between Pythagoras and trigonometry | 勾股定理与三角比的选择
A common exam pitfall is using the wrong tool. Use Pythagoras' theorem when you have two sides and need the third, and the triangle is right‑angled. Use trigonometry when you have an angle (other than the right angle) and a side, or need to find an angle. Some mixed problems require both – for example, finding a missing side with trigonometry first, then using Pythagoras to verify or find another side.
常见的考试陷阱是选错工具。当已知两条边求第三条边，且三角形是直角三角形时，用勾股定理。当你已知一个角（非直角）和一条边，或需要求一个角时，用三角函数。有些混合题需要两者兼用——例如，先用三角比求一条未知边，再用勾股定理验证或求另一条边。
12. Exam strategy and common mistakes | 考试策略与常见错误
Always clearly label your triangle and write the formula you plan to use. Show full working, including substitution steps, to gain method marks even if the final answer is wrong. Common mistakes include misidentifying the hypotenuse, confusing opposite and adjacent, forgetting to square for Pythagoras, using degrees when the calculator is in radians, and rounding too early in a multi‑step calculation. If a question asks for a bearing, remember the three‑digit format and direction from north.
始终清楚地标记三角形并写出你打算使用的公式。展示完整的运算过程，包括代入步骤，以便即使最终答案错误也能拿到方法分。常见错误包括：错误识别斜边、混淆对边和邻边、忘记给勾股定理平方、在计算器为弧度模式时使用角度，以及在多步运算中过早舍入。如果题目要求求方位角，记住三位数格式以及从正北开始的方向。
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