📚 IGCSE Maths: Pythagoras' Theorem and Trigonometry | IGCSE 数学：勾股定理与三角函数
Pythagoras' theorem and basic trigonometry are essential topics in IGCSE Mathematics. They are used to find missing side lengths and angles in right-angled triangles, and they appear frequently on both Core and Extended papers. This article explains the key ideas, worked examples, and common exam pitfalls to help you revise efficiently.
勾股定理和基本三角函数是 IGCSE 数学中的重要内容。它们用于求解直角三角形中缺失的边长和角度，在核心卷和扩展卷中都频繁出现。本文讲解核心概念、典型例题和常见考试易错点，帮助你高效复习。
1. Pythagoras' Theorem Statement | 勾股定理表述
In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is always the longest side and is opposite the right angle.
在任何直角三角形中，斜边的平方等于另外两条直角边的平方之和。斜边总是最长边，并且位于直角的对边。
c² = a² + b²
Here, c represents the hypotenuse, while a and b represent the two shorter sides. You can rearrange this formula to find any side if you know the other two.
这里，c 表示斜边，a 和 b 表示两条较短边。如果你已知另外两条边，可以变形这个公式来求任一边。
2. Finding the Hypotenuse | 求斜边
To find the hypotenuse, you add the squares of the two shorter sides and then take the square root. Always check that the hypotenuse is the side you are looking for.
要求斜边，你需要先求出两条较短边的平方和，再开平方。始终要确认你要求的是斜边。
Example: If a = 6 cm and b = 8 cm, then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
例如：如果 a = 6 厘米，b = 8 厘米，那么 c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 厘米。
Do not forget to include units in your final answer. In an exam, missing units can cost you a mark, especially when the question asks for a real-life measurement such as length or distance.
答案中不要忘记写单位。在考试中，漏写单位可能会被扣分，尤其是题目要求长度或距离等实际测量值时。
3. Finding a Shorter Side | 求直角边
nuse. Then take the square root of the result. The formula can be written as a² = c² − b².
要求一条直角边，你需要用斜边的平方减去已知直角边的平方，然后对结果开平方。这个公式可以写作 a² = c² − b²。
Example: If c = 13 m and b = 5 m, then a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 m.
例如：如果 c = 13 米，b = 5 米，那么 a² = 13² − 5² = 169 − 25 = 144，所以 a = √144 = 12 米。
Many students accidentally add instead of subtracting when they are looking for a shorter side. Underline the word 'shorter side' in the question to remind yourself to subtract.
许多学生在求直角边时误将加法当作减法。你可以在题目中圈出“直角边”或“较短边”的字样，提醒自己要用减法。
4. Converse of Pythagoras' Theorem | 勾股定理逆定理
The converse of Pythagoras' theorem says that if the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is right-angled. This is useful when you are asked to prove whether a triangle contains a right angle.
勾股定理的逆定理指出：如果一个三角形最长边的平方等于另外两条边的平方和，那么这个三角形是直角三角形。当你需要证明一个三角形是否含有直角时，这一定理非常有用。
² = 225 and 9² + 12² = 81 + 144 = 225. Since they are equal, the triangle is right-angled.
例如：对于边长 9 厘米、12 厘米和 15 厘米的三角形，检验 15² = 225，而 9² + 12² = 81 + 144 = 225。两者相等，所以该三角形是直角三角形。
Always compare the square of the longest side with the sum of the squares of the two shorter sides. If the values are not equal, the triangle is not right-angled.
始终将最长边的平方与两条较短边平方之和进行比较。如果数值不相等，则该三角形不是直角三角形。
5. Introduction to Trigonometric Ratios | 三角函数比引入
Trigonometry connects the angles of a right-angled triangle to the ratios of its sides. The three basic trigonometric ratios are sine, cosine and tangent. You must learn the definitions of these ratios.
三角函数将直角三角形的角度与其边之比联系起来。三个基本的三角函数比是正弦、余弦和正切。你必须牢记这些比的定义。
First label the sides relative to the angle you are working with: the side opposite the angle, the side next to the angle but not the hypotenuse, and the hypotenuse. The adjacent side is always next to the angle and the right angle.
首先，根据你正在使用的角来标注各条边：该角的对边、该角旁边但不是斜边的边，以及斜边。邻边总是紧挨着该角和直角。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Here θ (theta) is one of the acute angles in the right-angled triangle. The hypotenuse is the same side for both acute angles, but the opposite and adjacent sides change when you switch angles.
这里 θ（西塔）是直角三角形中的一个锐角。对于两个锐角来说，斜边是同一条边，但当你更换角度时，对边和邻边会发生变化。
6. Sine Ratio | 正弦比
The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. It is written as sin θ. You can use the sine ratio to find a missing side or angle in a right-angled triangle.
一个角的正弦值等于对边长度与斜边长度之比，记作 sin θ。你可以使用正弦比来求直角三角形中缺失的边或角。
, then sin 30° = opposite / 10. Since sin 30° = 0.5, the opposite side is 0.5 × 10 = 5 cm.
例如：如果 θ = 30°，斜边为 10 厘米，那么 sin 30° = 对边 / 10。因为 sin 30° = 0.5，所以对边长度为 0.5 × 10 = 5 厘米。
side is 4 cm and the hypotenuse is 8 cm, then sin θ = 4/8 = 0.5, so θ = sin⁻¹(0.5) = 30°.
要求角度，可以在计算器上使用反正弦函数。例如，如果对边为 4 厘米，斜边为 8 厘米，那么 sin θ = 4/8 = 0.5，所以 θ = sin⁻¹(0.5) = 30°。
7. Cosine Ratio | 余弦比
The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. It is written as cos θ. Use this ratio when the question involves the adjacent side and the hypotenuse.
一个角的余弦值等于邻边长度与斜边长度之比，记作 cos θ。当题目涉及邻边和斜边时，可以使用这个比。
, then cos 60° = adjacent / 12. Since cos 60° = 0.5, the adjacent side is 0.5 × 12 = 6 cm.
例如：如果 θ = 60°，斜边为 12 厘米，那么 cos 60° = 邻边 / 12。因为 cos 60° = 0.5，所以邻边长度为 0.5 × 12 = 6 厘米。
When the angle is unknown, rearrange the ratio and use the inverse cosine function. Always check whether you should be using sine or cosine by looking at which two sides are given or required.
当角度未知时，重新整理比例并使用反余弦函数。始终通过观察题目中已知或要求的是哪两条边来判断应该使用正弦还是余弦。
8. Tangent Ratio | 正切比
The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. It is written as tan θ. The tangent ratio is helpful when you know the two shorter sides and need to find an angle.
一个角的正切值等于对边长度与邻边长度之比，记作 tan θ。当你已知两条直角边并需要求角度时，正切比非常有用。
de is 12 cm, then tan θ = 9/12 = 0.75. Using the inverse tangent, θ = tan⁻¹(0.75) ≈ 36.9°.
例如：如果对边为 9 厘米，邻边为 12 厘米，那么 tan θ = 9/12 = 0.75。使用反正切，θ = tan⁻¹(0.75) ≈ 36.9°。
Do not confuse tan θ with sin θ or cos θ. An easy way to remember the three ratios is the phrase 'SOH CAH TOA', where S stands for sine, C for cosine, and T for tangent.
不要把 tan θ 与 sin θ 或 cos θ 混淆。记住这三个比的简单方法是短语“SOH CAH TOA”，其中 S 代表正弦，C 代表余弦，T 代表正切。
9. Angles of Elevation and Depression | 仰角与俯角
Angles of elevation and depression are often tested in IGCSE trigonometry problems. The angle of elevation is the angle measured upwards from the horizontal line of sight. The angle of depression is the angle measured downwards from the horizontal.
仰角和俯角在 IGCSE 三角问题中经常出现。仰角是从水平视线向上测量的角，俯角是从水平线向下测量的角。
Always draw a clear diagram and mark the horizontal line, the line of sight and the angle. In right-angled triangle problems, the horizontal and vertical distances form the two shorter sides, while the line of sight is the hypotenuse.
始终画出清晰的示意图，标记水平线、视线和角度。在直角三角形问题中，水平距离和垂直距离构成两条直角边，而视线是斜边。
ation of the ladder is 65°, so you can find the height reached using tan 65° = height / 2.
例如：一架梯子斜靠在墙上，与地面成 65° 角。如果梯子的底端离墙 2 米，那么水平距离为 2 米，梯子的仰角为 65°，因此你可以用 tan 65° = 高度 / 2 来求出达到的高度。
10. Solving Right-Angled Triangle Problems | 直角三角形问题解决
In exam questions, you often need to combine Pythagoras' theorem and trigonometric ratios to solve multi-step problems. Start by identifying the right-angled triangle and the information given. Then decide whether to use Pythagoras or a trigonometric ratio.
在考试题中，你经常需要结合勾股定理和三角函数比来解决多步骤问题。首先识别直角三角形和给出的信息，然后决定使用勾股定理还是三角函数比。
- Label all sides and angles clearly. | 清楚地标注所有的边和角。
- Choose the correct method based on the given and required information. | 根据已知和所求信息选择正确的方法。
- Write the formula you are using before substituting values. | 在代入数值前写出所使用的公式。
- Check that your answer is reasonable, for example, the hypotenuse should be the longest side. | 检查答案是否合理，例如斜边应当是最长边。
the opposite side, so use sin 40° = opposite / 15. Then opposite = 15 × sin 40° ≈ 9.64 cm.
例如：在一个直角三角形中，一个角为 40°，斜边为 15 厘米，求对边。已知斜边并要求对边，所以使用 sin 40° = 对边 / 15。因此对边 = 15 × sin 40° ≈ 9.64 厘米。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many marks are lost in IGCSE trigonometry and Pythagoras questions because of small mistakes. Pay attention to units, decimal places, and whether your calculator is in degree mode.
在 IGCSE 三角和勾股定理题目中，许多失分是因为小错误。注意单位、小数位数，以及你的计算器是否处于角度模式。
- Do not mix up the opposite and adjacent sides. | 不要混淆对边和邻边。
- When using inverse trigonometric functions, round only the final answer, not intermediate steps. | 使用反三角函数时，只对最终答案四舍五入，不要对中间步骤取整。
- Pythagoras' theorem only works for right-angled triangles. | 勾股定理只适用于直角三角形。
- Always write down the formula before substituting numbers. | 在代入数字前先写出公式。
- If the question asks for a side length, use the normal trig ratio. If it asks for an angle, use the inverse trig function. | 如果题目要求边长，使用普通的三角函数比；如果要求角度，使用反三角函数。
When drawing diagrams, use a ruler and label the vertices and given lengths. A clear diagram reduces the chance of using the wrong side or angle.
画图时要使用直尺，并标注顶点和已知长度。清晰的示意图可以减少使用错误边或角的可能性。
12. Summary | 总结
Pythagoras' theorem and the sine, cosine and tangent ratios are powerful tools for solving right-angled triangle problems. Remember that Pythagoras deals with side lengths only, while trigonometric ratios connect angles with side lengths.
勾股定理以及正弦、余弦和正切比是解决直角三角形问题的有力工具。记住勾股定理只涉及边长，而三角函数比将角度与边长联系起来。
In the exam, take time to identify the type of triangle, label the sides, and choose the correct formula. Practice past paper questions regularly to build confidence and speed.
在考试中，花时间识别三角形类型，标注各边，并选择正确的公式。定期练习历年真题，以增强信心并提高解题速度。
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