📚 Pythagoras Theorem and Trigonometric Ratios | 勾股定理与三角比
In IGCSE Mathematics, right-angled triangles appear constantly in geometry, mensuration and real-world problems. Two powerful sets of tools allow us to find unknown sides and angles: Pythagoras' theorem, which links the three sides, and the trigonometric ratios sine, cosine and tangent, which connect angles to side lengths. This revision guide explains both topics step by step, with worked examples and exam tips.
在 IGCSE 数学中，直角三角形在几何、测量和实际问题中经常出现。我们有两组强大的工具来求未知边和角：勾股定理把三条边联系起来，而正弦、余弦和正切三个三角比把角度与边长联系起来。本复习指南将逐步解释这两个主题，并给出例题和考试技巧。
1. Right-Angled Triangles and Labelling | 直角三角形与边标记
A right-angled triangle has one 90° angle. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides are called the 'legs' or shorter sides. When using trigonometry, the sides are named relative to a chosen acute angle θ: the side opposite θ is the opposite, and the side next to θ (not the hypotenuse) is the adjacent.
直角三角形有一个 90° 角。直角对面的边是最长的边，称为斜边。另外两条边称为直角边。使用三角函数时，边要相对于选定的锐角 θ 来命名：θ 对面的边是对边，与 θ 相邻的边（不是斜边）是邻边。
Always draw and label the triangle clearly before starting a calculation. If the same triangle is used with a different acute angle, the opposite and adjacent labels swap over, but the hypotenuse never changes.
在开始计算前，一定要先画出三角形并清楚标出各边。如果同一个三角形换用另一个锐角，对边和邻边的标记会互换，但斜边永远不变。
- Hypotenuse: opposite the right angle, always the longest | 斜边：直角对面，永远是最长边
- Opposite: opposite the chosen angle θ | 对边：选定角 θ 对面的边
- Adjacent: next to θ, not the hypotenuse | 邻边：与 θ 相邻，但不是斜边
2. Pythagoras' Theorem: The Core Formula | 勾股定理：核心公式
For any right-angled triangle with side lengths a, b and hypotenuse c, Pythagoras' theorem states:
对于任意直角三角形，若直角边长为 a 和 b，斜边长为 c，则勾股定理为：
a² + b² = c²
The theorem only works for right-angled triangles. It says the square on the hypotenuse equals the sum of the squares on the other two sides. You can use it to find a missing side when the other two sides are known.
该定理只适用于直角三角形。它表明斜边的平方等于两条直角边的平方和。已知另外两条边时，可以用它来求缺失的边。
Pythagoras' theorem is named after the Greek mathematician Pythagoras, but the relationship was known in earlier civilisations such as Babylon and Egypt. In IGCSE exams, it is usually applied in pure number form or within word problems.
勾股定理以古希腊数学家毕达哥拉斯命名，但这一关系在巴比伦、埃及等更早的文明中已被认识。在 IGCSE 考试中，它通常以纯数字形式或出现在文字应用题中。
3. Calculating the Hypotenuse | 计算斜边
If a = 3 cm and b = 4 cm, substitute into a² + b² = c²:
若 a = 3 厘米，b = 4 厘米，代入 a² + b² = c²：
3² + 4² = c² → 9 + 16 = c² → c² = 25 → c = 5 cm
Always take the positive square root because a length cannot be negative. In exam questions, remember to include the correct units.
总是取正的平方根，因为长度不能为负。考试中要记得写上正确的单位。
s the question asks for a decimal approximation. For example, if a = 2 and b = 3, then c = √13 cm.
如果结果不是完全平方数，除非题目要求小数近似值，否则应将答案保留为根式形式。例如，若 a = 2，b = 3，则 c = √13 厘米。
4. Calculating a Shorter Side | 计算直角边
e the formula to isolate the unknown shorter side. For example, if c = 13 cm and a = 5 cm:
移项公式以分离未知直角边。例如，若 c = 13 厘米且 a = 5 厘米：
b² = c² - a² = 13² - 5² = 169 - 25 = 144 → b = 12 cm
Subtract the known square from the hypotenuse square, never add the two given sides unless you are sure which side is missing.
从斜边的平方中减去已知直角边的平方，不要将两条已知边相加，除非你确定缺失的是哪条边。
. Remember that the hypotenuse squared is always the largest value, so you must subtract when finding a shorter side.
一个常见错误是写成 b² = c² + a²。记住斜边的平方总是最大的值，所以求直角边时必须用减法。
5. Converse of Pythagoras' Theorem | 勾股定理的逆定理
for the side lengths of a triangle, then the triangle is right-angled. This is useful for proving that a given triangle contains a 90° angle.
逆定理指出，如果三角形三边长满足 a² + b² = c²，那么该三角形是直角三角形。这常用于证明某个三角形含有 90° 角。
mple, a triangle with sides 6, 8, 10 satisfies 6² + 8² = 100 = 10², so it is right-angled.
例如，边长为 6、8、10 的三角形满足 6² + 8² = 100 = 10²，因此它是直角三角形。
² holds, and then conclude that the triangle is right-angled by the converse of Pythagoras' theorem.
在考试中，要清楚地写明 a² + b² = c² 成立，然后根据勾股定理的逆定理得出三角形是直角三角形的结论。
6. Introducing Trigonometric Ratios | 引入三角比
In a right-angled triangle, three basic ratios relate an acute angle θ to pairs of sides:
在直角三角形中，三个基本三角比把锐角 θ 与边的组合联系起来：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
The mnemonic SOH CAH TOA helps you remember which sides to use. 'SOH' means Sine is Opposite over Hypotenuse, 'CAH' means Cosine is Adjacent over Hypotenuse, and 'TOA' means Tangent is Opposite over Adjacent.
记忆口诀 SOH CAH TOA 帮助你记住该用哪些边。SOH 表示正弦是对边比斜边，CAH 表示余弦是邻边比斜边，TOA 表示正切是对边比邻边。
These ratios depend only on the angle θ, not on the size of the triangle. That is why two right-angled triangles with the same acute angle are similar, and their corresponding side ratios are equal.
这些比值只取决于角 θ 的大小，而与三角形的大小无关。因此，具有相同锐角的两个直角三角形是相似的，它们对应边的比值相等。
7. Finding a Missing Side with Trigonometry | 用三角函数求缺失边
the equation. For example, if θ = 35° and the hypotenuse is 12 cm, find the opposite side.
选择包含已知边和待求边的三角比，然后解方程。例如，若 θ = 35°，斜边为 12 厘米，求对边。
sin 35° = x / 12 → x = 12 × sin 35° ≈ 6.88 cm (3 s.f.)
Always ensure your calculator is in degree mode, not radians. Show the multiplication step to earn method marks in IGCSE exams.
务必确保计算器处于角度模式而不是弧度模式。在 IGCSE 考试中写出乘法步骤有助于获得方法分。
ow the opposite and adjacent sides, use tan θ = opposite / adjacent and solve accordingly.
如果是求邻边，则用 cos θ = 邻边 / 斜边。如果已知对边和邻边，则用 tan θ = 对边 / 邻边并相应求解。
8. Finding an Unknown Angle | 求未知角
ometric functions sin⁻¹, cos⁻¹ or tan⁻¹. For example, if opposite = 5 and hypotenuse = 13:
当你知道两条边而需要求角时，可使用反三角函数 sin⁻¹、cos⁻¹ 或 tan⁻¹。例如，若对边 = 5，斜边 = 13：
θ = sin⁻¹(5/13) ≈ 22.6° (1 d.p.)
Write the inverse function step clearly. Some exam papers accept the notation arcsin, but sin⁻¹ is standard. Make sure you are using the correct inverse for the ratio you have.
清楚地写出反三角函数步骤。有些试卷接受 arcsin 记号，但 sin⁻¹ 是标准写法。确保你使用的反函数与你所拥有的比值相对应。
= tan⁻¹(7/10) ≈ 35.0°. Choosing the wrong inverse function is a frequent source of error.
例如，若对边 = 7，邻边 = 10，则 θ = tan⁻¹(7/10) ≈ 35.0°。选错反函数是一个常见错误来源。
9. Exact Trigonometric Values | 特殊角的精确三角值
IGCSE exams often require exact values for 30°, 45° and 60° without a calculator. These values come from two special right-angled triangles: the 45°-45°-90° isosceles triangle and the 30°-60°-90° triangle.
IGCSE 考试常常要求不借助计算器写出 30°、45° 和 60° 的精确三角值。这些值来自两个特殊直角三角形：45°-45°-90° 等腰直角三角形和 30°-60°-90° 三角形。
| Angle θ | 角度 θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
广告 / Sponsored 【现货】剑桥国际高中教材 Cambridge IGCSE & O Level Additional Mathemati… 数学 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥198.0
更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法  在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao |