📚 Right-Angled Triangles: Pythagoras and Trigonometry | 直角三角形：勾股定理与三角比
In IGCSE Mathematics, right-angled triangles appear constantly in geometry, mensuration, bearings and real-world problems. This revision guide brings together Pythagoras' theorem, sine, cosine, tangent, exact values and angle applications so you can solve these questions confidently.
在 IGCSE 数学中，直角三角形经常出现在几何、测量、方位角和实际问题中。本复习指南将汇总勾股定理、正弦、余弦、正切、特殊角精确值以及角度应用，帮助你自信地解答相关题目。
1. The Right-Angled Triangle and Its Labels | 直角三角形及其边名
Before using any formula, you must identify the hypotenuse, the opposite side and the adjacent side relative to the angle you are working with. The hypotenuse is always the longest side and lies opposite the right angle.
在使用任何公式之前，你必须先找出斜边、对边和邻边。斜边总是最长边，并且位于直角的对面。
The opposite side is directly across from the given acute angle, while the adjacent side is next to that angle and is not the hypotenuse. Labelling correctly prevents most careless mistakes.
对边是给定锐角正对面的边，邻边则是紧挨该角且不是斜边的边。正确标注边长可以避免大多数粗心错误。
ou use angle A, then side BC is the opposite, side AC is the adjacent and side AB is the hypotenuse.
在三角形 ABC 中，若角 C = 90°，当使用角 A 时，边 BC 是对边，边 AC 是邻边，边 AB 是斜边。
2. Pythagoras Theorem: Finding a Missing Side | 勾股定理：求未知边
For any right-angled triangle, Pythagoras' theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
对于任意直角三角形，勾股定理指出斜边的平方等于另外两条边的平方和。
c² = a² + b²
If you need the hypotenuse c, write c = √a² + b². If you need one shorter side a, rearrange to a = √c² − b².
如果需要求斜边 c，则写作 c = √a² + b²。如果需要求较短边 a，则变形为 a = √c² − b²。
Always check that the longest side is on one side of the equation by itself. Substituting correctly into the formula is more important than remembering which letter is which.
一定要检查最长边是否单独在等式一边。正确代入公式比记住字母对应关系更重要。
3. Converse of Pythagoras: Testing for Right Angles | 勾股定理逆定理：判断直角
The converse of Pythagoras' theorem allows you to test whether a triangle with three known sides is right-angled. If the square of the longest side equals the sum of the squares of the other two sides, the triangle is right-angled.
勾股定理的逆定理可用于判断已知三边的三角形是否为直角三角形。如果最长边的平方等于另外两边的平方和，则该三角形是直角三角形。
with sides 5 cm, 12 cm and 13 cm is right-angled because 5² + 12² = 25 + 144 = 169 = 13².
例如，边长为 5 cm、12 cm 和 13 cm 的三角形是直角三角形，因为 5² + 12² = 25 + 144 = 169 = 13²。
If the equation does not balance, the triangle is not right-angled. This is a common IGCSE question type and requires no angle measurement.
如果等式不成立，则该三角形不是直角三角形。这是 IGCSE 常见题型，不需要测量角度。
4. Introducing Sine, Cosine and Tangent | 正弦、余弦与正切介绍
The three trigonometric ratios connect an acute angle in a right-angled triangle to the ratios of two side lengths. You must know the three definitions accurately.
三个三角比将直角三角形中的锐角与两边长度之比联系起来。你必须准确掌握这三个定义。
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
A common memory aid is SOH CAH TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
常用的记忆口诀是 SOH CAH TOA：正弦等于对边比斜边，余弦等于邻边比斜边，正切等于对边比邻边。
These ratios only apply when the triangle has a right angle. They do not work for non-right-angled triangles unless you use the sine rule or cosine rule.
这些比值仅适用于直角三角形。对于非直角三角形，它们不适用，除非使用正弦定理或余弦定理。
5. Choosing the Correct Trigonometric Ratio | 选择正确的三角比
To choose the correct ratio, first tick the side you need to find and the side you already know. Then decide whether the pair involves the hypotenuse and the opposite, the hypotenuse and the adjacent, or the opposite and the adjacent.
要选择正确的三角比，先标出你需要求的边和已知边。然后判断这对边是“斜边与对边”、“斜边与邻边”还是“对边与邻边”。
If you know or need the opposite and hypotenuse, use sine. If you know or need the adjacent and hypotenuse, use cosine. If you know or need the opposite and adjacent, use tangent.
如果已知或需要求对边和斜边，使用正弦。如果已知或需要求邻边和斜边，使用余弦。如果已知或需要求对边和邻边，使用正切。
Write the formula with the unknown side in the correct position before substituting numbers. This reduces rearrangement errors.
在代入数值前，先把公式中的未知边写在正确位置。这样可以减少移项错误。
6. Finding Missing Sides Using Trigonometry | 用三角函数求未知边
When one acute angle and one side are known, you can find another side by setting up an equation with the appropriate trigonometric ratio.
当一个锐角和一条边已知时，你可以通过列出一个合适三角比的方程来求另一条边。
then the opposite side can be found from sin 30° = x ÷ 10, giving x = 10 × sin 30° = 5 cm.
例如，若 θ = 30°，斜边为 10 cm，则对边可由 sin 30° = x ÷ 10 求得，即 x = 10 × sin 30° = 5 cm。
Remember to use the correct mode on your calculator. IGCSE trigonometry uses degrees, so your calculator must show DEG, not RAD or GRAD.
记得在计算器上使用正确的角度模式。IGCSE 三角学使用角度制，因此计算器必须显示 DEG，而不是 RAD 或 GRAD。
Always give answers to an appropriate degree of accuracy, usually three significant figures unless the question states otherwise.
答案通常保留三位有效数字，除非题目另有要求。
7. Finding Missing Angles Using Inverse Trigonometry | 用反三角函数求未知角
If two sides are known and you need to find an angle, use the inverse trigonometric functions sin⁻¹, cos⁻¹ or tan⁻¹.
如果已知两边并需要求角，可使用反三角函数 sin⁻¹、cos⁻¹ 或 tan⁻¹。
θ = sin⁻¹(opposite ÷ hypotenuse)
θ = cos⁻¹(adjacent ÷ hypotenuse)
θ = tan⁻¹(opposite ÷ adjacent)
jacent side is 5 m, then θ = tan⁻¹(4 ÷ 5). On a calculator this gives approximately 38.7°.
例如，若对边为 4 m，邻边为 5 m，则 θ = tan⁻¹(4 ÷ 5)。计算器计算结果约为 38.7°。
The inverse function is sometimes written as arcsin, arccos or arctan. You should be able to recognise both notations.
反函数有时写作 arcsin、arccos 或 arctan。你应能识别这两种表示法。
8. Exact Trigonometric Values for Special Angles | 特殊角的精确三角值
IGCSE exams often ask for exact values for 30°, 45° and 60°. You can derive these from the unit circle or special triangles, but you should memorise them.
IGCSE 考试经常要求给出 30°、45° 和 60° 的精确值。你可以从单位圆或特殊三角形推导，但应当记住它们。
The exact values are shown in the table below. These values must be given without a calculator when the question asks for an exact answer.
精确值如下表所示。当题目要求精确答案时，必须在不使用计算器的情况下写出这些值。
| Angle 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 |
60°, which often helps you check your memory. Tan 45° = 1 is also very frequently tested.
注意 sin 30° = cos 60°，cos 30° = sin 60°，这常可以帮助你检查记忆是否正确。tan 45° = 1 也是常考内容。
9. Angles of Elevation and Depression | 仰角与俯角
An angle of elevation is the angle measured upwards from the horizontal line of sight. An angle of depression is measured downwards from the horizontal.
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。
These angles are always made with the horizontal, not with a vertical wall or sloping edge. In diagrams, draw a thin horizontal line at the observer's eye level.
这些角始终与水平线形成，而不是与竖直墙壁或斜边形成。在图中，应在观察者视线高度画一条水平线。
A common IGCSE problem gives an angle of depression from a window to a car and asks for the horizontal distance. The angle of depression equals the angle of elevation at the car because they are alternate angles.
常见的 IGCSE 题目给出从窗口到汽车的俯角，要求水平距离。俯角等于汽车处的仰角，因为它们是内错角。
Convert the word problem into a right-angled triangle, label the known side, the unknown side and the angle, then use SOH CAH TOA.
将文字题转化为直角三角形，标出已知边、未知边和角度，然后使用 SOH CAH TOA。
10. Pythagoras and Trigonometry Combined | 勾股定理与三角函数的综合应用
Many exam questions require you to use Pythagoras' theorem to find one side first, and then use trigonometry to find an angle, or the other way round.
许多考题要求先用勾股定理求出一条边，再用三角函数求角度，或者反过来。
angle with opposite 6 cm and adjacent 8 cm, you can find the hypotenuse by Pythagoras: c = √6² + 8² = 10 cm, then find an angle using tan⁻¹(6 ÷ 8).
例如，在直角三角形中，对边为 6 cm，邻边为 8 cm，可用勾股定理求斜边：c = √6² + 8² = 10 cm，然后用 tan⁻¹(6 ÷ 8) 求角。
Three-dimensional problems often involve finding a diagonal on one face first, then using that diagonal to find an angle in the 3D shape.
三维问题通常先求一个面上的对角线，然后利用该对角线求三维图形中的角。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is labelling the opposite and adjacent sides from the wrong angle. Always stand at the angle you are using before deciding which side is opposite.
一个常见错误是依据错误的角度标注对边和邻边。务必先确定所使用的角，再判断哪一条边是对边。
Another mistake is using Pythagoras' theorem to find a shorter side without rearranging. Remember to subtract the square of the known shorter side from the square of the hypotenuse.
另一个错误是在求较短边时没有正确移项使用勾股定理。记住用斜边的平方减去已知较短边的平方。
Check that your calculator is in degree mode. A single question answered in radian mode can lose many marks.
检查计算器是否处于角度模式。一道题若使用弧度模式可能会导致大量失分。
Always show your working by writing the ratio, substituting numbers and then solving. Even if the final answer is wrong, method marks can be awarded.
始终展示解题步骤：写出三角比、代入数值、再求解。即使最终答案有误，也可以获得方法分。
12. Worked Example and Final Practice | 例题与最终练习
Worked example: A ladder 5 m long leans against a wall. The foot of the ladder is 2 m from the wall. Find the angle between the ladder and the ground.
例题：一架 5 m 长的梯子斜靠在墙上。梯子底端距离墙 2 m。求梯子与地面的夹角。
Use cosine: cos θ = adjacent ÷ hypotenuse = 2 ÷ 5 = 0.4. Therefore θ = cos⁻¹(0.4) ≈ 66.4°.
将地面距离标记为邻边 2 m，梯子为斜边 5 m。使用余弦：cos θ = 邻边 ÷ 斜边 = 2 ÷ 5 = 0.4。因此 θ = cos⁻¹(0.4) ≈ 66.4°。
For practice, try finding the height the ladder reaches on the wall using Pythagoras' theorem: h = √5² − 2² = √21 ≈ 4.58 m. This double check uses both key skills.
作为练习，可尝试用勾股定理求梯子在墙上的高度：h = √5² − 2² = √21 ≈ 4.58 m。这种双重检验同时使用了两个核心技能。
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