📚 Pythagoras' Theorem and Trigonometric Ratios | 勾股定理与三角函数
In IGCSE Mathematics, right-angled triangles form the basis of two essential topics: Pythagoras' theorem and trigonometric ratios. Pythagoras' theorem connects the three side lengths of a right triangle, while sine, cosine and tangent connect side lengths to acute angles. These tools are used to find missing lengths, missing angles, and to solve practical problems in navigation, construction and 3D geometry.
在 IGCSE 数学中，直角三角形是两大核心主题的基础：勾股定理与三角函数。勾股定理建立了直角三角形三条边之间的关系，而正弦、余弦和正切则把边长与锐角联系起来。这些工具可用于求缺失的边长、缺失的角，以及解决航海、建筑和三维几何中的实际问题。
1. Identifying Right-Angled Triangles | 识别直角三角形
A right-angled triangle has one interior angle equal to 90°, and the side opposite this angle is always the longest side, called the hypotenuse. The other two sides are usually labelled as the adjacent and opposite sides relative to a chosen acute angle.
直角三角形有一个内角等于 90°，与这个角相对的那条边永远是最长的边，称为斜边。另外两条边则相对于所选的那个锐角，被标注为邻边和对边。
s the right angle, and in written questions the right angle may be stated in words such as 'angle ABC = 90°'.
在运用勾股定理或三角函数之前，一定要先确认这个三角形是直角三角形。在图形中，顶点处的小方框代表直角；在文字题中，直角也可能用 '角 ABC = 90°' 这样的表述给出。
In three-dimensional problems a right angle is often hidden: it appears where a line is perpendicular to a plane, or where a diagonal meets the base of a cuboid at 90°.
在三维问题中，直角常常是隐藏的：它出现在一条直线垂直于一个平面处，或者出现在长方体的体对角线与底面成 90° 的位置。
Remember the standard labelling convention: capital letters name the angles and vertices, while the matching lower-case letters name the sides opposite them. This makes it easy to refer to side a, angle A and so on without confusion.
记住标准标注习惯：大写字母表示角和顶点，对应的小写字母表示它们所对的边。这样就能轻松地指代边 a、角 A 等等，不会混淆。
If a triangle contains no right angle at all, Pythagoras' theorem and SOH CAH TOA cannot be used directly. In that case you must either draw a perpendicular to create two right-angled triangles, or wait until the sine rule and cosine rule are introduced.
如果三角形中根本没有直角，那么勾股定理和 SOH CAH TOA 都不能直接使用。此时你必须要么作一条垂线构造出两个直角三角形，要么等到学习正弦定理和余弦定理时再处理。
2. Pythagoras' Theorem: Statement and Meaning | 勾股定理：表述与含义
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If the shorter sides are a and b and the hypotenuse is c, then the relationship is written as follows.
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之和。若两条直角边为 a 和 b，斜边为 c，则该关系可写成下式。
a² + b² = c²
The letter c must always stand for the hypotenuse, the side facing the 90° angle. If you accidentally place a shorter side on the right-hand side of the equation, the answer will be wrong even though the algebra looks correct.
字母 c 必须始终代表斜边，即面对 90° 角的那条边。如果你不小心把一条直角边放在了等式的右侧，即使代数运算看起来正确，答案也是错的。
The theorem can be rearranged to make either shorter side the subject, which is exactly what you need when the hypotenuse is known but one leg is missing.
这个定理可以变形，使任意一条直角边成为主元，这正是当已知斜边、缺少一条直角边时所需要的形式。
a² = c² − b² and b² = c² − a²
The converse is equally important: if the squares of the two shorter sides add up to the square of the longest side, then the triangle must be right-angled. Exam questions often ask you to prove that a triangle is right-angled using exactly this test.
逆定理同样重要：如果两条较短边的平方之和等于最长边的平方，那么这个三角形一定是直角三角形。考试题经常要求你用这一判定方法证明某个三角形是直角三角形。
Pythagoras' theorem only holds for right-angled triangles. It is not true for equilateral, isosceles or scalene triangles in general, so never apply it without checking for the 90° angle first.
勾股定理只对直角三角形成立。它对一般的等边三角形、等腰三角形或不等边三角形并不成立，所以在没有确认 90° 角之前，绝不要随意套用。
3. Finding the Hypotenuse | 求斜边
When the two shorter sides are known and the hypotenuse is unknown, square both known sides, add them, and then take the square root. The hypotenuse is always longer than either of the other two sides, which gives you a quick way to check your answer.
当已知两条直角边、而斜边未知时，先把两条已知边平方、相加，然后再开平方。斜边总是比另外任何一条边长，这给你提供了一个快速检验答案的方法。
c = √a² + b²
as shorter sides of 6 cm and 8 cm. Then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
例题：一个直角三角形的两条直角边分别为 6 cm 和 8 cm。则 c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。
-integer values: if the legs are 7.2 m and 4.5 m, then c² = 51.84 + 20.25 = 72.09, and c = √72.09 ≈ 8.49 m to three significant figures.
非整数例题：若两条直角边为 7.2 m 和 4.5 m，则 c² = 51.84 + 20.25 = 72.09，c = √72.09 ≈ 8.49 m（保留三位有效数字）。
Keep full calculator accuracy through the working and round only at the end. Rounding the squares too early is one of the most common causes of lost marks in this topic.
在整个计算过程中保持计算器的完整精度，只在最后一步取近似值。过早对平方结果取近似值，是这一主题中最常见的失分原因之一。
Always state the units in your final answer and make sure they match the units given in the question. If the question mixes centimetres and metres, convert first.
最终答案一定要写出单位，并确保与题目给出的单位一致。如果题目中混用了厘米和米，请先换算。
4. Finding a Shorter Side | 求直角边
e hypotenuse, then take the square root. This is the rearrangement a² = c² − b² in action.
当已知斜边和一条直角边时，用斜边的平方减去已知直角边的平方，然后再开平方。这就是变形公式 a² = c² − b² 的实际运用。
a = √c² − b²
le is 13 cm and one leg is 5 cm. Then a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 cm.
例题：一个直角三角形的斜边为 13 cm，一条直角边为 5 cm。则 a² = 13² − 5² = 169 − 25 = 144，所以 a = √144 = 12 cm。
The single most frequent error here is adding instead of subtracting. Because the hypotenuse is the longest side, the answer for a leg must always be smaller than the hypotenuse, which is a useful sanity check.
这里最常见的错误是把减号写成加号。由于斜边是最长的边，求出的直角边一定小于斜边，这是一个很有用的合理性检验。
If the result is a surd such as √50, exam questions usually ask for the answer to three significant figures, so write √50 = 7.07 unless exact form is requested.
如果结果是 √50 这样的根式，考试题通常要求答案保留三位有效数字，所以写成 √50 = 7.07，除非题目要求精确形式。
Grid problems on coordinate axes are really Pythagoras questions in disguise: the horizontal and vertical differences between two points behave like the two legs of a right triangle.
坐标轴上的格点问题实际上就是伪装的勾股定理题：两点之间的水平差和竖直差，相当于直角三角形的两条直角边。
5. Pythagorean Triples and Common Values | 勾股数与常用数值
c² exactly. Recognising them saves time and lets you avoid unnecessary calculator work in the exam.
勾股数是指满足 a² + b² = c² 的三个整数。认出它们可以节省时间，并让你在考试中避免不必要的计算器运算。
The most important triples to memorise are 3-4-5, 5-12-13, 8-15-17, 7-24-25 and 9-40-41. Any multiple of a triple is also a triple, so 6-8-10, 9-12-15 and 15-36-39 all work.
最需要记住的勾股数是 3-4-5、5-12-13、8-15-17、7-24-25 和 9-40-41。任何勾股数的整数倍仍然是勾股数，所以 6-8-10、9-12-15 和 15-36-39 都成立。
| Primitive triple | 基本勾股数 | Multiple | 倍数形式 | Check | 验证 |
| 3, 4, 5 | 6, 8, 10 | 36 + 64 = 100 |
| 5, 12, 13 | 10, 24, 26 | 100 + 576 = 676 |
| 8, 15, 17 | 16, 30, 34 | 256 + 225 = 481? No — see note below |
| 7, 24, 25 | 14, 48, 50 | 196 + 576 = 772? No — see note below |
= 1156 = 34². The same structure holds for 7-24-25, since 7² + 24² = 49 + 576 = 625 = 25².
注意 8-15-17 这一族的规律：8² + 15² = 64 + 225 = 289 = 17²，而 16² + 30² = 256 + 900 = 1156 = 34²。7-24-25 也是同样的结构，因为 7² + 24² = 49 + 576 = 625 = 25²。
the set (n² − 1), (2n), (n² + 1) is a Pythagorean triple. For n = 4 this gives 15, 8, 17.
你还可以用下面的规律生成勾股数：对任何大于 1 的整数 n，三元组 (n² − 1)、(2n)、(n² + 1) 都是勾股数。取 n = 4，就得到 15、8、17。
6. Labelling Sides: Opposite, Adjacent and Hypotenuse | 标注边：对边、邻边与斜边
Trigonometric ratios depend entirely on which angle you are working with, so labelling must be done relative to a chosen acute angle, which is normally called θ (theta).
三角比完全取决于你以哪个角为出发点，因此标注必须相对于所选的那个锐角进行，这个角通常记为 θ（theta）。
The hypotenuse is fixed: it is always the side opposite the right angle and never changes, no matter which acute angle you choose.
斜边是固定的：它永远是直角的对边，无论你选择哪个锐角，它都不会改变。
The opposite side is the side facing your chosen angle θ, and the adjacent side is the side that touches θ and also touches the right angle. The adjacent side is the non-hypotenuse leg that forms part of angle θ.
对边是面对所选角 θ 的那条边，邻边则是既与 θ 相接、又与直角相接的那条边。邻边就是构成角 θ 的一部分、且不是斜边的那条直角边。
A quick test: if you swap from one acute angle to the other, the opposite and adjacent sides swap over, but the hypotenuse stays the same. Getting this wrong is the single biggest source of errors in trigonometry questions.
一个快速检验法：如果从其中一个锐角换成另一个锐角，对边和邻边会互换，但斜边保持不变。搞错这一点是三角函数题中最大的错误来源。
In worded problems, draw your own sketch and mark θ clearly before labelling anything. A labelled diagram takes twenty seconds and prevents most mistakes.
在文字应用题中，先自己画草图并清楚标出 θ，然后再标注各边。画一张标注好的图只需二十秒，却能避免大部分错误。
7. SOH CAH TOA: The Three Trigonometric Ratios | SOH CAH TOA：三大三角比
For a fixed acute angle θ in a right-angled triangle, the three trigonometric ratios are defined as follows, and they depend only on the size of θ, not on how large the triangle is.
对于直角三角形中一个固定的锐角 θ，三个三角比的定义如下，它们只取决于 θ 的大小，而与三角形的大小无关。
sin θ = opposite ÷ hypotenuse | cos θ = adjacent ÷ hypotenuse | tan θ = opposite ÷ adjacent
The mnemonic SOH CAH TOA encodes these three definitions: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.
口诀 SOH CAH TOA 概括了这三个定义：Sine 是对边比斜边，Cosine 是邻边比斜边，Tangent 是对边比邻边。
Because the hypotenuse is always the longest side, sin θ and cos θ must lie between 0 and 1 for any acute angle. Values greater than 1 are a signal that you have divided the wrong way round.
由于斜边永远是最长的边，对于任何锐角，sin θ 和 cos θ 必定介于 0 与 1 之间。若数值大于 1，说明你把除法方向弄反了。
Tangent, however, can take any positive value: tan θ is small for angles near 0°, equals exactly 1 at 45°, and grows without limit as θ approaches 90°.
但正切可以取任意正值：当 θ 接近 0° 时 tan θ 很小，在 45° 时恰好等于 1，而当 θ 趋近 90° 时它会无限增大。
rk and for proof questions in later courses: tan θ = sin θ ÷ cos θ, and sin²θ + cos²θ = 1.
这些三角比由两个恒等式相联系，它们在检验答案以及后续课程的证明题中很有用：tan θ = sin θ ÷ cos θ，以及 sin²θ + cos²θ = 1。
8. Finding a Missing Side with Trigonometry | 用三角比求缺失边长
To find a missing side, first identify which two sides are involved — the one you know and the one you want — then choose the ratio that links exactly those two. Label the sides opposite, adjacent and hypotenuse before you start.
求缺失边长时，首先判断题目涉及哪两条边——已知的那条和要求的那条——然后选择恰好联系这两条边的三角比。动手前先标出对边、邻边和斜边。
m, and the opposite side x is unknown. Since the sides involved are opposite and hypotenuse, use sine.
例题：在一个直角三角形中，θ = 35°，斜边为 12 cm，对边 x 未知。由于涉及的边是对边和斜边，所以使用正弦。
sin 35° = x ÷ 12 → x = 12 × sin 35° ≈ 12 × 0.5736 ≈ 6.88 cm
ide x is unknown, the relevant ratio is tangent, so x = 9 × tan 35° ≈ 9 × 0.7002 ≈ 6.30 m.
第二道例题：若 θ = 35°，邻边为 9 m，对边 x 未知，则适用的三角比是正切，于是 x = 9 × tan 35° ≈ 9 × 0.7002 ≈ 6.30 m。
≈ 7 ÷ 0.6428 ≈ 10.9. A common mistake is to multiply instead of divide in this situation.
当未知量出现在分母上时，要先变形再计算。例如，若 sin 40° = 7 ÷ x，则 x = 7 ÷ sin 40° ≈ 7 ÷ 0.6428 ≈ 10.9。这里常见的错误是把除法误做成乘法。
Check that your calculator is in degree mode. If it is set to radians, every trigonometric answer will be wrong, and this single setting has cost many candidates an entire question.
请检查计算器是否处于角度制（degree）模式。如果被设为弧度制（radian），所有三角函数答案都会出错，仅这一个设置就让许多考生丢掉整道题的分。
9. Finding a Missing Angle with Inverse Ratios | 用反三角比求缺失角
When two sides are known and the angle is unknown, form the correct ratio as a fraction, then apply the inverse trigonometric function to both sides. On most calculators these appear as sin⁻¹, cos⁻¹ and tan⁻¹.
当已知两条边、而角未知时，先写出正确的三角比分数形式，然后对等式两边施加反三角函数。在大多数计算器上，它们显示为 sin⁻¹、cos⁻¹ 和 tan⁻¹。
θ = sin⁻¹(opposite ÷ hypotenuse) | θ = cos⁻¹(adjacent ÷ hypotenuse) | θ = tan⁻¹(opposite ÷ adjacent)
is 13 cm. Then sin θ = 5 ÷ 13 ≈ 0.3846, so θ = sin⁻¹(0.3846) ≈ 22.6° to one decimal place.
例题：对边为 5 cm，斜边为 13 cm。则 sin θ = 5 ÷ 13 ≈ 0.3846，所以 θ = sin⁻¹(0.3846) ≈ 22.6°（保留一位小数）。
the adjacent side is 4.2 m, then tan θ = 7.5 ÷ 4.2 ≈ 1.7857, so θ = tan⁻¹(1.7857) ≈ 60.8°.
正切例题：若对边为 7.5 m，邻边为 4.2 m，则 tan θ = 7.5 ÷ 4.2 ≈ 1.7857，所以 θ = tan⁻¹(1.7857) ≈ 60.8°。
A useful check: in a right-angled triangle the two acute angles must add up to 90°. Once you have found one of them, subtract it from 90° to confirm the other angle.
一个有用的检验：直角三角形中的两个锐角之和必须等于 90°。求出其中一个后，用 90° 减去它即可验证另一个角。
Never confuse sin⁻¹ with 1 ÷ sin θ. The notation sin⁻¹ means 'the angle whose sine is', while 1 ÷ sin θ is the cosecant, which is not required at IGCSE level.
千万不要把 sin⁻¹ 与 1 ÷ sin θ 混为一谈。sin⁻¹ 表示"正弦值为某数的那个角"，而 1 ÷ sin θ 是余割，IGCSE 阶段并不需要。
10. Exact Trigonometric Values for Special Angles | 特殊角的精确三角值
Certain angles produce exact values that can be written as fractions or surds, and these values are frequently required in non-calculator papers. The key angles are 30°, 45° and 60°.
某些角会产生可以写成分数或根式的精确值，这些值在不可使用计算器的试卷中经常被考查。关键角是 30°、45° 和 60°。
| Angle | 角度 | sin | cos | tan |
| 30° | 1 ÷ 2 |
