📚 G-4 Pythagoras' Theorem and Trigonometry | IGCSE 数学：勾股定理与三角函数
In IGCSE Mathematics, right-angled triangle problems connect algebra, geometry, and real-life measurement. Pythagoras' theorem gives a rule for the three sides of a right triangle, while sine, cosine, and tangent relate the angles to pairs of sides. This unit is essential for both Core and Extended candidates and often appears in multi-step questions involving bearings, elevation, and 3D shapes.
在 IGCSE 数学中，直角三角形问题将代数、几何与现实测量联系在一起。勾股定理给出了直角三角形三条边的规则，而正弦、余弦和正切将角度与边的比值联系起来。本单元对核心和扩展考生都至关重要，经常出现在涉及方位角、仰角和三维图形的多步问题中。
1. Right-Angled Triangles and Notation | 直角三角形与符号标记
A right-angled triangle contains one 90° angle. The side opposite the right angle is called the hypotenuse, and it is always the longest side. When we choose one of the other two acute angles and call it θ, the side directly facing θ is the opposite side, and the side next to θ that is not the hypotenuse is the adjacent side.
直角三角形包含一个 90° 的角。直角对面的边称为斜边，它总是最长的边。当我们选择另外两个锐角之一并称它为 θ 时，正对 θ 的边就是对边，紧挨着 θ 且不是斜边的边就是邻边。
In any triangle, the largest side lies opposite the largest angle. Since the 90° angle is always the largest in a right triangle, the hypotenuse is always longer than either of the two legs. This helps you check whether a side labelled as the hypotenuse is reasonable.
在任意三角形中，最大的边对着最大的角。由于 90° 角在直角三角形中总是最大的，因此斜边总是比任何一条直角边长。这有助于你检查标记为斜边的边是否合理。
2. Pythagoras' Theorem Statement | 勾股定理的表述
For any right-angled triangle with legs a and b and hypotenuse c, the theorem states:
对于任意直角三角形，设两条直角边为 a 和 b，斜边为 c，定理表述为：
c² = a² + b²
This means the square of the hypotenuse equals the sum of the squares of the other two sides. It only applies to right-angled triangles, so always check for the 90° angle before using it.
这意味着斜边的平方等于另外两条边的平方和。它只适用于直角三角形，因此使用前一定要确认存在 90° 角。
es satisfy a² + b² = c², then the triangle must contain a right angle opposite the side c.
这个定理也可以反过来检验一个三角形是否为直角三角形：如果三条边满足 a² + b² = c²，那么边 c 所对的角一定是直角。
3. Finding the Missing Hypotenuse | 求缺失的斜边
a² + b² and take the positive square root. For example, if a = 6 cm and b = 8 cm, then c = √6² + 8² = √36 + 64 = √100 = 10 cm.
如果已知两条较短的边，将它们代入 c² = a² + b²，再取正的平方根。例如，如果 a = 6 cm，b = 8 cm，那么 c = √6² + 8² = √36 + 64 = √100 = 10 cm。
Always give the positive square root because a length cannot be negative. In exact-answer questions, leave your answer as a simplified surd, such as √50 = 5√2. Do not round unless the question gives a specific rounding instruction.
始终取正的平方根，因为长度不能为负。在要求精确答案的题目中，请把答案写成化简后的根式，例如 √50 = 5√2。除非题目给出明确的四舍五入要求，否则不要取近似值。
4. Finding a Missing Shorter Side | 求缺失的短边
When the hypotenuse and one shorter side are given, rearrange the formula. If the missing leg is a, then:
当已知斜边和一条短边时，需要对公式进行变形。如果缺失的直角边为 a，那么：
a = √c² - b²
For example, if c = 13 cm and b = 5 cm, then a = √13² - 5² = √169 - 25 = √144 = 12 cm. Note the subtraction inside the square root, not addition.
例如，如果 c = 13 cm，b = 5 cm，那么 a = √13² - 5² = √169 - 25 = √144 = 12 cm。注意平方根内是减法，而不是加法。
A common error is writing a = √c² + b², but this would make the missing side longer than the hypotenuse, which is impossible. Always set up the equation so that the hypotenuse squared is on one side and the two legs squared are on the other.
一个常见错误是写成 a = √c² + b²，但这会让缺失的边比斜边还长，这是不可能的。列方程时要始终
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