📚 IGCSE Maths: Trigonometry in Right-Angled Triangles | IGCSE 数学：直角三角形中的三角比
Trigonometry is the branch of mathematics that studies the relationships between the angles and side lengths of triangles. In IGCSE Mathematics, the focus is mainly on right-angled triangles, where one angle is exactly 90°.
三角学是研究三角形的角度与边长之间关系的数学分支。在 IGCSE 数学中，我们主要关注直角三角形，即其中一个角恰好为 90°。
These relationships allow us to find unknown sides or angles when some measurements are known. This skill is widely used in surveying, navigation, construction and physics.
这些关系使我们能够在已知部分测量值时求出未知的边或角。这一技能广泛应用于测量、航海、建筑和物理中。
1. What Is Trigonometry? | 什么是三角学？
Trigonometry is the branch of mathematics that studies the relationships between the angles and side lengths of triangles. In IGCSE Mathematics, we focus mainly on right-angled triangles, where one angle is 90°.
三角学是研究三角形的角度与边长之间关系的数学分支。在 IGCSE 数学中，我们主要关注直角三角形，即其中一个角为 90°。
These relationships let us find unknown sides or angles when some measurements are known, which is useful in surveying, navigation and construction.
这些关系使我们能够在已知部分测量值时求出未知的边或角，在测量、航海和建筑中非常有用。
Although trigonometry can be applied to any triangle using the sine and cosine rules, this topic focuses on the simpler right-angled case that appears frequently in IGCSE examination papers.
虽然三角学可以借助正弦定理和余弦定理应用于任意三角形，但本主题只关注更简单的直角三角形情形，这在 IGCSE 考试中经常出现。
2. Labelling a Right-Angled Triangle | 直角三角形的边标记
Before using trigonometric ratios, you must label the three sides of a right-angled triangle relative to a chosen acute angle θ.
在使用三角比之前，必须相对于所选的锐角 θ 标记直角三角形的三条边。
The hypotenuse is the longest side, opposite the right angle.
斜边是最长边，对着直角。
The opposite side is the side directly facing the angle θ.
对边是正对着角 θ 的边。
The adjacent side is the side next to θ, but not the hypotenuse.
邻边是紧挨着 θ 的边，但不是斜边。
If you choose the other acute angle instead of θ, the opposite and adjacent sides swap labels, but the hypotenuse remains the same.
如果你选择另一个锐角而不是 θ，对边和邻边的标记会互换，但斜边不变。
Clear labelling is the first step in any trigonometry question, because choosing the wrong side is one of the most common errors.
清晰标记是解答任何三角学题目的第一步，因为选错边是最常见的错误之一。
3. The Three Basic Ratios: SOHCAHTOA | 三个基本比：SOHCAHTOA
The three trigonometric ratios connect an acute angle to pairs of sides in a right-angled triangle. They are summarised by the memory word SOHCAHTOA.
三个三角比将锐角与直角三角形中的两条边联系起来。它们可以用记忆词 SOHCAHTOA 来概括。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
SOH means sine equals opposite divided by hypotenuse.
SOH 表示正弦等于对边除以斜边。
CAH means cosine equals adjacent divided by hypotenuse.
CAH 表示余弦等于邻边除以斜边。
TOA means tangent equals opposite divided by adjacent.
TOA 表示正切等于对边除以邻边。
For any acute angle θ in a right-angled triangle, the values of sin θ and cos θ are always between 0 and 1, while tan θ can be any positive number.
对于直角三角形中的任意锐角 θ，sin θ 和 cos θ 的值始终在 0 到 1 之间，而 tan θ 可以是任意正数。
4. Choosing the Correct Ratio | 选择正确的三角比
To choose the correct ratio, first identify the given side and the required side relative to θ, or decide whether you are finding a side or an angle.
要选择正确的比，首先要确定相对于 θ 的已知边和所求边，或者判断你是求边还是求角。
Step 1: Label the sides. Step 2: Tick the side you know and the side you need. Step 3: Use SOHCAHTOA to select the ratio involving those two sides.
第一步：标记各边。第二步：勾出已知边和所求边。第三步：利用 SOHCAHTOA 选择涉及这两条边的比。
If you know the opposite and hypotenuse, use sine. If you know the adjacent and hypotenuse, use cosine. If you know the opposite and adjacent, use tangent.
如果已知对边和斜边，使用正弦；如果已知邻边和斜边，使用余弦；如果已知对边和邻边，使用正切。
For finding an angle, use the inverse functions sin⁻¹, cos⁻¹ or tan⁻¹.
求角时，使用反函数 sin⁻¹、cos⁻¹ 或 tan⁻¹。
5. Finding a Missing Side | 求未知边
When you know one side and one acute angle, you can find another side by setting up an equation with the appropriate ratio.
当已知一条边和一个锐角时，可以通过建立合适的三角比方程来求另一条边。
A right-angled triangle has hypotenuse 12 cm and angle θ = 35°. Find the side opposite θ.
示例：一直角三角形斜边为 12 cm，角 θ = 35°，求 θ 的对边。
sin 35° = opposite / 12
Multiply both sides by 12:
两边同时乘以 12：
opposite = 12 × sin 35° ≈ 12 × 0.574 = 6.88 cm
Therefore the opposite side is about 6.88 cm.
因此对边约为 6.88 cm。
ple: If the adjacent side is unknown and the hypotenuse is 15 cm with θ = 40°, use cosine.
另一个示例：如果邻边未知，斜边为 15 cm，θ = 40°，则使用余弦。
cos 40° = adjacent / 15
adjacent = 15 × cos 40° ≈ 15 × 0.766 = 11.5 cm
Always use the degree mode on your calculator and round sensibly.
计算器必须处于角度模式，并合理取近似值。
6. Finding a Missing Angle | 求未知角
If two sides are known, you can find an acute angle by using the inverse trigonometric ratio.
如果已知两条边，可以利用反三角比求锐角。
Example: In a right-angled triangle, the opposite side is 5 cm and the adjacent side is 9 cm. Find θ.
示例：一直角三角形中，对边为
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