📚 B8i Part 3: Advanced Trigonometry and 3D Problems | B8i 第3部分：高级三角学与三维问题
This revision guide covers the skills that usually appear in the third part of a trigonometry unit: the sine rule, the cosine rule, area of a non-right-angled triangle, bearings, angles of elevation and depression, and how to apply Pythagoras and trigonometry in three-dimensional solids. Each section pairs an English explanation with a Chinese explanation, so you can learn the ideas and the key vocabulary at the same time.
本复习指南涵盖三角函数单元第三部分常见技能：正弦定理、余弦定理、非直角三角形面积、方位角、仰角与俯角，以及如何将毕达哥拉斯定理和三角函数应用于三维几何体。每节都提供英文解释与中文解释配对，帮助你同时掌握思路和关键术语。
1. Right-Angled Trigonometry Recap | 直角三角形三角比回顾
θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse and tan θ = opposite / adjacent.
在使用正弦定理和余弦定理之前，请确保你能够快速标记直角三角形中的对边、邻边和斜边。三个基本三角比是：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
These ratios only work when the triangle contains a 90° angle. For non-right-angled triangles you need two stronger tools: the sine rule and the cosine rule.
这些比值只在三角形含有 90° 角时成立。对于非直角三角形，你需要两个更强的工具：正弦定理和余弦定理。
2. Labelling Non-Right-Angled Triangles | 非直角三角形的标记方法
In any triangle, label the vertices with capital letters A, B and C. The side opposite vertex A is written as lowercase a, the side opposite vertex B is b, and the side opposite vertex C is c. Angles and their opposite sides are always paired in this way.
在任意三角形中，用大写字母 A、B、C 标记顶点。顶点 A 的对边写作小写字母 a，顶点 B 的对边写作 b，顶点 C 的对边写作 c。角和它的对边始终这样一一对应。
This labelling is essential because the sine rule and cosine rule link each angle to its opposite side. If you label the triangle incorrectly, the formula will produce the wrong answer.
这种标记方式非常重要，因为正弦定理和余弦定理将每个角与其对边联系起来。如果标记错误，公式就会给出错误答案。
3. The Sine Rule | 正弦定理
The sine rule connects the ratios of side lengths to the sines of their opposite angles. The standard formula is:
正弦定理将边长与对角的正弦比联系起来。标准公式为：
a / sin A = b / sin B = c / sin C
You use the sine rule when you know two angles and one side, or when you know two sides and a non-included angle. It is especially useful for finding a missing side or a missing angle without needing a right angle.
当你已知两个角和一个边，或已知两个边和一个非夹角时，可以使用正弦定理。它特别适用于在不需要直角的情况下求缺失的边或角。
b is unknown, then write b / sin 60° = 7 / sin 40°, so b = 7 × sin 60° / sin 40° ≈ 9.4 cm.
示例：若 A = 40°，a = 7 cm，B = 60°，b 未知，则可写 b / sin 60° = 7 / sin 40°，因此 b = 7 × sin 60° / sin 40° ≈ 9.4 cm。
Always check whether the angle is obtuse in the ambiguous case, because sin θ can give two possible angles between 0° and 180°. Exam questions usually tell you whether the triangle is acute or obtuse.
在歧义情形中始终要检查角是否可能为钝角，因为 sin θ 在 0° 到 180° 之间可能给出两个角度。考试题通常会说明三角形是锐角还是钝角。
4. The Cosine Rule | 余弦定理
The cosine rule is used when the sine rule cannot be applied directly, especially when you know three sides or two sides and the included angle. The formula for finding a side is:
当正弦定理无法直接使用时，尤其是已知三条边或两边及其夹角时，可以使用余弦定理。求边的公式为：
a² = b² + c² − 2bc cos A
To find an angle, rearrange the formula as:
要求角时，将公式变形为：
cos A = (b² + c² − a²) / 2bc
angled triangles. When the angle is 90°, cos 90° = 0 and the formula reduces to Pythagoras' theorem.
这个定理非常可靠，因为它适用于所有三角形，包括直角三角形。当角为 90° 时，cos 90° = 0，公式就简化为毕达哥拉斯定理。
10 cm and A = 60°, then a² = 8² + 10² − 2 × 8 × 10 × cos 60° = 64 + 100 − 80 = 84, so a = √84 ≈ 9.2 cm.
示例：若 b = 8 cm，c = 10 cm，A = 60°，则 a² = 8² + 10² − 2 × 8 × 10 × cos 60° = 64 + 100 − 80 = 84，所以 a = √84 ≈ 9.2 cm。
5. Area of a Triangle | 三角形面积
When you know two sides and the angle between them, the area of any triangle can be found using the sine formula:
当你已知两条边及其夹角时，任何三角形的面积都可以用正弦公式计算：
Area = ½ ab sin C
Here a and b are the two known sides, and C is the included angle. This formula is particularly useful because it does not require a perpendicular height.
这里的 a 和 b 是两条已知边，C 是它们的夹角。这个公式特别实用，因为它不需要知道垂直高度。
if a = 6 cm, b = 9 cm and C = 30°, then Area = ½ × 6 × 9 × sin 30° = 27 × 0.5 = 13.5 cm².
示例：若 a = 6 cm，b = 9 cm，C = 30°，则面积 = ½ × 6 × 9 × sin 30° = 27 × 0.5 = 13.5 cm²。
6. Choosing the Right Formula | 选择正确的公式
Selecting between the sine rule, cosine rule and area formula is a key exam skill. The table below summarises the correct choice for each type of information given.
在正弦定理、余弦定理和面积公式之间做出选择是一项关键考试技能。下表总结了针对每种已知信息应选择的正确公式。
| Given information | 已知信息 | Use | 使用 |
|---|---|
| Two angles and one side | 两角一边 | Sine rule | 正弦定理 |
| Two sides and a non-included angle | 两边和一个非夹角 | Sine rule | 正弦定理 |
| Three sides | 三边 | Cosine rule | 余弦定理 |
| Two sides and the included angle | 两边及其夹角 | Cosine rule for side, or area formula | 余弦定理求边，或面积公式 |
Read the question carefully and identify which pieces of information are given before you start writing equations. This prevents you from using the wrong rule and losing accuracy.
开始列方程之前，请仔细阅读题目并识别已知信息。这可以防止你误用定理并降低准确性。
7. Bearings and Scale Drawings | 方位角与比例图
Bearings measure direction as a three-digit angle taken clockwise from north. For example, a bearing of 075° means 75° clockwise from north, and a bearing of 240° points southwest.
方位角是从正北方向顺时针测量的三位数角度。例如，075° 表示从北顺时针转 75°，240° 指向西南方向。
In navigation problems, two journeys often form a triangle with the north lines parallel. The angle inside the triangle can be found by subtracting or adding bearings, and then the sine or cosine rule is used to find missing distances.
在航海问题中，两段航程常常与平行北线构成三角形。三角形内的角可以通过加减方位角来求得，然后用正弦定理或余弦定理求缺失的距离。
Always write bearings with three digits, such as 008° instead of 8°. This is an easy mark to lose in an exam if you forget the leading zeros.
方位角始终要写成三位数，例如 008° 而不是 8°。如果忘记在数字前补零，在考试中很容易丢分。
8. Angles of Elevation and Depression | 仰角与俯角
The angle of elevation is the angle measured upwards from a horizontal line to a line of sight. The angle of depression is measured downwards from a horizontal line to a line of sight. These two angles are alternate angles, so they are equal when drawn between two horizontal lines.
仰角是从水平线向上到视线的夹角。俯角是从水平线向下到视线的夹角。这两类角互为内错角，因此在两条水平线之间它们相等。
In exam problems, draw the horizontal line first, mark the position of the observer, and then form a right-angled triangle or a non-right-angled triangle. This helps you decide whether to use basic SOHCAHTOA, the sine rule or the cosine rule.
在考试题中，先画出水平线，标出观察者位置，然后构造直角三角形或非直角三角形。这有助于你决定使用基本 SOHCAHTOA、正弦定理还是余弦定理。
9. 3D Pythagoras | 三维毕达哥拉斯定理
In a cuboid, the length of the space diagonal can be found by applying Pythagoras' theorem twice. If the cuboid has side lengths a, b and c, then the diagonal d is given by:
在长方体中，空间对角线的长度可以通过两次使用毕达哥拉斯定理求得。如果长方体的边长分别为 a、b、c，则对角线 d 满足：
d² = a² + b² + c²
This formula is a three-dimensional extension of the usual Pythagorean theorem. It is often the first step in a 3D trigonometry question.
这个公式是普通毕达哥拉斯定理的三维推广。它通常是三维三角函数题的第一步。
Example: a box has length 3 cm, width 4 cm and height 12 cm. The space diagonal is d = √3² + 4² + 12² = √9 + 16 + 144 = √169 = 13 cm.
示例：一个盒子的长为 3 cm、宽为 4 cm、高为 12 cm。空间对角线为 d = √3² + 4² + 12² = √9 + 16 + 144 = √169 = 13 cm。
10. 3D Trigonometry: Angle Between Line and Plane | 三维三角函数：直线与平面的夹角
To find the angle between a line and a plane, find the projection of the line onto the plane. The angle is then measured between the original line and this projection.
要求直线与平面的夹角，先找出该直线在平面上的投影。这个角就是在原直线与投影之间测量的角。
sing length; third, apply a basic trigonometric ratio such as tan θ = opposite / adjacent.
解题方法通常分三步：第一，找出包含所求夹角的直角三角形；第二，使用三维毕达哥拉斯定理求缺失长度；第三，应用基本三角比，例如 tan θ = 对边 / 邻边。
Clear diagrams are essential. Draw the plane as a flat surface, mark the line and its projection, and shade or label the angle you need to find.
清晰的示意图至关重要。把平面画成平坦表面，标出直线及其投影，并涂色或标记你需要求的角。
11. Mixed Exam-Style Strategy | 混合题型解题策略
When a question mixes bearings, area, sine rule and 3D reasoning, follow a consistent strategy. Read the whole question first, then draw a labelled diagram, write down the given information, choose the correct formula, substitute carefully, and check the reasonableness of your answer.
当题目混合方位角、
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