📚 Bearings | 方位角
Bearings are a fundamental tool in navigation and geometry, used to describe direction as a three-figure angle measured clockwise from North. Mastering bearings is essential for solving many IGCSE Mathematics problems involving maps, triangles, and real-world contexts.
方位角是导航和几何中的基础工具，用从正北方向顺时针测量的三位角度来描述方向。掌握方位角对于解决 IGCSE 数学中涉及地图、三角形和实际情境的许多问题至关重要。
1. What are Bearings? | 什么是方位角？
A bearing is an angle, measured in degrees, taken clockwise from the North direction. It is always written as a three-figure number, for example 058°.
方位角是从正北方向顺时针测量的角度，以度为单位，并且总是写成三位数字，例如 058°。
- North is the reference direction (000°).
- Measurement is always clockwise.
- Bearings range from 000° to 360°.
- 正北是参考方向（000°）。
- 始终顺时针测量。
- 方位角的范围是 000° 到 360°。
2. The Three-Figure Rule | 三位数字表示法
Every bearing must have exactly three digits. If the angle is less than 100°, a zero is added at the front. For example, 58° becomes 058°, and 8° becomes 008°.
每个方位角必须恰好有三位数字。如果角度小于 100°，则在前面补零。例如，58° 写成 058°，8° 写成 008°。
| Angle | Bearing | Direction |
|---|---|---|
| 0° | 000° | North |
| 58° | 058° | Northeast |
| 90° | 090° | East |
| 135° | 135° | Southeast |
| 270° | 270° | West |
3. Measuring Bearings with a Protractor | 用量角器测量方位角
To measure a bearing from point A to point B, first draw a North line at A. Place the protractor so that 0° lies on the North line, then read the angle clockwise to the line AB.
要测量从点 A 到点 B 的方位角，首先在 A 点画一条正北线。将量角器放在 A 点，使 0° 与正北线对齐，然后顺时针读取到直线 AB 的角度。
- Always draw the North line from the vertex of the angle.
- Place the protractor's centre exactly at the vertex.
- Read the outer or inner scale that starts at 0° on the North line.
- 始终从角的顶点画出正北线。
- 量角器的中心必须精确对准顶点。
- 读取从正北线上的 0° 开始的外圈或内圈刻度。
4. Bearings from a Point | 从一点测量方位角
The bearing of B from A is the angle measured clockwise from North at A to the ray AB. It is important to identify which point is the observer and which is the target.
点 B 相对点 A 的方位角，是在点 A 处从正北方向顺时针到射线 AB 的角度。明确哪个点是观察点、哪个点是目标点非常重要。
Bearing of B from A = clockwise angle from North at A to AB
For example, if the angle is 58°, the bearing of B from A is written as 058°.
例如，如果角度为 58°，则 B 相对 A 的方位角写为 058°。
5. Back Bearings | 反方位角
The back bearing is the bearing taken from the target back to the original point. It differs from the forward bearing by exactly 180°.
反方位角是从目标点返回原始测量点的方位角。它与正向方位角相差正好 180°。
- If the bearing is less than 180°, add 180°.
- If the bearing is greater than or equal to 180°, subtract 180°.
- 如果方位角小于 180°，则加 180°。
- 如果方位角大于或等于 180°，则减 180°。
Example: rom A is 058°. The back bearing is 058° + 180° = 238°. So the bearing of A from B is 238°.
示例： B 相对 A 的方位角是 058°。反方位角为 058° + 180° = 238°。因此 A 相对 B 的方位角是 238°。
6. Bearings and Angles at a Point | 方位角与点上的角
Bearings are often combined with angle facts: angles on a straight line add to 180°, angles around a point add to 360°, and alternate interior angles are equal when lines are parallel.
方位角常与角度性质结合：直线上的角之和为 180°，点周围的角之和为 360°，两条平行线被截时内错角相等。
When two North lines are parallel, they create equal corresponding angles. If you draw two North lines at different points, the line connecting those points forms equal alternate angles with them.
当两条正北线平行时，它们产生相等的同位角。如果在不同点画出两条正北线，连接这两点的直线与它们形成相等的内错角。
7. Calculating Bearings | 计算方位角
In many problems, you must calculate a bearing using known angles rather than measuring it. The key is to find the angle between North and the required direction, then express it as a three-figure bearing.
在许多问题中，你需要利用已知角度来计算方位角，而不是用量角器测量。关键是找出正北方向与目标方向之间的角度，然后将其表示为三位方位角。
For example, if an angle inside a triangle is 58°, and the North line is parallel to one side, the bearing may be equal to that angle or a supplementary angle. Always draw a small North line at the vertex to avoid confusion.
例如，如果三角形中的一个内角为 58°，且正北线与某一边平行，则方位角可能等于该角或其补角。始终在顶点处画一条小的正北线以避免混淆。
8. Using the Sine and Cosine Rules | 正弦定理和余弦定理
When a bearing problem involves a triangle where distances or angles are missing, the sine rule or cosine rule is often needed.
当方位角问题涉及三角形，且缺少距离或角度时，通常需要使用正弦定理或余弦定理。
Sine rule: a ⁄ sin A = b ⁄ sin B = c ⁄ sin C
Cosine rule: a² = b² + c² − 2bc cos A
Use the sine rule when you know two angles and one side (AAS/ASA) or two sides and a non-included angle (SSA). Use the cosine rule when you know two sides and the included angle (SAS) or three sides (SSS).
已知两角和一边（AAS/ASA）或两边和其中一边的对角（SSA）时，用正弦定理。已知两边及其夹角（SAS）或三边（SSS）时，用余弦定理。
9. Common Exam Mistakes | 常见考试错误
- Forgetting to write a leading zero, such as writing 58° instead of 058°.
- Measuring anticlockwise instead of clockwise from North.
- Using the bearing of B from A when asked for the bearing of A from B.
- Omitting the North line when sketching a diagram.
- Using the wrong scale on the protractor.
- 忘记在前面补零，例如把 058° 写成 58°。
- 从正北方向逆时针测量而非顺时针。
- 题目要求 A 相对 B 的方位角，却用了 B 相对 A 的方位角。
- 画图时漏画正北线。
- 用量角器时读错刻度圈。
10. Worked Example | 例题讲解
Problem: A ship sails 6 km from P to Q on a bearing of 058°. Then it sails 4 km from Q to R on a bearing of 150°. Find the distance PR and the bearing of R from P.
题目：一艘船从 P 点出发，以方位角 058° 航行 6 km 到达 Q 点；然后以方位角 150° 航行 4 km 到达 R 点。求 PR 的距离以及 R 相对 P 的方位角。
Solution: At Q, the angle between QP (direction 238°) and QR (direction 150°) is 238° − 150° = 88°.
解答：首先画出三角形 PQR。P 相对 Q 的反方位角为 058° + 180° = 238°。在 Q 点，QP 方向（238°）与 QR 方向（150°）之间的夹角为 238° − 150° = 88°。
Use the cosine rule with sides PQ = 6, QR = 4, and included angle 88°:
使用余弦定理，其中 PQ = 6，QR = 4，夹角为 88°：
PR² = 6² + 4² − 2 × 6 × 4 × cos 88°
Since cos 88° ≈ 0.0349,
因为 cos 88° ≈ 0.0349，
PR² = 36 + 16 − 48 × 0.0349 ≈ 52 − 1.675 = 50.325
PR ≈ √50.325 ≈ 7.09 km
Now use the sine rule to find angle at P:
再用正弦定理求 P 处的角：
sin P / 4 = sin 88° / 7.09
sin P = 4 × sin 88° / 7.09 ≈ 4 × 0.9994 / 7.09 ≈ 0.5637
P ≈ 34.3°
s 058°. Point R lies to the east of Q, so the bearing of R from P is 058° + 34.3° = 92.3°.
在 P 点，Q 相对 P 的方位角是 058°。点 R 位于 Q 的东侧，因此 R 相对 P 的方位角为 058° + 34.3° = 92.3°。
Answer: PR ≈ 7.09 km; bearing of R from P is 092.3° (or 092° rounded to 3 significant figures).
答案： PR ≈ 7.09 km；R 相对 P 的方位角约为 092.3°（保留三位有效数字为 092°）。
11. Practice Questions | 练习题
1. Convert 74° to a three-figure bearing.
1. 将 74° 转换为三位方位角。
2. The bearing of B from A is 152°. Find the bearing of A from B.
2. B 相对 A 的方位角为 152°，求 A 相对 B 的方位角。
ance XZ and the bearing of Z from X. (Hint: first work out the angle at Y in triangle XYZ.)
3. 从点 X 看，Y 的方位角为 035°；从点 Y 看，Z 的方位角为 110°。若 XY = 5 km，YZ = 7 km，求 XZ 的距离以及 Z 相对 X 的方位角。（提示：先求出三角形 XYZ 中 Y 处的角。）
Answers: 1. 074° 2. 332° 3. XZ ≈ 9.74 km, bearing ≈ 077.5°
答案： 1. 074° 2. 332° 3. XZ ≈ 9.74 km，方位角约为 077.5°
12. Summary | 总结
Bearings are always written as three figures and measured clockwise from North. Always draw a North line at the point of measurement, identify the correct order (A from B vs. B from A), and use angle facts and trigonometry when solving more complex problems.
方位角始终写成三位数字，并从正北方向顺时针测量。在测量点一定要画正北线，明确方位角的顺序（A 相对 B 还是 B 相对 A），在解决复杂问题时要结合角度性质与三角函数。
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