Bearings | 方位角 - aleveler.com

📚 Bearings | 方位角

In navigation and geometry, a bearing describes the direction of one point from another using a three-figure angle measured clockwise from north. Mastering bearings is essential for solving real-world problems and a reliable source of marks in the IGCSE exam.

在导航和几何中，方位角用“从正北方向顺时针量得的三个数字角度”来描述一个点相对于另一个点的方向。掌握方位角是解决实际问题的基础，也是 IGCSE 考试中稳定的得分点。

1. What Is a Bearing? | 什么是方位角？

A bearing is an angle, measured in degrees, taken clockwise from the north direction. It is always written with exactly three digits, so a bearing of 15° is written as 015°.

方位角是从正北方向开始、按顺时针方向量得的角度，单位为度。它必须写成三位数，例如 15° 应写作 015°。

Bearing = angle from north, clockwise.
方位角 = 从北方向顺时针测得的角。
Always three figures: 000° to 360°.
总是三位数：000° 到 360°。
North is 000° or 360°; East is 090°; South is 180°; West is 270°.
北为 000° 或 360°；东为 090°；南为 180°；西为 270°。

2. Drawing a Bearing | 画方位角

To draw a point B from A with a given bearing, first draw a north line at A. Then measure the angle clockwise from that north line, and draw a ray in that direction.

要从点 A 画点 B 的方位角，先在 A 处画一条指北线，然后从这条指北线顺时针量出指定角度，并沿该方向画一条射线。

Example: A bearing of 120° means start at north, turn 120° clockwise.
示例：方位角 120° 表示从北方向顺时针旋转 120°。

Remember to place the protractor correctly: the 0° mark must align with the north line, and the angle is read clockwise.

记住正确放置量角器：0° 刻度必须与指北线对齐，并且角度按顺时针方向读取。

3. Measuring a Bearing | 测量方位角

When measuring a bearing from a diagram, draw the north line at the starting point, then measure the clockwise angle to the target line. If the angle is drawn anticlockwise, subtract it from 360°.

在图中测量方位角时，在起点画指北线，然后顺时针量到目标线的角度。如果图中给出的是逆时针角，则用 360° 减去该角。

Bearing = 360° − anticlockwise angle
方位角 = 360° − 逆时针角

Always check that your answer is a three-digit number; if not, add leading zeros.

始终检查答案是否为三位数；如果不是，在前面补零。

4. Back Bearings (Reverse Bearings) | 反方位角

The back bearing is the bearing of the starting point from the destination. It differs from the original bearing by 180°.

反方位角是指从目的地看向起点的方位角。它与原方位角相差 180°。

If the bearing is less than 180°, add 180°.
如果方位角小于 180°，则加 180°。
If the bearing is greater than or equal to 180°, subtract 180°.
如果方位角大于或等于 180°，则减 180°。

Back bearing = (bearing + 180°) mod 360°
反方位角 =（原方位角 + 180°）对 360° 取余

For example, the back bearing of 078° is 078° + 180° = 258°.

例如，078° 的反方位角是 078° + 180° = 258°。

5. Bearings and Parallel Lines | 方位角与平行线

North lines are parallel to each other. Therefore, corresponding angles, alternate angles, and co-interior angles can be used when working with bearings.

所有指北线彼此平行。因此在处理方位角时，可以利用同位角、内错角和同旁内角的关系。

In the diagram below, the angle between the north line at A and the line AB is 078°. The angle between the north line at B and BA is the alternate angle, also 078°.

在下图中，A 处指北线与 AB 线的夹角是 078°。B 处指北线与 BA 线所成的内错角也等于 078°。

Angle between two north lines = 0° (parallel)
两条指北线之间的角 = 0°（平行）

This idea is often used to calculate bearings involving two points without drawing a full triangle.

这个思路常用于仅涉及两个点、无需画出完整三角形的方位角计算。

6. Using Bearings in Triangles | 在三角形中运用方位角

When two bearings and a distance are given, you can form a triangle and use sine or cosine rules to find missing distances or bearings.

当给定两个方位角和一段距离时，可以构成一个三角形，并利用正弦定理或余弦定理求缺失的距离或方位角。

Cosine rule: c² = a² + b² − 2ab cos C
余弦定理：c² = a² + b² − 2ab cos C

Sine rule: a/sin A = b/sin B = c/sin C
正弦定理：a/sin A = b/sin B = c/sin C

First find the interior angles of the triangle. The angle at each point is the difference between the two directions from that point.

首先求出三角形内角。每个点处的内角等于从该点出发的两条方向线之间的夹角差。

7. Calculating Interior Angles from Bearings | 由方位角计算内角

Suppose point A has a bearing to B of 078°, and A has a bearing to C of 130°. The angle BAC inside the triangle is 130° − 78° = 52°.

假设从 A 到 B 的方位角为 078°，从 A 到 C 的方位角为 130°，则三角形内的角 BAC = 130° − 78° = 52°。

If the two bearings lie on opposite sides of north, add them. For example, a bearing of 320° and a bearing of 040° give an angle of 360° − 320° + 40° = 80°.

如果两个方位角位于正北方向的两侧，则相加。例如，方位角 320° 与 040° 所夹的角为 360° − 320° + 40° = 80°。

8. Word Problems: Navigation | 实际应用：导航问题

Bearing problems often appear in the context of ships, aeroplanes, or hikers. You must read carefully to identify the starting point and the direction of the movement.

方位角问题常以轮船、飞机或徒步者为背景。仔细审题，确定起点和运动方向。

“From A, the bearing of B is 078°” means start at A.
“从 A 点看 B 点的方位角是 078°”说明起点是 A。
“A ship sails on a bearing of 078°” means the ship travels in direction 078°.
“轮船沿 078° 方位角航行”意思是船的运动方向为 078°。
Draw a clear diagram, mark all given angles.
画清晰的示意图，标出所有已知角。

Use the sine and cosine rules only after the interior angles are correctly found.

正确求出内角后，才能使用正弦定理和余弦定理。

9. Common Mistakes and Tips | 常见错误与要点

Many marks are lost in bearing questions due to small but avoidable errors.

许多学生在方位角题目上失分，是因为一些细小但可以避免的错误。

Mistake / 错误	Correction / 纠正
Writing 78° instead of 078°	Always use three digits: 078°
Measuring anticlockwise	Measure clockwise from north
Not drawing the north line at the correct point	Draw north lines at every reference point
Confusing back bearing with original bearing	Add or subtract 180° correctly

Always sketch a rough diagram first, and verify that your final angle is between 000° and 360°.

先画草图，并检查最终角度是否在 000° 到 360° 之间。

10. Worked Example | 完整例题

Example: From point A, the bearing of B is 078°. From point B, the bearing of A is 258°. A ship starts at A and sails 5 km on a bearing of 120° to C. Find the distance from B to C, rounded to one decimal place.

例题：从 A 点看 B 点的方位角为 078°。从 B 点看 A 点的方位角为 258°。一艘船从 A 出发，沿 120° 方位角航行 5 km 到达 C。求 B 到 C 的距离（精确到 1 位小数）。

At B, the angle between BA (bearing 258°) and BC is needed. First find the bearing of C from B. Since C is 5 km from A in direction 120°, we can use a triangle ABC.

在 B 点，需要求出 BA（方位角 258°）与 BC 之间的夹角。首先求从 B 看 C 的方位角。已知 C 在 A 的 120° 方向上，距离 5 km，于是在三角形 ABC 中求解。

Angle BAC is the difference between 120° and 078°, so ∠BAC = 120° − 78° = 42°. AB is unknown, but we can first find angle ABC using the back bearing: the bearing of A from B is 258°, so at B the direction BA is 258°. The direction BC is unknown.

角 BAC = 120° − 78° = 42°。AB 未知，但可先求角 ABC：从 B 看 A 的方位角为 258°，即 B 点处 BA 的方向为 258°。BC 的方向未知。

Since only one side (AC = 5 km) and angle A are known, more information is needed. In a full exam, the position of C would allow finding ∠ABC via another bearing. Here we illustrate the method, not the final calculation.

由于目前只知一边 AC = 5 km 和角 A，还需更多信息。在完整考题中，C 的位置可通过另一个方位角确定 ∠ABC。此处只演示方法，不进行最终计算。

11. Practice Questions | 练习题目

Try these to consolidate your understanding.

通过以下练习巩固理解。

Write the bearing equivalent to 45° east of north.
写出“北偏东 45°”对应的方位角。
A bearing is 310°. Find its back bearing.
已知方位角为 310°，求反方位角。
From A, the bearing of B is 078°. From B, the bearing of A is?
从 A 看 B 的方位角为 078°，则从 B 看 A 的方位角是多少？
In triangle ABC, ∠A = 42°, ∠C = 68°, AC = 10 km. Find AB using the sine rule.
在三角形 ABC 中，∠A = 42°，∠C = 68°，AC = 10 km。用正弦定理求 AB。

Answers: 1) 045° 2) 130° 3) 258° 4) Use sine rule carefully.

答案：1) 045° 2) 130° 3) 258° 4) 请仔细使用正弦定理。

12. Summary | 总结

A bearing is a three-figure clockwise angle from north. Always draw north lines, compute interior angles carefully, and use sine or cosine rules when needed.

方位角是从正北方向顺时针量得的三位角度。务必画指北线，仔细计算内角，并在需要时使用正弦或余弦定理。

With regular practice, bearing problems become straightforward and a reliable source of marks in the Edexcel IGCSE Mathematics exam.

通过定期练习，方位角问题会变得简单清晰，成为 Edexcel IGCSE 数学考试中稳定的得分点。

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Bearings | 方位角