Bearings | 方位角 - aleveler.com

📚 Bearings | 方位角

In the Edexcel IGCSE Mathematics syllabus, bearings are a key topic in both Paper 1 and Paper 2. A bearing is a way of describing a direction using three digits and an angle measured clockwise from north. This article will explain the rules, worked examples, and common pitfalls so you can tackle any bearing question with confidence.

在 Edexcel IGCSE 数学考纲中，方位角是 Paper 1 和 Paper 2 的重要考点。方位角用三位数和一个从正北方向顺时针量出的角度来描述方向。本文将讲解相关规则、例题和常见易错点，帮助你自信应对任何方位角题目。

1. What Is a Bearing? | 什么是方位角？

A bearing is an angle, measured in degrees, from the north line in a clockwise direction. It is always written as three figures, for example 045°, 136°, or 270°. If the angle is less than 100°, a zero is placed in front, so 45° becomes 045°.

方位角是从正北方向线起顺时针量出的角，单位是度。它始终写成三位数，例如 045°、136° 或 270°。如果角度小于 100°，则在前方补零，因此 45° 写成 045°。

Bearing = angle measured clockwise from North, written as three digits.

方位角 = 从正北方向顺时针量出的角，写成三位数。

2. The Three Golden Rules | 三条黄金法则

To find or draw a bearing, you must always follow these rules:

要确定或画出方位角，你必须始终遵循以下规则：

Rule 1: Start from the north line at the point where you are standing.

规则一：从你所在位置的正北方向线开始。

Rule 2: Measure the angle clockwise (turning to the right).

规则二：顺时针量角（向右转）。

Rule 3: Write the angle as three digits, even if it is a whole number.

规则三：把角度写成三位数，即使是整数也要补零。

3. Drawing a Bearing | 画方位角

Suppose you are asked to draw a point B from A on a bearing of 136°. First, draw a vertical north line at A. Then place a protractor so that 0° points to north, and measure 136° clockwise. Mark this direction and draw a ray from A along this line. Point B lies anywhere on this ray, usually at a given distance.

假设要求你从点 A 画出点 B，方向为方位角 136°。首先，在 A 处画一条竖直的正北线。然后把量角器放在 A 处，使 0° 对准正北，顺时针量出 136°。沿着这个方向从 A 画一条射线，点 B 在这条射线上的某个位置，通常还会给出距离。

For example, if AB = 5 cm, measure 5 cm along the 136° ray and mark point B. Always write the angle next to the ray, and put a small north arrow at A.

例如，如果 AB = 5 cm，就在 136° 射线上量出 5 cm 并标记点 B。始终在射线旁标出角度，并在 A 处画一个小小的北箭头。

4. Finding a Bearing from a Diagram | 从图中求方位角

When a diagram shows a point B from A, you need to measure the angle between the north line at A and the line AB, going clockwise. If the diagram is drawn accurately, use a protractor. If it is not drawn to scale, you must use given angles and geometry.

当图中显示从 A 到 B 的方向时，你需要测量 A 处的正北线与 AB 连线之间的角，并按顺时针方向量取。如果图是精确绘制的，用量角器测量；如果不是按比例绘制的，则必须利用已知角和几何知识计算。

Remember: the north line is always vertical on a standard map, unless stated otherwise.

记住：在标准地图上，正北线始终是竖直的，除非另有说明。

5. Back Bearings (Reverse Bearings) | 反向方位角

The bearing of A from B is the bearing you would use to travel back from B to A. It differs from the bearing of B from A by 180°. To find the back bearing, add 180° to the original bearing. If the result is greater than 360°, subtract 360°.

从 B 到 A 的方位角是你从 B 返回 A 时所用的方向。它与从 A 到 B 的方位角相差 180°。求反向方位角时，给原方位角加上 180°；如果结果大于 360°，则减去 360°。

Back bearing = (bearing + 180°) mod 360°

反向方位角 = (原方位角 + 180°) 对 360° 取模

For example, if the bearing of B from A is 136°, then the bearing of A from B is 136° + 180° = 316°.

例如，如果 B 相对于 A 的方位角是 136°，那么 A 相对于 B 的方位角就是 136° + 180° = 316°。

6. Using Parallel Lines to Calculate Bearings | 利用平行线计算方位角

In many exam questions, north lines are drawn at different points. These north lines are parallel to each other. Therefore, angles such as alternate angles and corresponding angles can be used to find unknown bearings.

在许多考试题目中，不同点处都画有正北线。这些正北线彼此平行。因此，可以利用内错角、同位角等关系来求未知方位角。

Consider a point B due east of A. The bearing of B from A is 090°. If you stand at B, the north line is still vertical, so the bearing of A from B is 270° (since 090° + 180° = 270°).

考虑点 B 在 A 的正东方向。B 相对于 A 的方位角是 090°。当你站在 B 处时，正北线仍然是竖直的，所以 A 相对于 B 的方位角是 270°（因为 090° + 180° = 270°）。

When a diagram contains a triangle or a polygon, you can combine angle facts with bearings. For example, the interior angles of a triangle sum to 180°, and angles on a straight line sum to 180°.

当图中含有三角形或多边形时，你可以把角度知识与方位角结合。例如，三角形内角和为 180°，平角为 180°。

7. Worked Example: Bearing 136° | 例题：方位角 136°

Let us draw a full example. Ship S is at a harbour H. The bearing of a lighthouse L from H is 136°. The distance HL is 8 km. Find the bearing of H from L, and describe the position of L relative to H.

我们来看一个完整例题。船 S 位于港口 H。灯塔 L 相对于 H 的方位角是 136°，距离 HL = 8 km。求 H 相对于 L 的方位角，并描述 L 相对于 H 的位置。

Step 1: Draw a north line at H and measure 136° clockwise. Draw a ray for L.

第一步：在 H 处画正北线，顺时针量出 136°，画出指向 L 的射线。

Step 2: Mark L on that ray so that HL = 8 km.

第二步：在这条射线上标出 L，使 HL = 8 km。

Step 3: To find the bearing of H from L, draw a north line at L. The north line at L is parallel to the north line at H. The angle we need is measured clockwise from L’s north line to the line LH.

第三步：求 H 相对于 L 的方位角。在 L 处画正北线，它与 H 处的正北线平行。我们需要的角是从 L 的正北线顺时针到 LH 连线的角。

Notice that the line LH is the same straight line as HL, but the direction is reversed. Therefore, the bearing changes by 180°.

注意，直线 LH 与 HL 是同一条直线，但方向相反。因此，方位角相差 180°。

Bearing of L from H = 136° ⇒ Bearing of H from L = 136° + 180° = 316°

L 相对于 H 的方位角 = 136° ⇒ H 相对于 L 的方位角 = 136° + 180° = 316°

So L is located 8 km from H, in the direction 136° (south-east, closer to south). We can also state that L is south-east of H, but the bearing is the precise mathematical description.

因此，L 位于从 H 出发沿 136° 方向 8 km 处（东南方向，更偏南）。我们也可以说 L 在 H 的东南方，但方位角是精确的数学描述。

8. Calculating the Position of a Point Using Trigonometry | 用三角函数求点的位置

Sometimes a bearing question involves a right-angled triangle. You may need to find an unknown distance or angle using sine, cosine, or tangent. For example, a boat travels 5 km on a bearing of 136°. How far south does it travel?

有时方位角题目涉及直角三角形。你可能需要用正弦、余弦或正切来求未知距离或角度。例如，一艘船以方位角 136° 航行了 5 km。它向南航行了多远？

Draw a diagram. The bearing 136° is measured from north. The angle between the path and the south direction is 180° − 136° = 44° (because south is at 180°). The southward distance is the adjacent side to the 44° angle, and 5 km is the hypotenuse.

画一个图。方位角 136° 从正北量起。路径与正南方向的夹角是 180° − 136° = 44°（因为正南在 180°）。向南的距离是 44° 角的邻边，而 5 km 是斜边。

cos 44° = adjacent ÷ hypotenuse ⇒ southward distance = 5 × cos 44°

cos 44° = 邻边 ÷ 斜边 ⇒ 向南距离 = 5 × cos 44°

Using a calculator, 5 × cos 44° ≈ 5 × 0.7193 = 3.60 km (to 3 significant figures). Similarly, the eastward distance uses sin 44°.

用计算器计算，5 × cos 44° ≈ 5 × 0.7193 = 3.60 km（精确到 3 位有效数字）。同理，向东的距离用 sin 44°。

9. Bearing Word Problems: Air and Sea Navigation | 方位角应用题：空中与海上导航

A typical exam problem might say: “A plane flies from airport A for 200 km on a bearing of 136°, then turns and flies 150 km on a bearing of 220°. Find the final distance from A.”

一个典型考试题可能说：“一架飞机从机场 A 出发，以方位角 136° 飞行 200 km，然后转向，以方位角 220° 飞行 150 km。求最终距离 A 的距离。”

Such problems usually require resolving each leg into north and east components, then combining the components.

这类题通常需要把每段航程分解为向北和向东的分量，然后再合并分量。

For a leg of length d on bearing θ, the north component is d × cos θ, and the east component is d × sin θ. Use these with care: if the bearing is between 90° and 180°, the east component is positive but the north component is negative (because the direction is southward).

对于长度为 d、方位角为 θ 的航段，向北分量为 d × cos θ，向东分量为 d × sin θ。要小心使用：如果方位角在 90° 到 180° 之间，向东分量为正，但向北分量为负（因为方向偏南）。

Instead of memorising signs, it is safer to draw a large diagram and mark the acute angle with the nearest horizontal or vertical axis.

与其死记符号，更安全的方法是画一个较大的图，标出与最近的水平轴或竖直轴之间的锐角。

10. Common Mistakes | 常见错误

Mistake 1: Writing a bearing with only two digits. For example, writing 45° instead of 045°. This loses a mark in the exam.

错误一：把方位角写成两位数。例如写 45° 而不是 045°，考试中会丢分。

Mistake 2: Measuring anticlockwise instead of clockwise. Always measure from north towards the east, then south, then west.

错误二：逆时针量角而非顺时针。始终从正北向正东、再向正南、再向正西方向量。

Mistake 3: Forgetting that north lines are parallel when calculating back bearings.

错误三：计算反向方位角时忘记正北线互相平行。

Mistake 4: Using the 180° rule incorrectly. Add 180°, not 90° or 270°.

错误四：错误使用 180° 规则。应加 180°，而不是 90° 或 270°。

11. Exam Tips for Bearing Questions | 方位角题目考试技巧

Always draw a clear north line at each point.
始终在每个点处画清楚的正北线。
Write three-figure angles on the diagram, even if the question only asks for a calculation.
在图上标出三位数的角度，即使题目只要求计算。
Check whether your final answer has three digits.
检查最终答案是否为三位数。
If a diagram is not drawn to scale, do not measure it; use angle rules instead.
如果图形没有按比例绘制，不要用量角器测量，而应使用角度规则。
For “find the bearing of A from B”, put yourself at B and draw the north line at B.
对于“求 A 相对于 B 的方位角”，要把自己放在 B 处，并在 B 处画正北线。

12. Practice Question | 练习

Try this question by yourself:

请自己尝试这道题：

From point P, the bearing of Q is 136°. Given that Q is 12 km from P, find:

从点 P 测得 Q 的方位角是 136°。已知 Q 距离 P 12 km，求：

(a) the bearing of P from Q;

(a) P 相对于 Q 的方位角；

(b) the distance Q is south of P, correct to 1 decimal place.

(b) Q 位于 P 以南的距离，精确到 1 位小数。

Answers: (a) 316° (b) 12 × cos(180° − 136°) = 12 × cos 44° ≈ 8.6 km (1 d.p.)

答案：(a) 316° (b) 12 × cos(180° − 136°) = 12 × cos 44° ≈ 8.6 km（1 位小数）

Keep practising with past papers and you will soon master bearings.

多练习历年真题，你很快就能掌握方位角。

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Bearings | 方位角