Three-figure bearings | 三位数方位角

📚 Three-figure bearings | 三位数方位角

Bearings are a fundamental concept in navigation, surveying and trigonometry, especially within the IB Mathematics curriculum. A bearing specifies the direction of one point from another in a clockwise sense, always measured from the North direction. In this article, we will explore three-figure bearings, how to measure, draw and calculate them, and see their connection to trigonometry through practical examples.

方位角是导航、测量和三角学中的一个基础概念，在IB数学课程中尤其重要。方位角总是从正北方向开始，顺时针测量，用来表示一点相对于另一点的方向。本文我们将探讨三位数方位角，学习如何测量、绘制和计算方位角，并通过实际例子了解它们与三角学的联系。

1. Introduction to Bearings | 方位角简介

A bearing is an angle, measured in degrees, that indicates the direction of one point from a fixed reference point. In most mathematical contexts, bearings are measured clockwise from the North direction. This convention is used in real-world applications such as aviation, maritime navigation and land surveying.

方位角是以度为单位的角度，表示从某个固定参考点到另一点的方向。在大多数数学情境中，方位角是从正北方向开始顺时针测量的。这一惯例广泛应用于航空、航海和陆地测量等实际领域。

For example, if you stand at point O and face North, then turn 60 degrees clockwise to face point P, the bearing of P from O is 060°. Note that the angle is always given with three digits, which is discussed in the next section.

例如，如果你站在点O并面向正北，然后顺时针转60度面向点P，那么从O到P的方位角就是060°。注意这个角度总是用三位数表示，下一节将详细介绍这一点。

2. The Three-Figure Convention | 三位数表示法

All true bearings must be written using three digits. This means that an angle of 5° is written as 005°, 45° becomes 045°, and 90° is 090°. The purpose of this rule is to avoid ambiguity in communication, especially in critical situations like air traffic control or search operations.

所有真实方位角都必须使用三个数字书写。这意味着5° 写为 005°，45° 写为 045°，90° 写为 090°。这条规则旨在避免沟通中出现歧义，特别是在空中交通管制或搜索行动等关键场合。

If a bearing is exactly 0° (due North), it is written as 000°. A bearing of 360° is never used, because it is the same as 000°. The three-figure system ensures that every bearing is easily distinguishable, even when written quickly.

如果方位角恰好是0°（正北），则写为 000°。360° 从不使用，因为它等同于 000°。三位数编号系统确保每个方位角都易于识别，即便快速书写时也是如此。

3. Measuring Bearings from North | 从正北方向测量方位角

The reference direction for bearings is always North. The angle is measured clockwise from the North line. This is different from the way we usually measure angles in coordinate geometry, where angles are often measured anticlockwise from the positive x-axis.

方位角的参考方向始终是正北。角度从北向线开始顺时针测量。这不同于我们在坐标系中通常从正 x 轴逆时针测量角度的方法。

To find the bearing of point B from point A, draw a north-pointing line at A, then rotate clockwise until the line points directly toward B. The size of this rotation is the bearing. For instance, if B is located exactly to the East of A, the rotation is 90°, so the bearing is 090°.

要找出从点A到点B的方位角，在A处画一条指向正北的直线，然后顺时针旋转，直到该直线直接指向B。这个旋转量就是方位角。例如，如果B恰好位于A的正东方向，则旋转为90°，因此方位角是090°。

4. Drawing Bearings | 绘制方位角

When drawing a bearing diagram, start by marking the reference point and drawing a vertical line upward to represent North. Then, using a protractor, measure the required clockwise angle from this North line. Mark the direction and label the bearing with its three-digit value.

绘制方位角图时，先标出参考点，并向上画一条垂直线代表北方向。然后用量角器从这条北向线开始顺时针量取所需的角度。标出方向，并用三位数字标记方位角。

Always ensure that the North line is clearly indicated. If the bearing is greater than 180°, the clockwise turn goes past the South line, and you may find it easier to calculate the complementary acute angle with the South direction, but the bearing must still be expressed as the full three-figure clockwise angle from North.

务必确保北向线标示清楚。如果方位角大于180°，顺时针转动会越过正南方向，你可能会发现先计算与正南方向的补角更方便，但方位角仍须表示为从北向开始顺时针旋转的完三位数角度。

5. Calculating Bearings with Angles | 利用角度计算方位角

Often, bearings are not given directly but must be deduced from other known angles. A common scenario involves parallel north lines. Since all north lines are parallel, alternate and corresponding angle rules can be used to find unknown bearings.

方位角通常不会直接给出，而需要从其他已知角度推导。常见的情形涉及平行北向线。由于所有北向线都是平行的，可以利用内错角、同位角等规则求未知方位角。

For example, if the bearing of B from A is 070°, and you need the bearing of A from B, you can use the fact that north lines at A and B are parallel. By drawing the diagram, you can see that the angle between the two north lines and the line AB gives the back bearing, which is 070° + 180° = 250°.

例如，如果从A到B的方位角是070°，需要求从B到A的方位角，可以利用A和B处的北向线平行的特点。通过画图，可以得知两条北向线与线段AB之间的夹角关系，得出反方位角为 070° + 180° = 250°。

6. Bearings and Trigonometry | 方位角与三角学

Bearings frequently appear in problems involving non-right-angled triangles, where the sine rule or cosine rule is needed. In the IB Mathematics course, you are expected to combine bearing concepts with trigonometry to find distances and unknown angles.

方位角经常出现在非直角三角形的题目中，需要运用正弦定理或余弦定理。在IB数学课程中，你应将方位角概念与三角学结合起来，求解距离和未知角度。

Consider a ship that sails from port A on a bearing of 120° for 8 km to point B, then changes to a bearing of 210° for 5 km to point C. To find the distance from A to C directly, you can use the cosine rule: c² = a² + b² − 2ab cos θ, after working out the included angle at B.

考虑一艘船从港口A出发，沿120° 方位航行 8 km 到达点B，然后改沿210° 方位航行 5 km 到达点C。要直接求A到C的距离，可以先算出B处的夹角，然后使用余弦定理：c² = a² + b² − 2ab cos θ。

In many situations, the sine rule: a / sin A = b / sin B = c / sin C is used to determine a missing bearing after sides have been found. All angles are in degrees, and calculator settings must be in degree mode.

在许多情形下，求出边长后，会用正弦定理：a / sin A = b / sin B = c / sin C 来确定缺失的方位角。所有角度都以度为单位，计算器必须设置在度模式。

7. Back Bearings | 反方位角

The back bearing (or reverse bearing) is the bearing of the starting point from the destination point. It is used when you need to return along the same line. The rule is simple: if the original bearing is less than 180°, add 180°; if it is 180° or more, subtract 180°.

反方位角（逆向方位角）是从目的地看向起点的方位角。当你需要沿同一路线返回时就会用到。规则很简单：如果原始方位角小于180°，则加180°；如果大于或等于180°，则减180°。

For example, the back bearing of 045° is 045° + 180° = 225°. The back bearing of 300° is 300° − 180° = 120°. This ensures the resulting bearing remains within the 000°–359° range, preserving the three-figure format.

例如，045° 的反方位角是 045° + 180° = 225°。300° 的反方位角是 300° − 180° = 120°。这样能保证结果落在 000° 至 359° 范围内，并保持三位数字格式。

8. Solving Real-World Problems | 解决实际问题

Real-world bearing problems often involve multiple legs of a journey and require you to draw a diagram, identify triangles, and apply trigonometric methods. For instance, an airplane flies from airport P to Q on a bearing of 055° for 200 km, then to R on a bearing of 125° for 150 km. To find the bearing from R back to P, you need to compute the angles inside triangle PQR.

现实中的方位角问题通常涉及分段航程，需要你画图、识别三角形并运用三角学方法。比如，一架飞机从机场P沿055° 方位飞行200 km 到Q，再沿125° 方位飞行150 km 到R。要求从R返回P的方位角，就需要计算三角形PQR内的角。

Such multi-step problems test your ability to combine bearing conventions with the sine and cosine rules. Always sketch the north lines at each vertex; they act as parallel guides to relate the given bearings to the interior angles of the triangle.

这类多步骤问题考查你将方位角惯例与正弦、余弦定理结合起来的能力。务必在每个顶点画出北向线；它们作为平行参考线，能将已知方位角与三角形的内角联系起来。

9. Common Mistakes to Avoid | 常见错误避坑

A typical mistake is forgetting the three-figure rule, writing 60° instead of 060°. Another error is measuring the angle anticlockwise, or confusing the bearing from A to B with the bearing from B to A without applying the back bearing rule correctly.

一个典型错误是忘记三位数规则，把 60° 写成 60 而不是 060°。另一个错误是逆时针量角，或混淆从A到B和从B到A的方位角，而未正确使用反方位角规则。

Students also sometimes misinterpret the parallel north lines and end up adding or subtracting angles incorrectly. It is essential to draw a clear, labelled diagram and check that the sum of angles around a point or along a straight line aligns with geometric principles.

学生有时也会错误理解平行北向线，导致角度加减出错。必须绘制清晰、标注明确的示意图，并核对某一点周围或沿直线的角度和是否符合几何原理。

10. Practice Examples | 练习示例

Example 1: Find the bearing of point X from point Y if the bearing of Y from X is 200°. Solution: 200° − 180° = 020°. The bearing is 020°.

示例1：若从X到Y的方位角是200°，求从Y到X的方位角。解：200° − 180° = 020°。方位角为020°。

Example 2: A lighthouse is 5 km due East of a port. A ship sails from the port on a bearing of 030° for 6 km. How far is the ship from the lighthouse? Solution: Draw the triangle, use cosine rule with included angle 90° + 30° = 120°. Distance = √(5² + 6² − 2×5×6×cos 120°) = √(25 + 36 − 60 × (−0.5)) = √(61 + 30) = √91 ≈ 9.54 km.

示例2：一座灯塔位于港口正东方向5 km处。一艘船从港口沿030° 方位航行6 km。此时船距灯塔多远？解：画出三角形，夹角为 90° + 30° = 120°，应用余弦定理。距离 = √(5² + 6² − 2×5×6×cos 120°) = √(25 + 36 − 60 × (−0.5)) = √(61 + 30) = √91 ≈ 9.54 km。

Example 3: The bearing of a tower T from a point A is 140°. From a point B, which is 3 km due North of A, the bearing of T is 210°. Find the distance AT. Solution: Use the sine rule after determining all angles in triangle ABT. This gives AT ≈ 2.46 km.

示例3：从点A观测塔T的方位角为140°。从点B（在A正北3 km处）观测T的方位角为210°。求AT的距离。解：确定三角形ABT的所有角后，运用正弦定理，解得 AT ≈ 2.46 km。

11. Summary | 总结

Three-figure bearings provide a consistent, unambiguous way to express direction. They are measured clockwise from North and always written with three digits. By connecting bearings with triangle geometry, you can solve a wide range of practical problems. Mastering these techniques is essential for success in IB Mathematics assessments involving trigonometry and navigation.

三位数方位角为表达方向提供了一种一致且明确的方式。它们从正北开始顺时针测量，并始终用三位数字书写。通过将方位角与三角形几何联系起来，你可以解决各种实际问题。掌握这些技巧对于在IB数学评估中成功解答涉及三角学和导航的题目至关重要。

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Three-figure bearings | 三位数方位角