Bearings and the 110° Angle: A Complete Guide | 方位角与110°角：完全指南

📚 Bearings and the 110° Angle: A Complete Guide | 方位角与110°角：完全指南

Bearings are a fundamental topic in IGCSE Mathematics, especially for students following the Edexcel specification. A bearing is a three-digit angle measured clockwise from north, and it is used to describe direction in navigation, surveying, and many real-world applications. In this article, we will focus on the bearing 110°, exploring its definition, how to draw it, how to calculate back bearings, and how to solve geometry problems involving this angle.

方位角是IGCSE数学中的一个基础主题，尤其对于学习爱德思（Edexcel）考纲的学生而言。方位角是从正北方向顺时针测量的三数字角度，用于导航、测量和许多实际应用中的方向描述。本文将聚焦于110°这一方位角，探讨其定义、画法、如何计算反方位角，以及如何解决涉及该角的几何问题。

1. What Is a Bearing? | 什么是方位角？

A bearing is an angle, in degrees, measured clockwise from the north direction. It must always be written as three digits, for example, 000°, 045°, 110°, 270°. Leading zeros are used if the angle is less than 100°, so 10° is written as 010° and 5° as 005°. Bearings are always measured from north (000°) and rotate clockwise through 360°.

方位角是从正北方向顺时针测量的角度，单位是度。它必须始终写成三位数形式，例如000°、045°、110°、270°。如果角度小于100°，需要补前导零，因此10°写作010°，5°写作005°。方位角总是从正北（000°）开始，顺时针旋转，范围是0°到360°。

2. The Three Rules of Bearings | 方位角的三条规则

To solve bearing problems correctly, you must remember three essential rules:

Rule 1: Bearings are measured from the north line (000°).
Rule 2: Bearings are measured clockwise.
Rule 3: Bearings must be written with three digits (e.g., 110°, not 110° is fine, but 20° must be written as 020°).

要正确解决方位角问题，必须记住三条基本规则：

规则1：方位角从正北线（000°）开始测量。
规则2：方位角是顺时针测量的。
规则3：方位角必须以三位数书写（例如110°没问题，但20°必须写成020°）。

3. Measuring from North | 从正北方向度量

On a diagram, the north direction is usually shown as a vertical line pointing upwards. To measure a bearing of 110°, place a protractor with its centre at the point of interest, align the 0° mark with north, and measure 110° clockwise from north. This means the direction is slightly south of east, because east is 090° and south is 180°.

在图中，正北方向通常用一条竖直向上的线表示。要测量110°的方位角，将量角器的中心放在所关注的点上，让0°刻度对准正北，然后从正北方向顺时针量取110°。这意味着该方向在正东（090°）和正南（180°）之间，即偏东偏南。

Bearing = angle clockwise from north, written as a 3-digit number

方位角 = 从正北顺时针测量的角度，写成三位数字

4. Interpreting a Bearing of 110° | 110°方位角的实际含义

A bearing of 110° means that, starting from a reference point, you face north and then turn 110° to the right (clockwise). This direction is equivalent to 20° south of east (because 110° – 90° = 20°). In navigation terms, if a ship travels on a bearing of 110°, it is moving in a direction that is 20° towards the south from the east line.

110°的方位角意味着，从参考点出发，面向正北然后向右（顺时针）转110°。该方向相当于正南方向偏东20°（因为110° – 90° = 20°）。在导航术语中，如果一艘船以110°的方位角航行，它实际上是在朝东偏南20°的方向移动。

5. Back Bearings: 110° and 290° | 反方位角：110°与290°

The back bearing (or reverse bearing) is the direction opposite to a given bearing. To find the back bearing, add 180° to the original bearing if it is less than 180°, or subtract 180° if it is greater than 180°. For a bearing of 110°, we add 180° because 110° is less than 180°:

反方位角（或反向方位角）是与给定方向相反的方向。要求反方位角，如果原方位角小于180°，则加180°；如果大于180°，则减180°。对于110°的方位角，因为110°小于180°，所以加180°：

Back bearing = 110° + 180° = 290°

反方位角 = 110° + 180° = 290°

If you are travelling from A to B on a bearing of 110°, then travelling back from B to A requires a bearing of 290°. This is because the two directions differ by 180°.

如果你从A点以110°方位角前往B点，那么从B点返回A点的方位角是290°。因为这两个方向相差180°。

6. Using Trigonometry with a Bearing of 110° | 利用三角函数求解110°方位角问题

In IGCSE problems, you often need to find distances or coordinates from a bearing. Consider a point P that is 50 km from an origin O on a bearing of 110°. We can split the displacement into east and south components. The angle between the east direction (090°) and the bearing is 20°, so the east component is 50 × cos(20°) km, and the south component is 50 × sin(20°) km.

在IGCSE考试中，常需要根据方位角求距离或坐标。假设点P在原点O的110°方位角上，且距离为50 km。我们可以将位移分解为东向和南向分量。正东方向（090°）与方位角之间的夹角为20°，因此东向分量为50 × cos(20°) km，南向分量为50 × sin(20°) km。

East component = 50 cos 20° ≈ 46.98 km

South component = 50 sin 20° ≈ 17.10 km

If you are asked for the coordinates, remember that east is positive x and north is positive y, so the y-coordinate will be negative (since the point is south of O).

如果要求坐标，请记住东为正x方向，北为正y方向，因此y坐标为负（因为该点位于O的南方）。

7. Common Mistakes and Pitfalls | 常见错误与易错点

Many students make avoidable errors when dealing with bearings. Here are the most frequent ones with the bearing 110°:

许多学生在处理方位角时会犯可以避免的错误。以下是针对110°方位角最常见的错误：

Writing 110 instead of 110° – always include the degree symbol unless the question specifies otherwise.
Forgetting to measure clockwise – measuring anticlockwise from north would give 250°, not 110°.
Using the inside angle of a triangle incorrectly – the bearing is not the same as the interior angle of a triangle unless carefully identified.
Back bearing error – some students subtract 180° from 110° to get -70°, which is invalid; the correct back bearing is 290°.

写出110而不是110°——除非题目另有说明，否则务必加上度符号。

忘记顺时针测量——从正北逆时针测量会得到250°，而不是110°。

错误使用三角形内角——方位角与三角形内角并不相同，除非仔细确认。

反方位角错误——有些学生从110°减180°得到-70°，这是无效的；正确的反方位角是290°。

8. Practice Problems | 练习题目

Try these questions to test your understanding of bearings, particularly 110°.

尝试以下问题来检验你对方位角的理解，特别是110°。

Question	Answer hint
1. What is the back bearing of 110°?	290°
2. A boat travels 120 km on a bearing of 110°. How far east does it travel?	120 cos 20° ≈ 112.76 km
3. From point A, B is on a bearing of 290°. What is the bearing of A from B?	110° (reverse the back bearing)

9. Summary | 总结

Bearings are a core skill in IGCSE Mathematics. Always start from north, measure clockwise, and write the answer as three digits. The bearing 110° lies between east and south, and its back bearing is 290°. Using right-angled trigonometry, you can resolve a bearing into north/south and east/west components. With careful practice, you can avoid common mistakes and score full marks on bearing questions.

方位角是IGCSE数学的核心技能。始终从正北开始，顺时针测量，并将答案写成三位数字。110°方位角位于东和南之间，其反方位角为290°。利用直角三角形三角函数，你可以将方位角分解为南北和东西分量。通过仔细练习，你就能避免常见错误，在方位角题目上获得满分。

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Bearings and the 110° Angle: A Complete Guide | 方位角与110°角：完全指南