Mastering Bearings: The 190° Case | 掌握方位角：以190°为例

📚 Mastering Bearings: The 190° Case | 掌握方位角：以190°为例

In IGCSE Mathematics (Edexcel), bearings are a classic application of angles, trigonometry, and geometry. They appear in problem-solving questions involving navigation, maps, and relative positions. A bearing of 190° is a typical example that many students find tricky because it lies beyond 180° and requires careful interpretation.

在爱德思IGCSE数学中，方位角是角度、三角学和几何学的经典应用。它们出现在涉及导航、地图和相对位置的解题问题中。190°的方位角是一个典型的例子，许多学生觉得它棘手，因为它超过180°，需要仔细理解。

1. What Is a Bearing? | 什么是方位角？

A bearing is an angle measured clockwise from the north direction. It is always written as a three-digit number, from 000° to 360°. For example, north is 000°, east is 090°, south is 180°, and west is 270°.

方位角是从正北方向顺时针测量的角度。它总是写成三位数，范围从000°到360°。例如，正北是000°，正东是090°，正南是180°，正西是270°。

A bearing of 190° means a direction that is 190° clockwise from north, which lies between south (180°) and west (270°). More precisely, it is 10° west of due south.

190°的方位角是指从正北顺时针旋转190°的方向，它位于正南（180°）和正西（270°）之间。更准确地说，它是在正南偏西10°。

2. The Three-Digit Convention | 三位数规则

Why do we write 190° instead of 190°? In bearings, we always use three digits, adding leading zeros if necessary. So a bearing of 10° must be written as 010°, and a bearing of 190° already has three digits.

为什么我们写190°而不是190°？在方位角中，我们总是使用三位数，必要时补前导零。所以10°的方位角必须写成010°，而190°已经是三位数。

This convention avoids confusion with ordinary angles and ensures that every direction has a unique numeric code between 000 and 360.

这个规则避免了与普通角的混淆，并确保每个方向都有唯一介于000和360之间的数值代码。

3. Drawing a Bearing of 190° | 画出190°的方位角

To draw a bearing of 190°, start by drawing a vertical line pointing north from your point. Then use a protractor to measure 190° clockwise from north. Because 190° is greater than 180°, the arrow will point downwards and slightly to the left (west).

要画出190°的方位角，首先从你的点画一条指向正北的竖线。然后用量角器从正北顺时针测量190°。因为190°大于180°，箭头将指向下方并略微偏左（西）。

North (000°) → 190° clockwise → direction 10° west of south

北（000°）→ 顺时针190° → 正南偏西10°的方向

4. Converting Bearings to Ordinary Angles | 方位角与普通角的转换

Sometimes it helps to convert a bearing into an angle from a reference line. For a bearing between 180° and 270°, the angle west of south is: bearing − 180°.

有时将方位角转换为相对于参考线的普通角会有帮助。对于180°到270°之间的方位角，它偏离正南向西的角度为：方位角 − 180°。

190° − 180° = 10° west of south

190° − 180° = 正南偏西10°

Similarly, a bearing of 190° is equivalent to 90° + 100° from north, but the useful form is “10° west of south”.

类似地，190°的方位角等价于从北算起90°+100°，但更有用的形式是“正南偏西10°”。

5. The Back Bearing | 反方位角

The back bearing is the direction from the destination back to the starting point. It is found by adding or subtracting 180°. Since 190° + 180° = 370°, which is greater than 360°, we subtract 360° to get 010°.

反方位角是从目的地返回起始点的方向。通过加或减180°得到。因为190° + 180° = 370°，大于360°，所以减去360°得到010°。

If bearing A→B = 190°, then bearing B→A = 190° + 180° = 370° → 010°

如果从A到B的方位角为190°，那么从B到A的方位角为190° + 180° = 370°，即010°

Rule: if the result is ≥ 360°, subtract 360°; if the result is < 000°, add 360°.

规则：若结果≥360°，减去360°；若结果<000°，加上360°。

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

In many IGCSE questions, bearings are used with parallel north-south lines. The angle between two bearings can be found by looking at the difference or using alternate angles.

在许多IGCSE问题中，方位角与平行的南北线一起使用。两个方位角之间的角可以通过求差或利用内错角来找到。

For example, if point A has a bearing of 190° from O and point B has a bearing of 120° from O, the angle at O between A and B is 190° − 120° = 70°.

例如，若从O点看A的方位角为190°，看B的方位角为120°，则O处A与B之间的夹角为190° − 120° = 70°。

When drawing north lines at each point, remember that all these north lines are parallel. You can then use alternate angles to find unknown angles in the triangle formed.

在每一点画北线时，记住所有这些北线是平行的。然后你可以利用内错角来求出所形成三角形中的未知角。

7. Solving a Triangle Using Bearings | 利用方位角解三角形

Consider a typical problem: Ship A is 5 km from point O on a bearing of 190°. Ship B is 7 km from O on a bearing of 130°. Find the distance AB.

考虑一个典型问题：船A在点O的190°方位上，距离5 km；船B在点O的130°方位上，距离7 km。求AB的距离。

First, the angle between OA and OB is 190° − 130° = 60°. Then we use the cosine rule in triangle AOB:

首先，OA与OB之间的夹角为190° − 130° = 60°。然后在三角形AOB中使用余弦定理：

AB² = 5² + 7² − 2 × 5 × 7 × cos 60°

AB² = 25 + 49 − 70 × 0.5 = 74 − 35 = 39

AB = √39 ≈ 6.24 km

Always draw a diagram and mark the included angle carefully before applying the cosine rule.

在应用余弦定理之前，始终画图并仔细标出夹角。

8. Using the Sine Rule with Bearings | 在方位角问题中使用正弦定理

If a problem provides two angles and one side, the sine rule is useful. For example, from a point P, the bearing of Q is 190° and the bearing of R is 120°. If PQ = 8 km and angle PQR = 50°, you may need to find PR.

如果问题给出两个角和一个边，正弦定理很有用。例如，从点P看，Q的方位角为190°，R的方位角为120°。若PQ = 8 km，且角PQR = 50°，你可能需要求PR。

First find angle QPR = 190° − 120° = 70°. Then angle PRQ = 180° − 70° − 50° = 60°. Applying the sine rule:

首先求角QPR = 190° − 120° = 70°。然后角PRQ = 180° − 70° − 50° = 60°。应用正弦定理：

PR ÷ sin 50° = PQ ÷ sin 60°

PR ÷ sin50° = PQ ÷ sin60°

PR = 8 × sin 50° ÷ sin 60° ≈ 7.08 km

PR = 8 × sin50° ÷ sin60° ≈ 7.08 km

Check that your calculator is in degree mode and round to a sensible degree of accuracy.

检查计算器处于角度制，并按合理的精度进行四舍五入。

9. Common Mistakes with 190° | 处理190°的常见错误

Many students make these errors:

许多学生容易犯以下错误：

Reading 190° as 10° west of north instead of 10° west of south. | 将190°误读为正北偏西10°，而不是正南偏西10°。
Forgetting to write three digits, e.g., writing 90° instead of 090°. | 忘记写成三位数，例如写90°而不是090°。
Using compass points directly without converting to bearings. | 未转换为方位角而直接使用罗盘方位。
Adding 180° incorrectly for the back bearing. For 190°, remember to subtract 360° after adding. | 计算反方位角时加180°出错。对于190°，加180°后要减去360°。

10. Exam Tips for Edexcel IGCSE | 爱德思IGCSE考试提示

In the Edexcel IGCSE Mathematics exam, bearing questions often appear in Paper 1 (non-calculator) and Paper 2 (calculator). They may be combined with speed, distance, and time, or with map scales.

在爱德思IGCSE数学考试中，方位角问题经常出现在Paper 1（无计算器）和Paper 2（有计算器）中。它们可能结合速度、距离、时间或地图比例尺。

Always show your working clearly, especially how you found the angle between two bearings. Use a sharp pencil and a protractor for accurate diagrams.

始终清晰展示解题过程，尤其是如何求得两个方位角之间的角。使用削尖的铅笔和量角器画精确的图。

Marks are often awarded for correct substitution into the cosine or sine rule, even if the final answer is slightly off.

即使最终答案略有偏差，正确代入余弦或正弦定理通常也能得分。

11. Practice Problem | 练习题

Try this question: A boat sails from port P to point Q on a bearing of 190° for 12 km. It then sails from Q to R on a bearing of 280° for 9 km. Find the bearing of R from P.

尝试这个题目：一艘船从港口P沿190°方位角航行12 km到达点Q。然后从Q沿280°方位角航行9 km到达R。求从P看R的方位角。

Hint: Draw a triangle, find the internal angle at Q using the difference between bearings, then use the cosine rule to find PR, and the sine rule to find the angle at P.

提示：画三角形，利用方位角之差求Q处的内角，然后用余弦定理求PR，再用正弦定理求P处的角。

The final bearing should be approximately 250°. Let’s break it down:

最终方位角应约为250°。我们来分解一下：

At Q, the bearing from Q back to P is 190° − 180° = 010°. The bearing from Q to R is 280°. The angle at Q between QP and QR is 280° − 010° = 270°, but the interior angle of the triangle is 90° (since 270° is the reflex angle; the acute angle is 360° − 270° = 90°).

在Q处，从Q返回P的方位角为190° − 180° = 010°。从Q到R的方位角为280°。QP与QR在Q处的夹角为280° − 010° = 270°，但三角形的内角为90°（因为270°是优角，锐角为360° − 270° = 90°）。

PR² = 12² + 9² − 2 × 12 × 9 × cos 90° = 144 + 81 = 225 ⇒ PR = 15 km

PR² = 12² + 9² − 2 × 12 × 9 × cos90° = 144 + 81 = 225，所以PR = 15 km

Then the angle at P (between PR and the north line at P) can be found using the sine rule. The bearing turns out to be about 250°.

然后使用正弦定理求P处角（PR与P点北线之间的角）。最终方位角约为250°。

12. Summary | 总结

Bearings are a fundamental skill in IGCSE mathematics. A bearing of 190° is just one example, but mastering it prepares you for all directions around the compass. Remember the three-digit format, the clockwise-from-north rule, and how to convert between bearings and ordinary angles.

方位角是IGCSE数学的基本技能。190°的方位角只是一个例子，但掌握它能帮助你应对罗盘上的所有方向。记住三位数格式、从正北顺时针的规则，以及如何在方位角和普通角之间转换。

Practice drawing diagrams, use parallel north lines, and always check whether your answer makes sense in the context of the problem. With these skills, you will confidently solve any bearing question on the Edexcel IGCSE exam.

练习画图，利用平行的北线，并始终检查你的答案在问题情境中是否合理。掌握这些技能，你将能自信地解决爱德思IGCSE考试中的任何方位角问题。

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Mastering Bearings: The 190° Case | 掌握方位角：以190°为例