Define a 'Three-figure Bearing'.

A three-figure bearing is a direction measured in degrees clockwise from North, represented by a three-digit number between $000$ and $360$.

What is the difference between a bearing of $090^{\circ}$ and a bearing of $270^{\circ}$?

A bearing of $090^{\circ}$ indicates a direction due East, while $270^{\circ}$ indicates due West. They are opposite directions on the same horizontal line, separated by $180^{\circ}$.

How does the bearing of A from B relate to the bearing of B from A?

They are reverse bearings of each other, meaning they differ by exactly $180^{\circ}$. This relationship exists because the North lines at A and B are parallel, making the bearings related by co-interior or alternate angle properties.

When should you use the Sine Rule versus the Cosine Rule in bearing problems?

Use the Sine Rule when you have a known side-angle pair and one other piece of information. Use the Cosine Rule when you have two sides and the included angle (SAS) or all three sides (SSS) to find a missing distance or heading.

What error occurs if a student records a bearing as $45^{\circ}$ instead of $045^{\circ}$?

This is a notation error; bearings must always be expressed as three-figure strings. The leading zero ensures that the value is recognized specifically as a bearing and prevents confusion with other types of angular measurements.

Why is it incorrect to measure a bearing starting from the East line?

By definition, bearings must be measured from the North line. Measuring from East would violate the universal standard, leading to incorrect navigation and making it impossible to use standard reverse-bearing calculations.

What happens if you measure the angle in an anti-clockwise direction?

You will calculate the 'complementary' direction relative to the $360^{\circ}$ circle. For example, measuring $30^{\circ}$ anti-clockwise from North results in a heading of $330^{\circ}$ instead of the intended $030^{\circ}$.

What is a 'Reverse Bearing'?

A reverse bearing is the direction required to travel back to the starting point from the destination. It is calculated by adding or subtracting $180^{\circ}$ from the original bearing.

What is the formula for finding a reverse bearing if the original bearing $\theta$ is $210^{\circ}$?

Since $\theta > 180^{\circ}$, the formula is $\theta - 180^{\circ}$. Therefore, $210^{\circ} - 180^{\circ} = 030^{\circ}$.

What is the formula for finding a reverse bearing if the original bearing $\theta$ is $050^{\circ}$?

Since $\theta < 180^{\circ}$, the formula is $\theta + 180^{\circ}$. Therefore, $050^{\circ} + 180^{\circ} = 230^{\circ}$.

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Geometry & Measures

Bearings

Summary

Bearings are a standardized system of angular measurement used in navigation and surveying to define the direction of one point relative to another. By establishing a universal reference (North) and a consistent direction of rotation (clockwise), bearings allow for precise communication of headings and the application of geometric principles to solve spatial problems.

1. Definition & Core Concepts

A diagram showing a bearing measured clockwise from a North line to a point P.

2. Underlying Principles

The mathematical foundation of bearings relies on the property that all North lines are parallel to each other. This allows for the use of Euclidean geometry, specifically the properties of parallel lines intersected by a transversal, to calculate unknown angles.

When moving between two points, the North lines at each point create a pair of parallel lines. The path connecting the two points acts as a transversal, creating relationships such as alternate angles and co-interior angles.

Because North lines are parallel, the sum of the interior angles between two points (the bearing from A to B and the angle required to find the bearing from B to A) often involves the constant $180^{\circ}$ .

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Bearings

Q: What is the formula for finding a reverse bearing if the original bearing $\theta$ is $050^{\circ}$?

Since $\theta < 180^{\circ}$, the formula is $\theta + 180^{\circ}$. Therefore, $050^{\circ} + 180^{\circ} = 230^{\circ}$.

Summary

1. Definition & Core Concepts

A bearing is a precise way of expressing direction as an angle, measured in degrees from a fixed reference line. In standard navigation, this reference line is always the North line, which is represented vertically on most maps and diagrams.

Three-figure bearings (also known as true bearings) are the standard format, always expressed using three digits to avoid confusion. For example, a direction of $45^{\circ}$ is written as $045^{\circ}$ , and $9^{\circ}$ is written as $009^{\circ}$ .

The measurement must always be taken in a clockwise direction starting from North. This convention ensures that every direction from $000^{\circ}$ to $360^{\circ}$ corresponds to a unique heading, where $000^{\circ}$ is North, $090^{\circ}$ is East, $180^{\circ}$ is South, and $270^{\circ}$ is West.

A diagram showing a bearing measured clockwise from a North line to a point P.

2. Underlying Principles

3. Methods & Techniques

To find the Reverse Bearing (or back bearing) from point B back to point A, you must determine the direction relative to the North line at point B. If the original bearing from A to B is $\theta$ , the reverse bearing is calculated as $\theta + 180^{\circ}$ if $\theta < 180^{\circ}$ , or $\theta - 180^{\circ}$ if $\theta > 180^{\circ}$ .

When solving complex navigation problems involving multiple points, it is essential to use Trigonometry. The Sine Rule and Cosine Rule are frequently applied to find distances and internal angles of the triangles formed by different bearing paths.

The step-by-step approach involves: 1) Drawing a clear diagram with North lines at every vertex, 2) Labeling given bearings and distances, 3) Using parallel line theorems to find internal triangle angles, and 4) Applying trigonometric ratios to solve for the unknown.

4. Key Distinctions

It is vital to distinguish between Three-figure Bearings and Compass Bearings. While both describe direction, they use different notation and reference points.

Feature	Three-figure Bearing	Compass Bearing
Reference	Always North ( $000^{\circ}$ )	North or South
Direction	Always Clockwise	Toward East or West
Format	3 digits (e.g., $060^{\circ}$ )	Direction-Angle-Direction (e.g., $N60^{\circ}E$ )
Range	$000^{\circ}$ to $360^{\circ}$	$0^{\circ}$ to $90^{\circ}$ per quadrant

Another distinction is the Bearing of B from A versus the Bearing of A from B. The 'from' point is the location where the North line must be drawn and where the angle measurement begins.

5. Exam Strategy & Tips

Always start by drawing a large, clear diagram. Use a ruler for North lines and ensure they are perfectly vertical; this visual aid is the most effective way to identify alternate or co-interior angles that are not explicitly stated in the problem.

Check the 'from' keyword carefully. A common error is measuring the angle at the wrong point; the bearing of $X$ from $Y$ means you put your protractor or North line at $Y$ .

Verify that your final answer is a three-digit number. Even if your calculation results in $75$ , you must write $075^{\circ}$ to satisfy the standard convention for bearings in a mathematical or navigational context.

Perform a sanity check on your result. If the bearing is $200^{\circ}$ , the direction should be roughly South-West; if your diagram shows the point is North-East, you have likely measured anti-clockwise or from the wrong reference.

6. Common Pitfalls & Misconceptions

A frequent mistake is measuring the angle anti-clockwise from North. This results in a bearing that is the reflection of the correct path (e.g., recording $330^{\circ}$ instead of $030^{\circ}$ ).

Students often forget to include the leading zero for bearings less than $100^{\circ}$ . In professional and academic contexts, failing to use three digits is considered a notation error.

Misinterpreting the geometric relationship between points is common. For example, assuming the internal angle of a triangle is the bearing itself, rather than calculating the difference between the North line and the triangle side.

It is vital to distinguish between Three-figure Bearings and Compass Bearings. While both describe direction, they use different notation and reference points.

Feature	Three-figure Bearing	Compass Bearing
Reference	Always North ( $000^{\circ}$ )	North or South
Direction	Always Clockwise	Toward East or West
Format	3 digits (e.g., $060^{\circ}$ )	Direction-Angle-Direction (e.g., $N60^{\circ}E$ )
Range	$000^{\circ}$ to $360^{\circ}$	$0^{\circ}$ to $90^{\circ}$ per quadrant

Another distinction is the Bearing of B from A versus the Bearing of A from B. The 'from' point is the location where the North line must be drawn and where the angle measurement begins.

Check the 'from' keyword carefully. A common error is measuring the angle at the wrong point; the bearing of $X$ from $Y$ means you put your protractor or North line at $Y$ .

Students often forget to include the leading zero for bearings less than $100^{\circ}$ . In professional and academic contexts, failing to use three digits is considered a notation error.