# Bearing And Distance With Solved Examples

Bearing and distance are two of the applied knowledge of trigonometry. This topic can be found in most SS1, SS2, and SS3 textbooks under trig (1) and trig (2).

Bearing is the angular relationship between two or more places. The four main directions are North (N), South (S), West (W), and East (E). The sub-directions are North-West (NW), North-East (NE), South-West (SW), and South-East (SE). The angular relationship between two or more places can be expressed as a compass or three-digit notation.

**Techniques**

Firstly, you must know that Ox and Oy’s cartesian axes divide the plane into the 1st, 2nd, 3rd, and 4th quadrants.

For trigonometric purposes, an angle is defined as a measure of the rotation of a line about the origin, starting from the direction of Ox. The angle may be positive or negative depending on whether the rotation is anticlockwise or clockwise.

You must use either sine or cosine rules to answer any question about bearing and distances.

**Sine rule**: In any triangle ABC, the angles are usually denoted by the capital letters A, B, and C, and the sides opposite these angles by A, b, c respectively.

a/sinA = b/sinB = c/sin C

This formula is used for solving triangles that are not right-angled and in which either two angles and any side are given, or two sides and the angle opposite one of them are given.

Example 1

A point X is 34m due east of a point Y. The flagpole bearings from X and Y are N18^{o}W and N40^{o}E, respectively. Calculate the distance of the flagpole from Y

Solution

**Cosine rule**: this rule is for solving triangles that are not right-angled in which two sides and the included angle are given.

c^2 = a^2 + b^2 -2ab cosC

b^2 = a^2 + c^2 -2ac cosB

a^2 = b^2 + c^2 -2bc cos A

Example 2

From a point on the sea’s edge, one ship is 5km away on a bearing S50oE, and another is 2km away on a bearing S60^{o}W. How far apart are the ships?

Solution

Read: Make a the subject of the formula v = u + at

**Other Solved Examples**

**Compass Bearing**: This bearing can be represented in the sub-direction, i.e., North-West (NW), North-East (NE), South-West (SW), and South-East (SE). For example, N30^{o}E, S50^{o}W, S20^{o}E

Example 3

Express the true bearing 210^{o} as a compass bearing

**Three digit notation**: This is measured clockwise from the North. For example, 020^{o}, 215^{o}, 309^{o}, etc.

Example 4

What is the equivalent of S70^{o}E in three-digit notation?

Example 5

If the bearing of X from Y is N64^{0}W. Find the bearing of Y from X.

Example 6

If the bearing of P from R is 065^{0}, what is R from P? A. 230^{0} B. 245^{0 } C. 120^{0 } D 025^{0}