Bearing 1

Mathematics SSS 2 Third Term

WEEK 2

Bearing 1

Performance Objectives

Students should be able to;

1. Solve problems on trigonometry ratios
2. Solve problems on angle of Elevation and Depression
3. Define the 4 – 8 and 16 cardinal points
4. State two bearing notations

Angle of Elevation and Depression

Content

Angle of Elevation

Suppose that you are looking at an object in the distance.

If the object is above you, then the angle of elevation is the angle your eyes look up.

If the object is below you, the angle of depression is the angle your eyes look down.

Angles of elevation and depression are measured from the horizontal.

It is common mistake not to measure the angle of depression from the horizontal.

The angle that an observer would raise his or her line of sight above a horizontal line in order to see an object. The angle of elevation is the angle between the horizontal and the

observer’s line of sight

In the diagram above, the angle of elevation of the object from the observer is α⁰.

Angle of Depression

If the object is below the level of the observer, then the angle between the horizontal and the observer’s line of sight is called the angle of depression. If an observer were up above and

needed to look down, the angle of depression would be the angle that the person would need to lower his or her line of sight.

In the diagram above, the angle of depression of the object from the observer is β⁰.

NOTE: Using the angle of depression or elevation to an object, and knowing how far away the object is, enables us to find the height of the object using trigonometry.

The advantage of doing this is that it is very difficult to measure the height of a mountain or the depth of a canyon directly; it is much easier to measure how far away it is (horizontal distance) and to measure the angle of elevation or depression.

Measuring Angles of Elevation and Depression

Angles of elevation and depression can be measured with a simple instrument called Clinometers. Simple Clinometers is made from a chalk-board protractor in which a plumb-line hangs from the center of the protractor. The angle that the plumb-line makes with the 90⁰ vertical axes when the Clinometers is placed in the observer’s direction is the angle of elevation or

depression.

In the figure above, a plumb-line hangs from the center of the protractor at A. The observer sights an object along the line BAC. The angle of elevation e⁰ is the angle between AO and the plumb-line. The size of e⁰ can be read from the scale. Notice that e⁰ increases from 0⁰ at O to 90⁰ at B.

The Relationship between Angle of Elevation and Depression

In other words, as shown in the diagrams below, the angle formed with the horizontal when an observer looks up is called angle of elevation whereas, the angle formed with the horizontal when an observer looks down is called angle of depression.

 

There is a connection between angle of elevation and angle of depression. For example, the angle of elevation of Y from X is the same as the angle of depression of X from Y. Therefore angles of elevation and depression are alternate angles and are therefore equal in a given situation.

The Compass Directions

Major Compass/Cardinal Directions

There are four major directions used to describe locations. These cardinal directions are:

North (N)    South (S)    East (E)    West (W)

The four main directions, North, South, East and West, divide the angle at a point (360^o), into four equal parts and each is 90^o or a right angle.

Minor Cardinal Directions

Other minor cardinal directions are those that lie in the midpoints as follows:

North and East called North-East (NE)

South and East called South-East (SE)

South and West called South-West (SW)

North and West called North-West (NW)

These minor cardinal directions subdivide each right angle into two equal parts such that the angle between each major cardinal and minor direction is 45^o. The eight cardinal points are

illustrated below:

 

Bearing and its Types

Bearing is the direction of one point with respect to a given point.

If a line which points due North of a compass is fixed, the direction of any other line on the surface of the earth is given as the angle which it makes with the North-pointing line, this angle is called bearing. In particular, we must note that the bearing is measured from the line due North in a clockwise direction. Since bearings involve mainly finding directions, we use a compass to find them. 

The Compass and Compass/Directional Bearing

The Compass is an instrument used in finding directions. It is also used in erecting a wind-vane in the correct position. A wind vane is an instrument used in detecting the direction of the

wind. It can, therefore be used in place of a compass to determine cardinal directions.

Compass bearing is the direction of one point with respect to a given point given in terms of the major or minor cardinal direction of the relative point.

Consider the points X and Y in the diagram below:

The Acute Angle or Simple Bearing

This method involves using the acute angle which the line XY makes with the North or South (in the diagram above, the South Pole is appropriate) direction at X, Eastwards or Westwards. For example, in the diagram below, the bearing of Y from X is written as S 30^o E or (or South 30^o Eastwards) and is called the acute-angle bearing of Y from X.

In general, acute angle bearings are measured in relation to the North or South Pole and must therefore be greater than 0^o but less than 90^o as its name implies. If an angle related to the East or West pole is given, its complementary angle is used to give the acute-angle bearing.

 

The Three-figure or Surveyor’s Bearing

This method involves reading on the compass, the angle which the line XY makes with the North direction. This angle is the bearing of the object Y from the reference point X, and it is called the surveyor’s bearing of Y from X. The surveyor’s bearing is written with three digits known as three-figure bearings. When the angle is between 0^o and 90^o inclusive, say 7^o, the bearing is written as 007^o, that is, two zeros are added before the angle. Suppose the angle is 55^o, we write it as 055^o. 

Directional versus Three-figure versus Acute Angle Bearing

Examples:

1.

 

	Directional	Three-figure Bearing	Acute-Angle Bearing
	North	000o/360o	−
	North East	045o	N45oE
	East	090o	−
	South East	135o	S45oE
	South	180o	−
	South West	225o	S45oW
	West	270o	−
	North West	315o	N45oW

2. Directional/Compass = NE

The three-figure bearing = 037^o

Acute Angle bearing = N37^oE

3. The compass/directional bearing = SE

The three-figure bearing = 117^o

The Acute-angle bearing = S63^oE

4. The compass/directional bearing = SW

The three-figure bearing = 246^o

The Acute-angle bearing = S66^oW

 

5. The compass/directional bearing = NW

The three-figure bearing = 344^o

The Acute-angle bearing = N16^oW

 

6. The table below shows the angles when the turning is clockwise from the South direction to the other cardinal directions.

 

	Direction	N	E	W	NE	NW	SE	SW
	Clockwise Turning from South	180o	270o	90o	225o	135o	315o	45o

 

Reciprocal/Back Bearing

The bearing of B from A is 075^o, while the bearing of A from B is 255^o. 255^o is called the back/ reciprocal bearing of 075^o.

In general, if the bearing is less than 180^o we add 180^o to get the back bearing and if the bearing is greater than 180^owe subtract 180^o.

Example:

Find the bearing whose reciprocal/ back bearing is?

(i) 033^o

Solution

33^o is less than 180^o

So we add 180^o to 33^o

180^o + 33^o = 218^o

So the back bearing of 033^o is 218^o

 

(ii) 220^o

Solution

220^o is greater than 180^o

So we subtract 180^o from 220^o

220^o – 180^o = 40^o

So the back bearing of 220^o is 40

Scale Drawing

Example:

Ibadan is 116km on a bearing 027 ^o from Lagos. How far north of Lagos is Ibadan? How far west of Ibadan is Lagos?

Solution:

Make a scale drawing of the data.

Scale: 1cm to 20km

K is the point which is due north of Lagos and due west of Ibadan. LK represents the distance that Ibadan is north of Lagos.

By measurement, LK ≈ 5.2cm

The true distance LK ≈ 5.2 × 20km

= 104km

IK represents the distance that Lagos is west of Ibadan.

By measurement, IK ≈ 2.6cm

The true distance IK ≈ 2.6 × 20km

= 52km

Thus Ibadan is approximately 104km north of Lagos and Lagos is approximately 104km north of Lagos and Lagos is approximately 52km west of Ibadan.

Example:

Use scale drawing to answer the questions that follow the diagram:

1. What is the angle of depression (θ) of the top of the building from the observer at point A?
2. Find the height of the building in the figure above.

STEP 1: Draw a sketch of the figure in the form of a right-angled triangle.

STEP 2: Choose an appropriate scale, considering the measurements on the given lengths in the original diagram. From 12m, a scale of 1cm represent 2m can be used to draw 12m as 6cm.

STEP 3: Draw the appropriate diagram, measure and mark the given angle.

STEP 4: Measure length AB in cm then convert it back to m to get the height of the building