Directions and Bearings
Directions and Bearings
The direction to a point is stated as the number of degrees east or west of north or south.
For instance, the direction of A from O is N30ºE.
B is N60ºW from O.
C is S70ºE from O.
D is S80ºW from O.
Note:
N30ºE means the direction is 30º east of north.
The bearing to a point is the angle measured in a clockwise direction from the north line.
For instance, the bearing of P from O is 065º.
The bearing of Q from O is 300º.
Note:
The direction of P from O is N65ºE.
The direction of Q from O is N60ºW.
A bearing is used to represent the direction of one point in relation to another point.
For instance, the bearing of A from B is 065º.
The bearing of B from A is 245º.
Note:
• Three figures are used to give bearings.
• All bearings are measured in a horizontal plane.
Example
A boat sails from a particular port in the direction N30ºW. After the boat has sailed 20 km, how far is it west of the port?
Solution:
Let the boat be x km west of the port.
Therefore, the boat is 10 km west of the port.
Example
A cyclist travels 10 km south, then 8 km east. Find the cyclist's bearing from her starting point to the nearest degree.
Answer:
Therefore, the cyclist's bearing is 141º from her starting point.
Bearings
A directional compass is shown below. It is used to find a direction or bearing .
The four main directions of a compass are referred as cardinal points. They are north (N), east (E), south (S) and west (W). Sometimes, the half-cardinal points of north-east (NE), north-west (NW), south-east (SE) and south-west (SW) are shown on the compass. The above compass shows degree measurements from 0° to 360° in 10° intervals with:
• north representing 0° or 360°
• east representing 90°
• south representing 180°
• west representing 270°
When making use of a directional compass, hold the compass in order that the point marked north points directly away from you. Observe that the magnetic needle at all times points to the north.
Bearing
The true bearing to a point is the angle measured in degrees in a clockwise direction from the north line. We will refer to the true bearing merely as the bearing.
For example, the bearing of point P is 065º which is the number of degrees in the angle measured in a clockwise direction from the north line to the line linking the centre of the compass at O with the point P (i.e. OP).
The bearing of point Q is 300º which is the number of degrees in the angle measured in a clockwise direction from the north line to the line linking the centre of the compass at O with the point Q (i.e. OQ).
Note:
The bearing of a point is the number of degrees in the angle measured in a clockwise direction from the north line to the line linking the centre of the compass with the point.
A bearing is used to represent the direction of one point I relation to another point.
For instance, the bearing of A from B is 065º. The bearing of B from A is 245º.
Observe:
• Three figures are used to give bearings.
• All bearings are measured in a horizontal plane.
Example
State the bearing of the point P in each of the following diagrams:
Solution:
a. Mark the angle in a clockwise direction by showing the turn between the north line and the line linking the centre of the compass to the point P.
The bearing of point P is 048°.
b. Mark the angle in a clockwise direction by showing the turn between the north line and the line linking the centre of the compass to the point P.
The cardinal point S corresponds to 180°. It is clear from the diagram that the required angle is 60° bigger than 180°. Therefore, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 180° + 60° = 240°.
Thus, the bearing of point P is 240°.
c. Mark the angle in a clockwise direction by showing the turn between the north line and the line linking the centre of the compass to the point P.
The cardinal point S corresponds to 180°. It could be seen from the diagram that the needed angle is 40° less than 180°. Thus, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 180° –40° = 140°.
Therefore, the bearing of point P is 140°.
d. Mark the angle in a clockwise direction by showing the turn between the north line and the line linking the centre of the compass to the point P.
The cardinal point W corresponds to 270°. It is evident from the diagram that the required angle is 20° bigger than 270°. Thus, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 270° + 20° = 290°.
Therefore, the bearing of point P is 290°.
Direction
The conventional bearing of a point is specified as the number of degrees east or west of the north-south line. We will refer to the conventional bearing merely as the direction.
To state the direction of a point, write:
• N or S which is determined by the angle that is measured
• the angle between the north or south line and the point, measured in degrees
• E or W which is determined by the location of the point in relation to the north-south line
E.g. In the above diagram, the direction of:
• A from O is N30ºE.
• B from O is N60ºW.
• C from O is S70ºE.
• D from O is S80ºW.
Note:
N30ºE means the direction is 30º east of north.
Example
Describe each of the following bearings as directions.
a. 076°
b. 150°
c. 225°
d. 290°
Solution:
a. The position of a point P on a bearing of 076° is illustrated in the following diagram.
The position of the point P is 76° east of north. Thus, the direction is N76°E.
b. The position of a point P on a bearing of 150° is illustrated in the diagram below:
The position of the point P is 180° – 150° = 30° east of south. Thus, the direction is S30°E.
c. The position of a point P on a bearing of 225° is illustrated in the diagram below:
The position of the point P is 225° – 180° = 45° west of south. Thus, the direction is S45°W.
d. The position of a point P on a bearing of 290° is illustrated in the diagram below:
The position of the point P is 360° – 290° = 70° west of north. Thus, the direction is N70°W.
Bearings
A bearing is an angle, measured clockwise from the north direction. Below, the bearing of B from A is 025 degrees ( observe that 3 figures are always given). The bearing of A from B is 205 degrees.
Example
A, B and C are three ships. The bearing of A from B is 045º. The bearing of C from A is 135º. If AB= 8km and AC= 6km, what is the bearing of B from C?
Solution
tanC = 8/6, so C = 53.13º
y = 180º - 135º = 45º (interior angles)
x = 360º - 53.13º - 45º (angles round a point)
= 262º (to the nearest whole number)
Calculating bearing and orienteering questions with law of sines and law of cosines
The reason for this section is due to the significance of it in trigonometry. of this section. The two trigonometry functions, law of cosines and law of sines, are of crucial importance in trigonometry based on the following reasons:
1. law of sines is essential for calculating the lengths of the unknown sides in a triangle if two angles and one side are given and
2. law of cosines can be used to calculate a side of a triangle if two sides and the angle between them are known. It can as well be used to find the cosines of an angle (and as a result the angles themselves) if the lengths of all the sides are known. In addition, bearings are important as they are used a lot to calculate the direction of an object. Conversely, a bearing will tell you the direction from one point to another relative point. Therefore, this section is very important in trigonometry.
Law of Cosines:
The law of cosines relates to the length of the sides of a plane triangle to the cosines of one of its angle.
a and b stand for the opposite side lengths of c, which are given to you. C stands for the angle between the two side lengths given. c stands for the side length, opposite of C.
The formula can be written in three different ways. This is depending on what sides and angles you are given and/or if you change which sides of the triangle play the role of a, b, and c.
Law of Sines:
The law of sines is an equation that relates the lengths of the sides of a triangle to the sines of its angle.
a, b, and c are the lengths of the sides of the triangles. A, B, and C are the opposite angles as illustrated above.
The law of sines can be used to calculate the remaining sides of a triangle when two angles and a side are known.
What is a bearing?
There are a lot of different types of bearings.
A bearing is an angle that is measured in relation to the fixed horizontal reference plane of true north, that is making use of the direction toward the geographic north pole as a reference. Instead of standard position, the angle is measured in degrees in a clockwise direction from the north line (this is as well referred to as a true bearing).
Bearings can be used for a lot of different things. In aircraft navigation, a bearing is the actual (corrected) compass direction of the forward course of the aircraft. In land navigation, a bearing is the angle between a line linking two points and a north-south line. To put it in simpler terms, a bearing will tell you the direction from one point to another relative point.
Bearings are significant due to the fact that true bearings are frequently being use rather than compass bearings, like at airports. At an airport, the numbers stand for four quadrants in which the difference between the numbers equal 9. What one may not realize though, is that the numbers on the runway are missing a zero, meaning the actual difference is 90. This would mean the total is 360, which is relative to the amount of degrees on a compass.
For instance: 16 - 7 = 9 → 160 -70 = 9025 - 16 = 9 → 250 - 160 = 9034 - 25 = 9 → 340 - 250 = 90
How do you solve a bearing?
Rather than beginning on the right horizontal line, like you would from standard position, 0 degrees begins at the top vertical line of the quadrants -North. From 0 degrees, you would move in a clockwise direction.
For instance:
The angle/bearing of point P is 48 degrees.
The angle/bearing of the plane is 30 degrees.
The angle/bearing for point A is 30 degrees.
The angle/bearing for point C is 110 degrees.
The angle/bearing for point D is 260 degrees.
The angle/bearing for point B is 300 degrees.
Example Questions:
Solve the bearing for the following questions.
What is the bearing for point P? a) 60 degrees
b) 210 degrees
c) 120 degrees
d) 240 degrees
What is the bearing for point P? a) 140 degrees
b) 40 degrees
c) -50 degrees
d) 310 degrees
What is the bearing for point P? a) 160 degrees
b) 290 degrees
c) 70 degrees
d) 20 degrees
Application of what we are given to solve bearing and orienteering questions with Law of Sines and Law of Cosines:
Example
Question:1.
In the example, the bearing of the plane is 270° and the bearing of the wind is 225°.
Redrawing the figure as a triangle using the tail-tip rule, the length (ground speed of the plane) and bearing of the resultant can be estimated.)
First, make use of the law of cosines to find the magnitude of the resultant.
Then, make use of the law of sines to calculate the bearing.
The bearing, β, is thus 270° − 4.64°, or roughly 265.4°.
2. A plane flies at 300 miles per hour. There is a wind blowing out of the southeast at 86 miles per hour with a bearing of 320°. At what bearing must the plane head in order to have a true bearing (relative to the ground) of 14°? What will be the plane's groundspeed?)
Make use of the law of sines to estimate the bearing and the groundspeed. Due to the fact that these alternate interior angles are congruent, the 54° angle is the sum of the 14° angle and the 40° angle.
Thus, the bearing of the plane ought to be 14° + 13.4° = 27.4°. The groundspeed of the plane is 342.3 miles per hour.
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