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India
Class XI

Introduction to bearings

Lesson

 

True Bearings

In surveying and air navigation a true bearing measures the angle that a path or line makes with a fixed north-south line. A true bearing measures clockwise from North ALWAYS. For example in a true bearing we would measure this obtuse angle and say it has a bearing of $\theta$θ$^\circ$°T.  

We also write true bearings using exactly three digits. So if $\theta$θ is less than $100$100 degrees, then the true bearing will look something like $040$040$^\circ$°T or $008$008$^\circ$°T. 

 

Compass Bearings

The four main directions of a compass are known as cardinal points. They are north (N), east (E), south (S) and west (W).  

A compass bearing is measured using the acute angle from the north south line.  

  we would write the bearing of B from A as S$\theta$θ$^\circ$°E.

 

Which one first?

The bearing needed or used completely depends on which position comes first. Have a look at the investigation below, it quickly shows you how the angle changes depending on if we are measuring the bearing of A from B or B from A.

Example

A slightly lost hiker walks $300$300 m east before turning south and walking another $800$800 m. What is its true bearing from their original position (to $1$1 decimal place)? And then what is this as a compass bearing?

Think: to do this questions a diagram is a great way to start. This is the diagram I drew.  

 

 

 

The angle we want to find is in green. It is equal to $90$90$^\circ$°$+$+the angle inside the triangle, which we can find using trigonometry.

 

 

Do: Bearing is

$90+\tan^{-1}\frac{800}{300}$90+tan1800300 $=$= $90+\tan^{-1}\frac{8}{3}$90+tan183
  $=$= $90+69.44$90+69.44
  $=$= $159.44$159.44

and to one decimal place the final bearing is $159.4$159.4$^\circ$°T.

As a compass bearing we need to know the acute angle with the North South line. This is $180-159.4=20.6$180159.4=20.6

The compass points we will use will be South first, and then East, so the compass bearing is S$20.6$20.6$^\circ$°E.

 

Practice questions

Question 1

Consider the point $A$A.

  1. Find the true bearing of $A$A from $O$O.

  2. What is the compass bearing of point $A$A from $O$O?

    $\editable{}$ $\editable{}$$^\circ$° $\editable{}$

Question 2

What is the true bearing of Southwest?

Question 3

The position of a ship S is given to be $20$20 kilometres from P, on a true bearing of $0$0$49$49$^\circ$°T.

The position of the ship can also be given by its $\left(x,y\right)$(x,y) coordinates.

  1. If the ship's $x$x-coordinate is $x$x, find $x$x to one decimal place.

  2. If the ship's y-coordinate is $y$y, find $y$y to one decimal place.

Question 4

In the figure below, point $B$B is due East of point $A$A. We want to find the position of point $A$A relative to point $C$C.

  1. Find the true bearing of point $A$A from point $C$C.

  2. What is the compass bearing of point $A$A from point $C$C?

    $\editable{}$ $\editable{}$$^\circ$° $\editable{}$

Outcomes

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

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