Trigonometry with bearings

pi crow

fort

Question 1

A carrier pigeon delivers a message to a fort which bears 315° from the pigeon's base.

The pigeon is flying at a speed of 50km per hour.

How far north and west of the base is the pigeon after 1 hour ?

Give your answers firstly to 1 decimal place, and then in rationalised surd form.

 

 

1hr

 

Firstly, split the bearing into known angles:

pig2

this gives a right angled triangle, which allows simple trigonometry to be used

pp8

 

all that is needed now is to find the x and y components :

pig4

 

To find how far north the pigeon has flown,

9

To get the answer in rationalised surd form, it is necessary to use the exact values of sin45° and cos45°.

is

sin45 co45

 

8

 

To find how far west the pigeon has flown,

 

11

and in surd form as

 

10

 

Question 2

Write down the exact co-ordinates of the pigeon, given that the base has co-ordinates (0,0) .

ex

Note that the x co-ordinate is negative, since it is west of the origin.

Question 3

A crow delivers a message to a fort which bears 135° from the same base.

The crow is flying at a speed of 100km per hour.

How far south and east of the base is the crow after 1 hour ?

Give your answers firstly to 1 decimal place, and then in rationalised surd form.

 

crow

Again, split the bearing into known angles:

c2

c3

To find how far south the crow has flown,

19

 

or in rationalised surd form;

 

12

To find how far east the crow has flown,

21

and in rationalised surd form;

20

 

Question 4

Write down the exact co-ordinates of the crow, given that the base has co-ordinates (0,0) .

22

Note that the y co-ordinate is negative, since it is south of the origin.

allc

Notice that the two bearings form vertically opposite angles!

Question 5

Another pigeon leaves the base and flies for 2 1/2 hours at a speed of 50 km/h on a bearing of 075°. How far is the pigeon from the crow ?

Give your answers firstly to 1 decimal place, and then in rationalised surd form.

Remember that bearings are measured clockwise from North !

all2

split the bearing into known angles:

3

which leaves

4

Since there are no right angles available, it is necessary to use either the sine rule or the cosine rule.

But which one is needed ?

Here, two sides and and included angle (SAS) are known.

Use the Cosine rule.

 

23

and in rationalised surd form;

24

 

Question 6

What are the co-ordinates of the pigeon,correct to 1 decimal place, given that the base has co-ordinates (0,0) .

 

 

25 26

 

The co-ordinates (to 1dp) are

27

 

Question 7

What bearing does the pigeon need to fly to reach the crow ?

Give your answer correct to 1 dp.

ff

The bearing will be 180 + α°

eq929

so bearing is 180 +25.9 = 205.9°

 

Cosine rule

12

Put in a north line and known angles

11

The bearing will be 180 + α°, where

α°+β°+15°=90°

Use the second form of the Cosine rule to find β°:

28

giving α°=90°-15°-49.1°=25.9°

so bearing is 180 +25.9 = 205.9°

 

Using vectors

Question 4 above required the exact co-ordinates of the crow, given that the base has co-ordinates (0,0) .

 

This can be demonstrated using the section formula (Higher):

ani

 

 

30