# Trigonometry with bearings

## 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.

Firstly, split the bearing into known angles:

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

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

To find how far north the pigeon has flown,

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

To find how far west the pigeon has flown,

and in surd form as

## Question 2

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

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.

Again, split the bearing into known angles:

To find how far south the crow has flown,

or in rationalised surd form;

To find how far east the crow has flown,

and in rationalised surd form;

## Question 4

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

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

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 !

split the bearing into known angles:

which leaves

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.

and in rationalised surd form;

## Question 6

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

The co-ordinates (to 1dp) are

## Question 7

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

The bearing will be 180 + α°

so bearing is 180 +25.9 = 205.9°

Cosine rule

Put in a north line and known angles

The bearing will be 180 + α°, where

α°+β°+15°=90°

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

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):