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
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.
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;
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!
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;
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
What bearing does the pigeon need to fly to reach the crow ?
Give your answer correct to 1 dp.
The bearing will be 180 + α°
so bearing is 180 +25.9 = 205.9°
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°
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 :