Worked examples
The question
Question: a ship goes 8 km on a bearing of 040°: how far north and how far east is it from the start?
1 Draw the triangle
Go straight north first, then straight east: a right triangle.
The path is the long side.
The angle at the start, between north and the path, is the bearing: 40°.
angle at the start = 40°
Careful!
The north side is next to the angle, so it uses cos.
The east side is across from it, so it uses sin.
Careful: north: next to the angle → cos; east: across → sin
2 How far north
cos is adjacent over hypotenuse.
So the north side is 8 times cos 40°: about 6.1 km.
north = 8 × cos 40° ≈ 6.1 km
3 How far east
sin is opposite over hypotenuse.
So the east side is 8 times sin 40°: about 5.1 km.
east = 8 × sin 40° ≈ 5.1 km
4 Check
Let's check with Pythagoras: the two legs squared add up to about 64, the path squared.
So it is right!
6.1² + 5.1² ≈ 63.2 ≈ 8² = 64
Let's look back.
First: draw the triangle.
Then: how far north.
Then: how far east.
Last: check.
Answer east = 8 × sin 40° ≈ 5.1 km
▶ Watch it solved
The question
Question: a ship goes 9 km on a bearing of 130°: how far south and how far east is it from the start?
1 Draw the triangle
Go straight south first, then straight east: a right triangle.
The path is the long side.
In the triangle we need the angle at the start, between the path and the south line: 50°.
angle with the north-south line = 180° − 130° = 50°
Careful!
The south side is next to the angle, so it uses cos.
The east side is across from it, so it uses sin.
Careful: south: next to the angle → cos; east: across → sin
2 How far south
cos is adjacent over hypotenuse.
So the south side is 9 times cos 50°: about 5.8 km.
south = 9 × cos 50° ≈ 5.8 km
3 How far east
sin is opposite over hypotenuse.
So the east side is 9 times sin 50°: about 6.9 km.
east = 9 × sin 50° ≈ 6.9 km
4 Check
Let's check with Pythagoras: the two legs squared add up to about 81, the path squared.
So it is right!
5.8² + 6.9² ≈ 81.3 ≈ 9² = 81
Let's look back.
First: draw the triangle.
Then: how far south.
Then: how far east.
Last: check.
Answer east = 9 × sin 50° ≈ 6.9 km
▶ Watch it solved
The question
Question: a ship goes 20 km on a bearing of 248°: how far south and how far west is it from the start?
1 Draw the triangle
Go straight south first, then straight west: a right triangle.
The path is the long side.
In the triangle we need the angle at the start, between the path and the south line: 68°.
angle with the north-south line = 248° − 180° = 68°
Careful!
The south side is next to the angle, so it uses cos.
The west side is across from it, so it uses sin.
Careful: south: next to the angle → cos; west: across → sin
2 How far south
cos is adjacent over hypotenuse.
So the south side is 20 times cos 68°: about 7.5 km.
south = 20 × cos 68° ≈ 7.5 km
3 How far west
sin is opposite over hypotenuse.
So the west side is 20 times sin 68°: about 18.5 km.
west = 20 × sin 68° ≈ 18.5 km
4 Check
Let's check with Pythagoras: the two legs squared add up to about 400, the path squared.
So it is right!
7.5² + 18.5² ≈ 398.5 ≈ 20² = 400
Let's look back.
First: draw the triangle.
Then: how far south.
Then: how far west.
Last: check.
Answer west = 20 × sin 68° ≈ 18.5 km
▶ Watch it solved
