Step by step
The question
Question: a ship goes 11 km on a bearing of 322°: how far north and how far west is it from the start?
1 Draw the triangle
Go straight north 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 north line: 38°.
angle with the north-south line = 360° − 322° = 38°
Careful!
The north side is next to the angle, so it uses cos.
The west side is across from it, so it uses sin.
Careful: north: next to the angle → cos; west: across → sin
2 How far north
cos is adjacent over hypotenuse.
So the north side is 11 times cos 38°: about 8.7 km.
north = 11 × cos 38° ≈ 8.7 km
3 How far west
sin is opposite over hypotenuse.
So the west side is 11 times sin 38°: about 6.8 km.
west = 11 × sin 38° ≈ 6.8 km
4 Check
Let's check with Pythagoras: the two legs squared add up to about 121, the path squared.
So it is right!
8.7² + 6.8² ≈ 121.9 ≈ 11² = 121
Let's look back.
First: draw the triangle.
Then: how far north.
Then: how far west.
Last: check.
Answer
west = 11 × sin 38° ≈ 6.8 km