Step by step
The question
Question: a boat goes 10 km on a bearing of 030°: 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: 30°.
angle at the start = 30°
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 10 times cos 30°: about 8.7 km.
north = 10 × cos 30° ≈ 8.7 km
3 How far east
sin is opposite over hypotenuse.
So the east side is 10 times sin 30°: about 5.0 km.
east = 10 × sin 30° ≈ 5.0 km
4 Check
Let's check with Pythagoras: the two legs squared add up to about 100, the path squared.
So it is right!
8.7² + 5.0² ≈ 100.7 ≈ 10² = 100
Let's look back.
First: draw the triangle.
Then: how far north.
Then: how far east.
Last: check.
Answer
east = 10 × sin 30° ≈ 5.0 km