Step by step
The question
Question: a boat goes 7 km on a bearing of 210°: 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: 30°.
angle with the north-south line = 210° − 180° = 30°
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 7 times cos 30°: about 6.1 km.
south = 7 × cos 30° ≈ 6.1 km
3 How far west
sin is opposite over hypotenuse.
So the west side is 7 times sin 30°: about 3.5 km.
west = 7 × sin 30° ≈ 3.5 km
4 Check
Let's check with Pythagoras: the two legs squared add up to about 49, the path squared.
So it is right!
6.1² + 3.5² ≈ 49.5 ≈ 7² = 49
Let's look back.
First: draw the triangle.
Then: how far south.
Then: how far west.
Last: check.
Answer
west = 7 × sin 30° ≈ 3.5 km