Step by step
The question
Question: a car goes 13 km on a bearing of 160°: 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: 20°.
angle with the north-south line = 180° − 160° = 20°
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 13 times cos 20°: about 12.2 km.
south = 13 × cos 20° ≈ 12.2 km
3 How far east
sin is opposite over hypotenuse.
So the east side is 13 times sin 20°: about 4.4 km.
east = 13 × sin 20° ≈ 4.4 km
4 Check
Let's check with Pythagoras: the two legs squared add up to about 169, the path squared.
So it is right!
12.2² + 4.4² ≈ 168.2 ≈ 13² = 169
Let's look back.
First: draw the triangle.
Then: how far south.
Then: how far east.
Last: check.
Answer
east = 13 × sin 20° ≈ 4.4 km