Step by step
The question
Question: a ship goes 12 km on a bearing of 065°: how far north and how far east is it from the start?
1 Draw the triangle
A bearing is an angle from north, turning clockwise.
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: 65°.
Name the sides from the angle.
The path is across from the right angle: the hypotenuse, H.
The north side is next to the angle: the adjacent side, A.
The east side is across from it: the opposite side, O.
Sin, cos and tan are calculator keys.
Each one links an angle to two sides.
SOH CAH TOA helps us remember: sin is O over H, cos is A over H, tan is O over A.
Careful!
We have H, the path.
The north side is A: CAH, so cos.
The east side is O: SOH, so sin.
Do not mix them up.
2 How far north
Cos is adjacent over hypotenuse: north over 12.
The north side is divided by 12.
To get it alone, we multiply both sides by 12.
The calculator gives cos 65°: about 0.4226.
Times 12: about 5.1 km.
3 How far east
Sin is opposite over hypotenuse: east over 12.
Again we multiply both sides by 12.
The calculator gives sin 65°: about 0.9063.
Times 12: about 10.9 km.
4 Check
Let's check with Pythagoras: the two short sides squared add up to about 144, the path squared.
So it is right!
Let's look back.
First: draw the triangle.
Then: how far north.
Then: how far east.
Last: check.