How do unit conversions work?

It involves simple algebra (multiplication or division) using conversion factor (numerical relationship between the two units).
Explanation:
Watch How to Convert between units here
Unit conversions use conversion factors to change the units of a measurement.
Explanation:

For example, you buy a 1.7 kg smoked ham. How much is this in pounds?
You use a conversion factor to change the units.

What is a conversion factor?
A conversion factor is a multiplying fraction that is alway equal to one.
You use the conversion factor that relates grams and pounds.
You may know that

1 kg = 2.2 lb

(Its actually 2.204 62 kg, but we can use the approximate number here.)
If you divide both sides of the equation by 2.2 lb, we get

1 kg/2.2 lb = 2.2 lb/2.2 lb =1

If you divide both sides by 1 kg, we get

1 kg/1 kg = 2.2 lb/1 kg = 1

Thus, the conversion factor can be either 1 kg/2.2 lb or 2.2 lb/1 kg because each equal to one.

How do I use a conversion factor for unit conversions?
Your problem is

1.7 kg = ? lb

Thus, the conversion factor is either

color(blue)(1 kg/2.2 lb) or color(blue)(2.2 lb/1 kg)

You use the one with the desired unit (lb) on top.
For this problem, you would write

1.7 color(red)(cancel(color(black)(kg))) × color(blue)(2.2 lb)/(color(blue)(1) color(red)(cancel(color(blue)(kg)))) = 3.7 lb

Notice that putting kg on the bottom makes the units cancel and gives an answer with the units of lb.

If you had used the other conversion factor, you would have gotten
1.7 kg × color(blue)(1 lb/2.2 kg) = 0.77 lb^2kg^-1
The units make no sense, so this choice is wrong.

Heres another way to think of using unit conversions.

150 is the mirror image of 30 across the x-axis if that makes sense…
Explanation:

If you look at the unit circle you can see that both angles are rotated by 60 from 90 (the x-axis) but in opposite directions.
30=90−60
150=90+60
This means that
sin(30))=sin(150)
and
cos(30)=−cos(150)