How do you find the exact value of cos 180degrees - sin 180degrees?

You can look at a unit circle for a reference. However, if you do not have one, I can tell you what to do.

As you may know, the unit circle first begins with ##0 °.## In ##0 °##, the point is ##(1,0).##

A little trick I have found is that, as you know it begins with ##(1,0).## As you go to the other side, on left where ##π## is located, which is also 180°, the 1 changes to become negative whereas the zero doesn't change. So the point in ##180##° is ##(-1, 0##). This also applies to ##π/2## and ##3π/2##, as in ##π/2## the point is ##(0, 1##) and in ##3π/2## the point is ##(0, -1).##

I have just given you the coordinate for 180°; ##(-1, 0 )##.

Now, cosine is the number on the left of the point and sine is the number on the right.

For example, in the coordinate ##(sqrt3/2 , 1/2)##, cosine is ##sqrt3/2## and sine is ##1/2##. (this is in 30° or ##π/6##)

Therefore, in 180°, cosine is ##-1## and sine is ##0##.