Let’s take a look at this pic below
Now as the angle x goes down and down toward 0, the side on the front of the angle also get shorten and shorten. When x goes negative the side in front of the angle point downward.
The side near the angle remain constant though.
As x goes down to exactly -x we got a congruent triangle except flipped outside down.
So the cosine remain the same, namely b/c. The sine is just the additive inverse. Before it’s a/c, now it’s -a/c.
Hence, we easily see sin(-x)=-sin(x) and cos(-x)= cos(x)
See?
What about sin (180-x)? Well, it’s cos (90°-(180°-x))=cos(x-90°)=cos(-(90°-x))=cos(90°-x)=sin(x).
And what about cos(180-x)? Well, it’s cos(x-180°). By the way, that’s the opposite angle. So it’s equal to -cos(x).
Another way is cos(180°-x)=sin(90°-(180°-x))=sin(x-90°)=sin(-(90°-x))=-sin(90°-x)=-cos(x).
Don’t we all love pencil pushing.
There should be another easier way to see this.