How can trigonometric functions be negative? I cannot understand why $\cos(180-\theta)$  say is $-\cos\theta$. This is probably because my teacher first introduced trigonometry in triangles. I do not understand it for obtuse angles because I cannot think of them in a right triangle. 
I realised that I couldn't feel what I had read "in my spleen" when I was looking at the proof for the law of cosines in an obtuse-angled triangle. I have spent quite some time thinking about how the "$-\cos\theta$" entered the derivation. I cannot fully understand, why the negatives which work in the $XY$-plane work in triangles. For instance, since in a triangle, all the sides are positive while taking the ratio of sides we do not get any negative values but how then does $\cos 120^{\circ}=-0.5$. My brain is in a mess right now. I would appreciate it if someone could help me out or suggest something that I can do.
Let me illustrate what I can't get around.

It is given that in the triangle $\angle BAC=120$ degrees,$|AC|=3$ and that D is the foot of the perpendicular from C to BD. Then $\cos\angle BAC=-0.5=\dfrac{AD}{AC} \implies AD=-1.5$ ?

Length can be negative.
How is that possible?
Because the negative and positive signs are always relative.
What do I mean by that?
In your setup, you start with A as origin (0,0). Anything on the left-hand side of this is a negative value. If you consider the triangle DCB and ignore rest. Then point D would become (0,0) in that setup and all values to the right-hand side would be positive and all values to the left of D would be negative. All ratios in that DCB triangle will be positive, try out the calculations.
The negative sign essentially denotes the direction. You select a point and assume that all values on one side would be positive and all values on the other side would be negative. It all sounds weird at first, but just practice more and one day it will all start making sense and your mind will actually be blown.
