If $\cos(-\theta) = \cos \theta$ then why is the value of $\cos(-\theta) $ negative when $\theta \lt -90^\circ$? I'm learning Trigonometry right now with myself and currently learning about trigonometric functions. I'm a little bit confused right now. 
If  $\cos(-\theta) = \cos \theta$, then why does $\cos \theta$ have  negative values when $\theta$ is less than $-90^\circ$? 
 A: Graph the function and look at where $\theta \lt -90^\circ$, then you will soon see why it is negative. I have split the graph up into 4 regions corresponding to the 4 quadrants of the unit circle.
To the left of the $\color{blue}{\mathrm{blue}}$ line $\cos\theta$ is below the $x$-axis ($\theta$ axis) so it is negative until $\theta\lt -270^\circ$, past  the $\color{#0f5}{\mathrm{green}}$ line where $\cos\theta$ becomes positive again. To the left of the green line corresponds to the fourth quadrant. 

Further to our conversation, yes the fourth quadrant will be positive for $\cos\theta$ as shown below:

A: One of the ways of defining the trigonometric functions is geometrically.  
See Wikipedia.

Taking the Unit Circle (the circle of radius $1$ with center the origin, describable by the equation $x^2+y^2=1$), given a particular angle $\theta$, the value of cosine of $\theta$ is the $x$ coordinate for where the ray with angle $\theta$ above the positive $x$-axis intersects the circle.
Notice that for $90^\circ<\theta<180^\circ$ you will be in the top left quadrant, and in particular will have negative $x$ coordinates.  Similarly for $-180^\circ<\theta<-90^\circ$ it will lie in the bottom left quadrant.  In particular for something like $-100^\circ$.
(there are more complicated definitions which fix some inconsistencies with this definition as well as extend cosine to more abstract settings, but for now this definition is good enough)
A: If $\cos(-\theta) = \cos \theta$ then why is the value of $\cos(-\theta) $ negative when $\theta \lt -90^\circ$?
Because the above is not an absolute rule but is relative to $ \theta$.
Note  $\cos(\theta)$ itself is negative when $\theta $ radius vector tip ends up in second or third quadrants i.e., on negative side of x-axis.
