Why consider only direction cosines? Why are these called direction angles? 


*

*Why do we consider only direction cosines and not direction sines or tans.

*What is its actual significance? 

*And How to use them?
Why are they called direction angles?
 A: Because direction cosines are, unlike sines and tans, even functions of the angle which makes the sign of the angle irrelevant and that's a good thing. 
More importantly, the direction cosines of a unit vector $\vec v$ end up being the coordinates $v_x,v_y,v_z$, respectively, so the direction cosines obey 
$$\cos^2 a+\cos^2 b+\cos^2 c = 1$$
which is nice. At the end, the special feature of the cosine is that it appears in the inner product of two vectors:
$$ |\vec u\cdot \vec v| = |u|\cdot |v| \cdot \cos\varphi$$
One may talk about "direction sines" or "tans" as well but they're calculable by more complicated formulae and/or obey other, more complicated identities, so they're less useful for direction calculations of interesting quantities. Sines and tans are not really banned; they are just not equally useful or natural.
A: We are going to describe the direction of a line through the origin.  I need to tell you how, starting at the origin, you walk so many steps parallel to the x-axis, then so many steps parallel to the y-axis, then so many steps parallel to the z-axis, to arrive at a point on the line distant exactly one from the origin.  If I stood at that point and dropped a perpendicular on each of the axes in turn, the perpendicular would hit the axis at the point with coordinates (cosΘx,0,0), (0,cosΘy,0),(0,0,cosΘz).  I need to say that Θx is the angle between the line and the x-axis, and similarly for the other angles.
If you need to walk on a line which doesn't go through the origin, then the same exercise is applied by walking from any point on the line, say (a,b,c), and the answers change to (a+cosΘx,b,c),(a,b+cosΘy,c),(a,b,c+cosΘz) which is seen by drawing a set of axes through (a,b,c) rather than the origin.
