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?
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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.
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.