Versors (Vectors) and Trigonometry I recently remebered, when I asked my physics high school teacher if unit vectors are somehow related to sine and cosines (or trigonometry in general). She replied to me that I was pretty lost and confuse if I was asking that...
The thing is, even when she told me that, I didn't change my opinion, and continue thinking about it. Let me explain myself:
2D Vectors:
If you have a vector (in 2D, for the moment):
$$\vec{v} = x_v \hat{i} + y_v \hat{j}$$
You can represent it in the plane:

For example, this is the vector given above
$$\vec{v} = \hat{i} + 2 \hat{j}$$
And if you normalize the vector, you simply divide by the lenght of the vector itself.
This is when I had an idea (back then). I thought this is pretty similar to the $\sin$ and $\cos$ definition:
$$
\cos{x} = \frac{adjacent}{hypotenuse}
$$
$$
\sin{x} = \frac{opposite}{hypotenuse}
$$
But instead, I'm doing:
$$
\frac{v}{|v|} = \frac{\hat{i} + 2\hat{j}}{\sqrt{1^2 + 2^2}} = \frac{1}{\sqrt{5}}\hat{i} + \frac{2}{\sqrt{5}}\hat{j}
$$
The $x$ component, is now similar to the cosine, beacuase I have divided the adjacent side, between the magnitude of the vector (that is like the hypotenuse). Same happens for the $y$.
3D Vectors:
So now, if you instead have a three-dimensional vector, you could also find the versor by doing the same thing:
$$
\hat{v} = \frac{x_v \hat{i} + y_v \hat{j} + z_v \hat{k}}{|v|}
$$
This was the concrete case I asked to my teacher, so after her ansewer, I still thinking about it, and maybe is not related with $sin$ and $cos$, but maybe to spherical coordinates or something similar (because the final unitary vector, will always lie on the spehere of radius 1).
So, this versors, have a relation with sines and cosines, or not?

Given any vector $$\underline{v}=a\underline{i}+b\underline{j}+c\underline{k},$$
The unit vector is $$\underline{\hat{v}}=\frac{a\underline{i}+b\underline{j}+c\underline{k}}{\sqrt{a^2+b^2+c^2}}$$
The coefficients $\frac{a}{\sqrt{a^2+b^2+c^2}}$ etc are sometimes referred to as direction cosines because these are the cosines of the angles that $\underline{v}$ makes with the coordinate axes.
I hope this helps.
