  In this paper, we describe the entire structure of the vector space $Sym_2^0$
of all symmetric matrices of size $2$ having trace zero. This is motivated by
the geometrical interpretation of any arbitrary element of $Sym_2^0$. We
further study the orbits and stable sets of these elements. As an application
of the obtained structure of $Sym_2^0$, we obtain the symmetric matrices of
size $2$, trace of whose product with any trace zero symmetric matrix is zero.
Finally some well known trigonometric formulas are interpreted geometrically
incorporating the anatomy of $Sym_2^0$.
