By righttriangle trigonometry and the
Pythagorean theorem: $$ \sin \theta \ = \
\frac{y}{r}, \quad \cos \theta \ = \ \frac{x}{r}\,, \quad r^2 \ = \
x^2 + y^2\,, \quad \tan \theta \ = \ \frac{y}{x}\,.$$ Thus for
a point $P$ the Cartesian and polar coordinate systems are related
by $$ (x,\, y) \ = \ (r \cos \theta,\, r \sin
\theta)\,, \quad (r,\, \theta) \ = \ \left(\sqrt{x^2+y^2},\
\tan^{1}\Bigl( \frac{y}{x}\Bigr)\right)\,.$$ 
