Coordinate systems/Derivation of formulas/Cartesian from cylindrical unit vectors

From Wolfram Mathworld, we see the following relations:

$\begin{array}{cc}\hat{\mathit{\rho }}& =\cos \varphi \widehat{𝐱}+\sin \varphi \widehat{𝐲}\\ \hat{\mathit{\varphi }}& =-\sin \varphi \widehat{𝐱}+\cos \varphi \widehat{𝐲}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$

Hence, by making $\widehat{𝐱}$ and $\widehat{𝐲}$ the subjects, we get:

$\begin{array}{cc}\widehat{𝐱}& =\cos \varphi \hat{\mathit{\rho }}-\sin \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐲}& =\sin \varphi \hat{\mathit{\rho }}+\cos \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$

Which was what we wanted to show.