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

From Wolfram Mathworld, we see the following relations:

$$\begin{align} \hat{\boldsymbol\rho} &= \cos\phi\hat{\mathbf x}+\sin\phi\hat{\mathbf y} \\ \hat{\boldsymbol\phi}&=-\sin\phi\hat{\mathbf x}+\cos\phi\hat{\mathbf y} \\ \hat{\mathbf z}&=\hat{\mathbf z} \end{align}$$

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

$$\begin{align} \hat{\mathbf x}&=\cos\phi\hat{\boldsymbol\rho}-\sin\phi\hat{\boldsymbol\phi} \\ \hat{\mathbf y}&=\sin\phi\hat{\boldsymbol\rho}+\cos\phi\hat{\boldsymbol\phi} \\ \hat{\mathbf z}&=\hat{\mathbf z} \end{align}$$

Which was what we wanted to show.