Question

To convert from cylindrical to rectangular coordinates:

$x=\_\_\_\_,y=\_\_\_\_,z=\_\_\_\_$

Short answer

$$\begin{gathered} x=\rho \cos \theta \\ y=\rho \sin \theta \\ z=z \end{gathered}$$

Step-by-step solution

Step 1:Given information

Given$\left(x,y,z\right)$are rectangular coordinates and$\left(\rho ,\theta ,z\right)$are cylindrical coordinates

Step 2:Simplification

$\left(\rho ,\theta ,z\right)$are cylindrical coordinates and $\left(x,y,z\right)$are rectangular coordinates

The relation is as below:

From the figure

$\cos \theta =\frac{x}{\rho }$

role="math" localid="1653036479004" $\Rightarrow x=\rho \cos \theta$

Similarly

$\sin \theta =\frac{y}{\rho }$

$\Rightarrow y=\rho \sin \theta$

And from the figure we can observe$z=z$