When given Cartesian coordinates of the form \(\displaystyle (x,y,z)\) to cylindrical coordinates of the form \(\displaystyle (r,\theta,z)\), the first and third terms are the most straightforward.

\(\displaystyle r=\sqrt{x^2+y^2}\)

\(\displaystyle z=z\)

Care should be taken, however, when calculating \(\displaystyle \theta\). The formula for it is as follows: 

\(\displaystyle \theta=arctan(\frac{y}{x})\)

However, it is important to be mindful of the signs of both \(\displaystyle y\) and \(\displaystyle x\), bearing in mind which quadrant the point lies; this will determine the value of \(\displaystyle \theta\):

It is something to bear in mind when making a calculation using a calculator; negative \(\displaystyle y\) values by convention create a negative \(\displaystyle \theta\), while negative \(\displaystyle x\) values lead to \(\displaystyle |\theta|>90^{\circ};\frac{\pi}{2}\)

For our coordinates \(\displaystyle (2,-6,13)\)

 \(\displaystyle \begin{align*}&r=\sqrt{(2)^2+(-6)^2}=6.32\\&\theta=arctan(\frac{-6}{2})=-71.57^{\circ}\\&z=13\end{align*}\)

When given Cartesian coordinates of the form \(\displaystyle (x,y,z)\) to cylindrical coordinates of the form \(\displaystyle (r,\theta,z)\), the first and third terms are the most straightforward.

\(\displaystyle r=\sqrt{x^2+y^2}\)

\(\displaystyle z=z\)

Care should be taken, however, when calculating \(\displaystyle \theta\). The formula for it is as follows: 

\(\displaystyle \theta=arctan(\frac{y}{x})\)

However, it is important to be mindful of the signs of both \(\displaystyle y\) and \(\displaystyle x\), bearing in mind which quadrant the point lies; this will determine the value of \(\displaystyle \theta\):

It is something to bear in mind when making a calculation using a calculator; negative \(\displaystyle y\) values by convention create a negative \(\displaystyle \theta\), while negative \(\displaystyle x\) values lead to \(\displaystyle |\theta|>90^{\circ};\frac{\pi}{2}\)

For our coordinates \(\displaystyle (-2,-39,30)\)

 \(\displaystyle \begin{align*}&r=\sqrt{(-2)^2+(-39)^2}=39.05\\&\theta=arctan(\frac{-39}{-2})=-92.94^{\circ}\\&z=30\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=75cos(-58^{\circ})=39.74\\&y=75sin(-58^{\circ})=-63.6\\&z=138\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=88cos(91^{\circ})=-1.54\\&y=88sin(91^{\circ})=87.99\\&z=-83\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=39cos(72^{\circ})=12.05\\&y=39sin(72^{\circ})=37.09\\&z=2\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=134cos(17^{\circ})=128.14\\&y=134sin(17^{\circ})=39.18\\&z=138\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=21cos(-88^{\circ})=0.73\\&y=21sin(-88^{\circ})=-20.99\\&z=-106\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=127cos(113^{\circ})=-49.62\\&y=127sin(113^{\circ})=116.9\\&z=-74\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=37cos(-54^{\circ})=21.75\\&y=37sin(-54^{\circ})=-29.93\\&z=129\end{align*}\)

\(\displaystyle \begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=30cos(42^{\circ})=22.29\\&y=30sin(42^{\circ})=20.07\\&z=-75\end{align*}\)