Question

Find the cylindrical coordinates of the points where the Cartesian coordinates are

i.(6,6,8)

ii.(√2,1,1)

Answer 1

Solution (i):

Given, $x=6,\ y=6, z=8$

In rectangular coordinates,

$x=r\cos\theta,\ y=r\sin\theta,\ z=z$

So, $6=r\cos\theta,\ 6=r\sin\theta,\ z=8$

Also, $r^2=x^2+y^2=6^2+6^2=36+36=72$

$\Rightarrow r=\sqrt{72}=6\sqrt2$ (Taking positive value only)

And, $\tan\theta=\dfrac yx=\dfrac 66=1$

$\Rightarrow \theta=\tan^{-1}1=\dfrac{\pi}4$

So, cylindrical coordinates are $(r,\theta,z)=(6\sqrt2,\dfrac {\pi}4,8)$

Solution (ii):

Given, $x=\sqrt2,\ y=1, z=1$

In rectangular coordinates,

$x=r\cos\theta,\ y=r\sin\theta,\ z=z$

So, $\sqrt2=r\cos\theta,\ 1=r\sin\theta,\ z=1$

Also, $r^2=x^2+y^2=(\sqrt2)^2+1^2=2+1=3$

$\Rightarrow r=\sqrt3$ (Taking positive value only)

And, $\tan\theta=\dfrac yx=\dfrac 1{\sqrt2}$

$\Rightarrow \theta=\tan^{-1}(\dfrac 1{\sqrt2})=0.6154$ rad

So, cylindrical coordinates are $(r,\theta,z)=(\sqrt3,0.6154,1)$