Convertion from and to spherical - cartesian

[M. next]

I googled it, and it says:
$\dot{x}$=$\dot{r}$sinθcos∅ + (rcosθcos∅)$\dot{θ}$ - (rsinθsin∅)$\dot{∅}$
.
.
and so on for $\dot{y}$ & $\dot{z}$

And then they wrote "We will also need the inverse transformation obtained by solving the equations above w.r.t $\dot{r}$, $\dot{θ}$, and $\dot{∅}$

for example they got:
$\dot{r}$=sinθcos∅$\dot{x}$+sinθsin∅$\dot{y}$ + cosθ$\dot{z}$
.
.
and so on, my question how did they deduced the latter from the former equations?

Here's the link if you want to see them clearer:
http://www.physics.sc.edu/~yar/Phys701_2009/homework/hw9_solutions.pdf

Just what's the procedure?
Thanks

 

[tiny-tim]

Hi M. next!

… for example they got:
$\dot{r}$=sinθcos∅$\dot{x}$+sinθsin∅$\dot{y}$ + cosθ$\dot{z}$
…
Just what's the procedure?

It's just algebra (plus standard trigonometric identities) …

for example, the r' component of the RHSs is r'(sin²θcos²∅ + sin²θsin²∅ + cos²θ) = r'

 

[M. next]

Hmm.. I mean how do we get r' from (x', y', z')?

 

[M. next]

I solved three equations with three unknowns and it worked but after so much effort! Thanks anyways Tim.