Question 852838
find the rectangle coordinates of the polar point (r,{{{theta}}}) = (-5,{{{5pi/2}}})
<pre>
This one is better done graphically than algebraically:

To plot the polar point (r,{{{theta}}}), start at the rectangular point
(0,r). Think of a radius drawn from the origin to that point on the x-
axis.  That's the green line on the graph below connecting the origin 
to the rectangular point (x,y) = (-5,0)

That is, we start with a radius r = x = -5 on the x-axis at the rectangular 
point (-5,0) like this:

{{{drawing(400,400,-7,7,-7,7, graph(400,400,-7,7,-7,7),
green(
circle(-5,0,0.15),circle(-5,0,0.13),circle(-5,0,0.11),circle(-5,0,0.09),circle(-5,0,0.07),circle(-5,0,0.05),circle(-5,0,0.03),circle(-5,0,0.01),
line(0,0,-5,0))  )}}}

{{{5pi/2}}} is {{{5*expr(pi/2)}}} which means it is 5 right angles, 
or 5 90-degree angles.

So we swing the green line counter-clockwise through the 
angle {{{5pi/2}}} (five 90° angles, or {{{1&1/4}}} revolutions,
indicated by the curved red line below:

{{{drawing(400,400,-7,7,-7,7, graph(400,400,-7,7,-7,7),

red(arc(0,0,3,-3.25,180,270),
arc(0,0,3.5,-3.25,270,360),
arc(0,0,3.5,-3.75,0,90),
arc(0,0,4,-3.75,90,180),
arc(0,0,4,-4.35,180,270)), 


green(line(0,0,0,-5),
circle(0,-5,0.15),circle(0,-5,0.13),circle(0,-5,0.11),circle(0,-5,0.09),circle(0,-5,0.07),circle(0,-5,0.05),circle(0,-5,0.03),circle(0,-5,0.01))  )}}}

So the green radius ended up with its tip at the rectangular point (0,-5).

So the polar point (r,{{{theta}}}) = (-5,{{{5pi/2}}}) is the exact same 
point as the rectangular point (x,y) = (0,-5).

Edwin</pre>