SOLUTION: Find the polar coordinates of the point whose rectangular coordinates are (4sqrt 3, -4) Find the rectangular coordinates of the point whose polar coordinates are (-4, π/6).

We will indicate the rectangular coordinates of point P in 
black as  and the polar coordinates of point P
in red as 

Plot P using its rectangular coordinates  = (:



The x-coordinate of point P is  and the y-coordinate
of point P is -4.

Draw a right triangle with this point P and the origin as
vertices and the right angle on the x-axis.  The legs of this right
triangle x and y are the RECTANGULAR coordinates of point P. The 
hypotenuse r of this right triangle is the first POLAR coordinate 
of P.  The angle  indicated by the counter-clockwise red 
arc is the second POLAR coordinate of the point  



We only need to calculate  and 










Therefore  in the 4th quadrant is , and

 = 

Let's draw the point  using its polar coordinates.

First we draw the angle  with a dotted line through
the origin:



Next we locate the value of r on the x-axis, then we swing an
arc from that point on the x-axis around to the dotted line,
like the green arc below swinging from -4 on the x-axis to
the dotted line, and mar that point .



Then we erase the green arc and draw a right triangle with 
this point P and the origin as vertices and the right angle 
on the x-axis, and indicate the RECTANGULAR coordinates x,
y, of the point P.  Since we swung the point from the point
x=-4, the hypotenuse .

  

Now we calculate x and y.













So  = 

Always substitute trig functions first, 

, , 

and always wait until last to replace  by 








You can then put it in standard form of a circle,



getting:



whose graph is a circle with center (h,k) = (,0) 

and radius : 



Edwin