Find the polar coordinates of the point whose rectangular coordinates are
(, -4)
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
=
Find the rectangular coordinates of the point whose polar coordinates are
.
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 =
Find a rectangular form of the equation
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