If #(color(red)(x),color(blue)(y)) = (color(red)(-7sqrt(3)/2),color(blue)(7/2))#
Then for an angle #theta# between the positive X-axis and the point #(color(red)(x),co9lor(blue)(y))# with vertex at the origin,
by definition of #tan#
#color(white)("XXXX")tan(theta) = (color(red)(-7sqrt(3)/2))/(color(blue)(7/2))#
#color(white)("XXXXXXXX")=-sqrt(3)#
This value for #tan(theta)# is one found in the standard #pi/3=6O^@# triangle
and tells us that the angle is either #-pi/3# or #2pi/3#
# (color(red)(-7sqrt(3)/2),color(blue)(7/2))# is in Quadrant IV, so #theta = -pi/3#
The radius for the polar coordinate is given by the Pythagorean Theorem as
#color(white)("XXXX")r=sqrt((color(red)(-7sqrt(3)/2))^2+(color(blue)(7/2))^2)#
#color(white)("XXXXX")=sqrt((color(red)(3*7^2)+color(blue)(7^2))/4#
#color(white)("XXXXX")=sqrt((cancel(4)*7^2)/cancel(4))#
#color(white)("XXXXX")=7#
The Polar coordinate #(theta,r) = (-pi/3,7)#
