How do you convert the Cartesian coordinates (-7,6) to polar coordinates?

1 Answer
Alan P.
Oct 22, 2015

#(x,y)=(-7,6) iff (r,theta)~=(9.22,2.43)#

Explanation:

#(-7,6)# is a point in Quadrant II
and if #theta# is the angle between the positive X-axis and #(-7,6)# with vertex at the origin:
#color(white)("XXX")tan(theta)=-6/7#

Using a claculator
#color(white)("XXX")hattheta=arctan(-6/7)#
gives us
#color(white)("XXX")hattheta = -0.70863#

#hattheta# is the same reference angle as the required #theta# but since #arctan(_)# is restricted to a range of #(-pi/2,pi/2]#
#hattheta# is in Quadrant IV

Therefore
#color(white)("XXX")theta = pi+arctan(-6/7) ~= 2.43#

#r# (the radius) can be calculated using the Pythagorean Theorem as
#color(white)("XXX")r=sqrt((-7)^2+6^2) ~= 9.22#

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