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Answer 1 · 2016-11-29T02:44:56

#(r, theta) = (13, arctan(-12/5)) ~~ (13, -67.380^@)#

There are two main sets of equations when converting between rectangular and polar coordinates. To go from polar to rectangular, we have

#{(x = rcos(theta)), (y = rsin(theta)):}#

and to go from rectangular to polar, we have

#{(r^2=x^2+y^2), (tan(theta) = y/x):}#

Using the second pair of equations with #x=5# and #y=-12#, we have

#r^2 = 5^2+(-12)^2 = 169#

#=> r = sqrt(169) = 13#

and

#tan(theta) = -12/5#

#=> theta = arctan(-12/5) ~~ -67.380^@#

Thus, the polar coordinates for #(x, y) = (5, -12)# are #(r, theta) = (13, arctan(-12/5)) ~~ (13, -67.380^@)#