What is the Cartesian form of #(36,pi)#?

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Ken C.
Jun 28, 2016

#(-36,0)#

Explanation:

The formulas used to convert between polar and Cartesian co-ordinates are:
#x=rcostheta#
#y=rsintheta#

We are given the polar co-ordinate point #(r,theta)->(36,pi)#, so #r=36# and #theta=pi#. Making substitutions in the above formulas:
#x=36cos(pi)=36*(-1)=-36#
#y=36sin(pi)=36(0)=0#

Therefore, the point in the Cartesian plane is #(-36,0)#.

Intuitively, this result makes sense because #pi# is a half-circle rotation, so in the Cartesian plane an angle of #pi# corresponds to a point on the #x#-axis (and therefore #y=0#). A radius of #36# means the point is #36# units to the left or right of the origin, and since an angle of #+pi# is a counterclockwise rotation, it would mean the point is #36# units to the left (and therefore #-36#).

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