How do you convert the Cartesian coordinates (2√3, 2) to polar coordinates?

1 Answer
Aldrich
Sep 5, 2015

#(4,pi/6)#.

Explanation:

Let #(x,y)# be a coordinate on the Cartesian plane.

The corresponding polar coordinate is #(r,theta)#, where:

# r = sqrt(x^2 + y^2) #

(You might notice that this is similar to the distance formula; that's not a coincidence, #r# is the distance from the point to the pole (a.k.a. the center) )

and:

#theta = tan^-1(y/x)#

So, given #(2sqrt(3),2)#:

#r = sqrt((2sqrt(3))^2 + 2^2)#

#r = sqrt(12+4)#

#r = sqrt(16)#

#r = 4#

#theta = tan^-1(2/(2sqrt(3)))#

#theta = tan^-1(1/sqrt(3))#

#theta = tan^-1(sqrt(3)/3)#

#theta = pi/6#

We can say that #theta = pi/6# (and not #(5pi)/6# , etc.) because #x# and #y# are both positive, which means the point is in the first quadrant.

Thus, the point in polar coordinates is #(4,pi/6)#.

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