What is the Cartesian form of #( -1, (4pi)/3 ) #?

2 Answers
Jake M.
Jul 18, 2018

#(1/2, sqrt(3)/2) #

Explanation:

We are given the polar form, so there is a radius and an angle. We want to convert to #x,y# coordinates.

So we can use Euler's formula (or at least the idea behind it) to convert between Cartesian and polar:

#x = r costheta #
#y = r sintheta #

From that, we just plug in the numbers, remembering our unit circle:

#cos((4pi)/3) = -1/2 and sin((4pi)/3) = - sqrt(3)/2#
therefore
#(x,y) = (1/2, sqrt(3)/2) #
You could also notice that a negative radius is the same as adding or subtracting #pi# to the angle, hence
#(-1, (4pi)/3) = (1, pi/3)#
which I think is a bit easier to think about.

Harish Chandra Rajpoot
Jul 18, 2018

#(1/2, \sqrt3/2)#

Explanation:

The Cartesian coordinates #(x, y)# of the point #(-1, {4\pi}/3)\equiv(r, \theta)# are given as follows

#x=r\cos\theta#

#=-1\cos({4\pi}/3)#

#=-\cos(\pi+\pi/3)#

#=\cos(\pi/3)#

#=1/2#

#y=r\sin\theta#

#=-1\sin({4\pi}/3)#

#=-\sin(\pi+\pi/3)#

#=\sin(\pi/3)#

#=\sqrt3/2#

hence, the Cartesian coordinates are #(1/2, \sqrt3/2)#

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