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

Answer 1

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

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.
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Answer 2

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

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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Answer 3

The Cartesian form of (-1, (4π)/3) is (-1, -√3).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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