What is the Cartesian form of #(12,(5pi )/3)#?

Answer 1

#(6,-6sqrt(3))#

Remember that polar coordinates are of the form #(r, theta)#.
Also, remember that our #x#-values correspond with cosine and #y#-values with sine.
Then, remember that our sine and cosine values come from the unit circle, where #r=1#. So, when changing our coordinates from polar to cartesian coordinates we are taking #(r, theta) -> (rcos(theta), rsin(theta))#.
Notice that #(5pi)/3=2pi-pi/3#. So, we can say that #theta=-pi/3#, which is in the fourth quadrant. This means that cosine is positive, and sine is negative.
Then we can essentially say that #(12, (5pi)/3) -> (12cos(pi/3),-12sin(pi/3))# #=(12(1/2),-12((sqrt(3))/2))=(6,-6sqrt(3))#
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Answer 2

The Cartesian form of the point ( (12, \frac{5\pi}{3}) ) is obtained by converting from polar coordinates to Cartesian coordinates. To do this, use the following conversions:

[ x = r \cos(\theta) ] [ y = r \sin(\theta) ]

Given the polar coordinates ( (r, \theta) = (12, \frac{5\pi}{3}) ), substitute these values into the formulas to find the Cartesian coordinates:

[ x = 12 \cos\left(\frac{5\pi}{3}\right) ] [ y = 12 \sin\left(\frac{5\pi}{3}\right) ]

Evaluate the trigonometric functions:

[ x = 12 \cos\left(\frac{5\pi}{3}\right) = 12 \cdot \left(-\frac{1}{2}\right) = -6 ] [ y = 12 \sin\left(\frac{5\pi}{3}\right) = 12 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -6\sqrt{3} ]

So, the Cartesian form of ( (12, \frac{5\pi}{3}) ) is ( (-6, -6\sqrt{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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