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

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

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Answer 1 · 2015-09-05T18:33:31

$(4,pi/6)$ .

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

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