How do I convert Cartesian coordinates to polar coordinates?

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How do I convert Cartesian coordinates to polar coordinates?

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Answer 1 · 2014-08-19T21:12:11

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Let's look at the trig formulas SYR, CXR, TYX:

$sin θ = \frac{y}{r}$ $sin theta = y/r$
$cos θ = \frac{x}{r}$ $cos theta = x/r$
$tan θ = \frac{y}{x}$ $tan theta = y/x$

Since we are given the Cartesian coordinates, we are given $x$ $x$ and $y$ $y$ . For polar coordinates, we need to figure out $r$ $r$ and $θ$ $theta$ . $r$ $r$ is easy, we just use Pythagorean:

$r = \sqrt{x^{2} + y^{2}}$ $r=sqrt(x^2+y^2)$

To figure out $θ$ $theta$ , I like to use cosine because the range of arccosine is in quadrants I and II and adjusting $theta'$ is easier. So,

$theta'=cos^(-1)x/r$

If $y>=0$ then $theta=theta'$ .
If $y<0$ then $theta=2 pi - theta'$ (in radians) or $theta=360-theta'$ (in degrees).

Our final answer is $(r, theta)$ .

Let's look at a concrete example: Convert $(-3, 3sqrt3)$ to polar coordinates:

$r=sqrt((-3)^2+(3sqrt3)^2)=sqrt(36)=6$
$theta'=cos^(-1)((-3)/6)=(2pi)/3$
$y<0$ so, $theta=2pi-(2pi)/3=(4pi)/3$

So the polar coordinates are $(6, (4pi)/3)$ .

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How do I convert Cartesian coordinates to polar coordinates?

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