How do you convert $r=2sin3(theta)$ to rectangular form?

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Answer 1 · 2016-04-29T14:46:41

$x^2+y^2=4 sin^2(tan^(-1)(y/x))$

Here, $|r|<=2, r=+-sqrt(x^2+y^2) and theta=tan^(-1)(y/x)$

So, the rectangular form is $+-sqrt(x^2+y^2)=2 sin (3 tan^(-1)(y/x))$

Remove the ambiguity in sign for r, by squaring both sides.

It is also possible to have a form, sans trigonometric functions, by

expanding $sin 3 theta$ in powers of $sin theta and cos theta$ and

substituting $sin theta =y/r and cos theta=x/r$ However, the form

given as answer appears to be elegant... .

How do you convert r=2sin3(theta)r=2sin3(theta) to rectangular form?