Question

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

Answer 1

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

Explanation:

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... .