How do I convert the Polar Equation $r=\sin(k \theta)$ to Cartesian Equation?
I understand that $r^2=x^2+y^2$ and that $x=r\cos\theta$ and $y=r\sin\theta$, but no matter how I try to arrange them it seems that I can never cancel out both r and $\theta$.
I've looked at Writing a Polar Equation for the Graph of an Implicit Cartesian Equation and several mathematics sites on the internet and some videos by PatrickJMT on this, but my knowledge of trigonometry is limited, and I haven't been able to find any sort of a way to get a general cartesian equation for a polar rose.
Also, the Parametric equations for polar rose such that $r=\cos(k\theta)$ are $x=\cos(k t)\sin t$ and $y=\cos(k t)\cos t$.
Best Answer
Each one may be solved individually: Polar to cartesian form of $ r = \sin(2\theta)$
$$r = \sin(2\theta) = 2\sin\theta\cdot \cos\theta$$ $$r^3 = 2(r\sin\theta)(r\cos\theta)$$
$$x = r\cos\theta$$ $$y = r\sin\theta$$
$$r^3 =2xy$$
$$r = (x^2 + y^2)^{\frac 12}$$
$$(x^2+y^2)^{\frac 32} =2xy$$ $$(x^2+y^2)^3=4x^2y^2$$
See Also: Polar to cartesian form of r=cos(2θ)