Question

Transform to Cartesian coordinates the equation:

$r(\cos 3\theta +\sin 3\theta )=5k\sin \theta \cos \theta$

Answer 1

Given equation

$r(\cos 3\theta +\sin 3\theta )=5k\cos \theta \sin \theta$ ..... $\left(i\right)$

We know, $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$ and $\cos 3\theta =-3\cos \theta +4{\cos }^3\theta$

Using above results in equation $\left(i\right)$

$\Rightarrow r\left[3\right(-\cos \theta +\sin \theta )+4({\cos }^3\theta -{\sin }^3\theta \left)\right]=5k\cos \theta \sin \theta$

But $x=r\cos \theta$ and $y=r\sin \theta$

Substituting in above eqn.

$\Rightarrow r[3(\frac{y}{r}-\frac{x}{r})+4({\left(\frac{x}{r}\right)}^3-{\left(\frac{y}{r}\right)}^3)]=5k\times \frac{x}{r}\times \frac{y}{r}$

$\Rightarrow 3r^2(y-x)+4(x^3-y^3)=5kxy$

$\Rightarrow 3(x^2+y^2)(y-x)+4(x^3-y^3)=5kxy$

$\Rightarrow x^3-3x{y}^2+3x^{2}y-y^3=5kxy$