[Math] Converting cardioid from parametric form into Cartesian form

Question

I'm trying to find the cartesian equation of the curve defined by the parametric equations:

$$x=2\cos(t) – \cos(2t), \qquad y=2\sin(t) – \sin(2t)$$

I feel stumped. How can I go about this?

Answer 1

Another way:

we have $x=2\cos t-\cos2t\ \ \ \ (1)$

$\implies x=2\cos t-(2\cos^2t-1)\iff x-1=2\cos t(1-\cos t)$

and $y=2\sin t-\sin2t\ \ \ \ (2)$

$\implies y=2\sin t -2\sin t\cos t=2\sin t(1-\cos t)$

On division, $\displaystyle\frac yx=\tan t$

$\displaystyle\implies \cos2t=\frac{1-\tan^2t}{1+\tan^2t}=\frac{x^2-y^2}{x^2+y^2}$

and $\displaystyle\cos t=\frac1{\sec t}=\pm\frac1{\sqrt{1+\tan^2t}}=\frac{\pm x}{\sqrt{x^2+y^2}}$

Put these values in $(1)$ and square to remove $\pm$