How to hand sketch the parametric curve $y=\sin3t$ and $x=\cos t$?

Question

I tried to eliminate the variable $t$ and I've got $y = \sin(t)(4x^2-1)$ I know what it looks like by plotting with computer. but how do i do it by hand?

Answer 1

In fact you can go further to obtain $y=\pm \sqrt{1-x^2}(4x^2 - 1)$ noting $\sin x = \pm \sqrt{1-\cos^2 x}$. If you are patient to get $y'$ and $y''$ then you can sketch the graph by monotonicity and convexity of $y$.