How to show that the deltoid is a plane algebraic curve of degree $4$? 
How can I show that a deltoid is a plane algebraic curve of degree 4?

I have searched that the parametric equation for deltoid is given by
$$\begin{align}
x&= 2 \cos t + \cos 2t \\
y&=2 \sin t - \sin 2t
\end{align}$$
So, by using some trigonometric properties, we can write
$$\begin{align}
x = 2 \cos t + 2 \cos^2 t - 1  &\quad\to\quad x^2 = \phantom{-}4 \cos^4t + 8\cos^3 t - 4 \cos t + 1 \\
y =2 \sin t -2\sin t \cos t\;\, &\quad\to\quad y^2=-4 \cos^4t+ 8 \cos^3t -8\cos t+4 
\end{align}$$

In M2 setting $c=\cos(t), s=\sin(t)$ using $c^2+s^2-1=0$:
R=QQ[c,s,x,y,MonomialOrder=>Lex]
I=ideal(c^2+s^2-1,x-(2*c+c^2-s^2),y-(2*s-2*s*c))
gens gb I

yields $x^4-8x^3+2x^2y^2+18x^2+24xy^2+y^4+18y^2-27$ or as wikipedia puts it (for $b=3, a=1$): $$(x^2+y^2)^2+18(x^2+y^2)-27=8(x^3-3xy^2).$$
