Equation of cardioid Can someone please explain to how
$$c = \frac{1}{2} e^{i \theta}- \frac{1}{4} e^{2 i \theta}, \quad 0 \leq \theta \leq 2 \pi$$
does represent an equation of cardioid  $r =  \frac{1}{2}  -  \frac{1}{4} \cos(\theta)$, $0\leq \theta \leq 2 \pi$?
I tried to write  $\cos(\theta) = \frac{e^{i \theta} + e^{-i \theta}}{2}$, but couldn't prove it is equal to $\frac{1}{2} e^{i \theta}- \frac{1}{4} e^{2 i \theta}$. Thanks in advance.

Plot the two functions. You will then note that:

*

*The formula for the radius is wrong. In a cardioid the radius is zero at $\theta=0$. Try $$r=\frac12(1-\cos\theta)$$

*The complex representation is a carioid shifted along the real axis by $1/4$
$$x_c=\frac12\cos\theta-\frac14\cos2\theta=\frac12\cos\theta-\frac12\cos^2\theta+\frac14=\frac12(1-\cos\theta)\cos\theta+\frac14\\y_c=\frac12\sin\theta-\frac14\sin2\theta=\frac12\sin\theta-\frac12\sin\theta\cos\theta=\frac12(1-\cos\theta)\sin\theta
$$
