Minoanman,
I haven't a clue as to what curve you have drawn. Could you be more explicit by giving parametric equations or an implicit equation for the curve? In what sense does your curve represent
$\displaystyle y=1-2\text{cos}(3\theta+\pi)$?
Minoan,
I'll try to make my question perfectly clear. Presumably, you used some software to generate the image in your first response. What was the software and exactly what did you type to get the curve?
Neither Geogebra nor Graph will accept the given equation: y=1-2cos(t+pi) (or theta for t). Wolfram Alpha interprets the given equation as plotting the function y=f(t) with parameter t; that is, the first graph of your second response.