Question

I want to test for symmetry when theta=pi-theta...........r=4cos 2theta

Answer 1

$\bf \textit{symmetry test with respect to the }\theta =\frac{\pi }{2}~line \\\\\\ r=4cos(2\theta )\implies r=4cos[2(\pi -\theta )]\implies r=4cos(2\pi -2\theta )\\\\ -------------------------------$

$\bf cos(2\pi -2\theta )\implies \stackrel{1}{cos(2\pi )}cos(2\theta )+\stackrel{0}{sin(2\pi )}\stackrel{}{sin(2\theta ) } \\\\\\ 1\cdot cos(2\theta )+0\cdot sin(2\theta )\implies cos(2\theta )+0\implies cos(2\theta )\\\\ -------------------------------\\\\ r=4cos(2\theta )\impliedby \textit{is symmetric with respect to }\theta =\frac{\pi }{2}~line$