r=16cos3theta
r=6sintheta
1)To test for symmetry with respect to theta= pi/2:
replace theta with pi-theta,
r=16cos3theta
r= 16cos3(pi-theta)= 16cos(3pi-3theta) = -16cos3theta
thus we get r= -16cos3theta,we do not get the same curve.
thus r= 16cos3theta is not symmetric about the line theta= pi/2
similarly let us check for symmetry for the curve r= 6sintheta:
replace theta with pi-theta,
r= 6sin(pi-theta)= 6sintheta, thus we get r= 6sintheta,
therefore the curve is symmetric about the line theta= pi/2
2) To test for symmetry about polar axis:
replace theta with -theta
r= 16cos3(pi-(-theta)) = 16cos(3pi+3theta) =16. (-1)^3cos3theta = -16cos3theta
thus we get r= -16cos3theta, we do not get the same curve,
therefore r= 16cos3theta is not symmetric about the polar axis
replace theta with -theta in r= 6sintheta
we get r= 6sin(-theta) = -6sintheta
r= -6sintheta, we do not get the same curve
therefore r=6sintheta is not symmetric about the polar axis
3) To test for symmetry about the pole:
replace r with -r in both the curves
we get -r= 16cos3theta,
-r = 6sintheta,
we do not get the same curves in both the cases hence they are not symmetric about the pole.