Question

So I have some polar symmetry questions.

Answer 1

\[3. r=4\cos(2\theta)+1\]
\[Pole symmetry: -r=4\cos(2\theta)+1\]
\[\pi/2 symmetry: -r=4\cos(-2\theta)+1 \to -r=4\cos(2\theta)+1\]
How is this polar equation symmetrical over pi/2 and the pole

Answer 2

\[4.r ^{2}=9\sin(2\theta)\]
\[r=\sqrt{9\sin2\theta}\]
What is the correct way to prove symmetry for this equation? I found that there's no symmetry over pi/2 and the polar axis, but I found there to be symmetry over the pole. That happens to be wrong as well, so can somebody show what steps should be taken?

Answer 3

\[6.r=1+2\sin(\theta/2)\]
\[Polar symmetry:r=1+2\sin(-\theta/2)\]
Can somebody prove how this equation is symmetrical over the polar axis?

Answer 4

3. has polar axis symmetry
it doesn't have symmetry about the line \(\theta = \pi/2\) or about the pole

Answer 5

It doesn't? I plugged in those exact equations on desmos, but they happen to overlap each other perfectly.

Answer 6

true
I think I'm missing something
I just replaced r with -r and theta with -theta to check for symmetry about pi/2

Answer 7

I'm not sure what's going on either, but when I divided -1 on both sides of the equations I plugged into desmos, I ended up getting a different graph

Answer 8

oh, I forgot to check for the case when you replace \((r, \theta)\) with \((r, \pi-\theta)\) to check for symmetry about the line \(\pi/2\)