In your opinion how to show symmetry in polar equations without graphing.
i thought of these methods :-
converting to cartesian then test .
check the period of the function .
please help me any other method or details would be help . thanks for advance :) .
$\endgroup$1 Answer
$\begingroup$Let $P$ be a point in the plane with polar coordinates $(r,\theta)$. Let $P_x$ be the reflection of $P$ across the $x$ axis; $P_y$ the reflection of $P$ across the $y$ axis; and $P_O$ be the reflection of $P$ through the origin.
$x$ axis symmetry: $P_x$ has polar coordinates $(r,-\theta)$, and so replacing $\theta$ with $-\theta$ into a polar equation and getting the same equation back again is an $x$-axis symmetry test. Because $P_x$ has many polar representations, other tests are possible. For instance $P_x$ also has polar coordinates $(-r,\pi-\theta)$, which would give another test.
$y$ axis symmetry: Same ideas, but noting $P_y$ has polar coordinates $(r, \pi-\theta)$, or $(-r,-\theta)$.
Origin symmetry: Same ideas. $P_O$ has polar coordinates $(r,\pi+\theta)$ or $(-r,\theta)$.
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