Question 1205368
 the polar equation {{{r = 3+2sin(theta)}}}


the {{{x}}} -axis  -  called the {{{polar}}} axis
the line {{{theta=pi/2}}} ({{{y}}}-axis)
the pole is origin 


 test for symmetry about polar-axis: replace {{{theta}}} with {{{-theta}}} and see is {{{r }}}unchanged 

{{{r = 3+2sin(-theta)}}}....since {{{sin(-theta)=-sin(theta)}}}, we have

{{{r = 3-2sin(theta)}}} =>not same as what we started with, {{{r}}} is changed; so, no symmetry about {{{x}}}-axis or  the polar axis


test the line {{{theta=pi/2}}} ( {{{y}}}-axis): replace ({{{theta}}}) with ({{{pi-theta}}}) and see is {{{r}}} unchanged

{{{r = 3+2sin(pi-theta)}}}...since {{{sin(pi-theta)=sin(theta)}}}

{{{r = 3+2sin(theta)}}}=> same as what we started with, {{{r}}} is {{{not }}}{{{changed}}}; so, there is symmetry about {{{y}}}-axis or  the the line {{{theta=pi/2}}}


test the pole is origin ({{{180}}}°  rotation), replace {{{r}}} with {{{-r}}} and see is {{{r}}} unchanged

.{{{-r = 3+2sin(theta)}}} = >  obviously {{{r}}} is changed, so no symmetry abut pole (origin)


graph it:

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