Question 1191547
Write each equation in rectangular-coordinate form

1. {{{r = 1/2+sin(theta )}}}


In polar coordinates, {{{x = r *cos(theta )}}} and {{{y=r*sin(theta)}}}.


=>{{{cos(theta ) =x/r}}}

=>{{{sin(theta)=y/r}}}


{{{r = 1/2+y/r}}}

{{{r = (r+y)/2r}}}

{{{2r^2=r+y}}}

{{{2r^2-r-y=0}}}.....In rectangular coordinates, {{{r= sqrt(x ^2+y ^2)}}}
 
{{{2(sqrt(x ^2+y ^2))^2-sqrt(x ^2+y ^2)-y=0}}}

​{{{2(x ^2+y ^2)-sqrt(x ^2+y ^2)-y=0}}}

{{{2x ^2+2y ^2-y-sqrt(x ^2+y ^2)=0}}}



2.

 {{{r = 6 tan(theta )sec(theta )}}}...................{{{tan(theta )=sin(theta )/cos(theta )}}} and {{{sec(theta )=1/cos(theta )}}}


{{{r =6(sin(theta )/cos(theta ))(1/cos(theta ))}}}

{{{r =6sin(theta )/cos^2(theta )}}}................{{{cos(theta )= x/r}}} and {{{sin(theta )=y/r}}}
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{{{r =6(y/r)/(x/r)^2}}}

{{{r=(6yr)/x^2}}}

{{{6yr=rx^2}}}...........simplify

{{{6y=x^2}}}

{{{y=x^2/6}}}