Question 1119696
<pre>
{{{r=a*sin(3theta)}}}

We use the identity for sin(3<font face="symbol">q</font>):

{{{sin(3theta) = 3sin(theta) - 4sin^3(theta)}}}

{{{r=a(3sin(theta) - 4sin^3(theta)^"")}}}

Then use this picture:

{{{drawing(200,100,-.5,2,-.5,1.5,
line(-3,0,3,0),line(0,-3,0,3), line(0,0,1.5,1),line(1.5,0,1.5,1),
locate(.4,.3,theta), locate(1.53,.62,y),locate(.77,0.05,x),
locate(.75,.85,r) )}}} 

Note: When trig ratios are involved, leave r as it is, and
wait until last to substitute for r:

{{{r=a(3*expr(y/r) - 4*(y/r)^3^"")}}}

{{{r=a((3y^"")/r^"" - 4(y^3/r^3)^""))}}}

{{{r=a((3y^"")/r^"" - 4y^3/r^3)^"")}}}

{{{r=(3ay^"")/r^"" - 4ay^3/r^3)^""}}}

Multiply through by r³

{{{r^4=3ayr^2 - 4y^3)}}}

Finally substitute for r:

          {{{r=sqrt(x^2+y^2)}}}
          {{{r^2=x^2+y^2}}}
          {{{r^4=x^4+2x^2y^2+y^4}}}

Substituting in

{{{r^4=3ayr^2 - 4y^3)}}}

We get

{{{x^4+2x^2y^2+y^4=3ay(x^2+y^2) - 4y^3}}}

Edwin</pre>