SOLUTION: Find the Cartesian equation of r = asin3θ

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Question 1119696: Find the Cartesian equation of r = asin3θ
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
r=a%2Asin%283theta%29

We use the identity for sin(3q):

sin%283theta%29+=+3sin%28theta%29+-+4sin%5E3%28theta%29

r=a%283sin%28theta%29+-+4sin%5E3%28theta%29%5E%22%22%29

Then use this picture:

 

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

r=a%283%2Aexpr%28y%2Fr%29+-+4%2A%28y%2Fr%29%5E3%5E%22%22%29

r=a%28%283y%5E%22%22%29%2Fr%5E%22%22+-+4%28y%5E3%2Fr%5E3%29%5E%22%22%29%29

r=a%28%283y%5E%22%22%29%2Fr%5E%22%22+-+4y%5E3%2Fr%5E3%29%5E%22%22%29

r=%283ay%5E%22%22%29%2Fr%5E%22%22+-+4ay%5E3%2Fr%5E3%29%5E%22%22

Multiply through by r³

r%5E4=3ayr%5E2+-+4y%5E3%29

Finally substitute for r:

          r=sqrt%28x%5E2%2By%5E2%29
          r%5E2=x%5E2%2By%5E2
          r%5E4=x%5E4%2B2x%5E2y%5E2%2By%5E4

Substituting in

r%5E4=3ayr%5E2+-+4y%5E3%29

We get

x%5E4%2B2x%5E2y%5E2%2By%5E4=3ay%28x%5E2%2By%5E2%29+-+4y%5E3

Edwin