document.write( "Question 1119696: Find the Cartesian equation of r = asin3θ \n" ); document.write( "
Algebra.Com's Answer #735485 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"r=a%2Asin%283theta%29\"\r\n" );
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document.write( "We use the identity for sin(3q):\r\n" );
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document.write( "\"sin%283theta%29+=+3sin%28theta%29+-+4sin%5E3%28theta%29\"\r\n" );
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document.write( "\"r=a%283sin%28theta%29+-+4sin%5E3%28theta%29%5E%22%22%29\"\r\n" );
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document.write( "Then use this picture:\r\n" );
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document.write( "Note: When trig ratios are involved, leave r as it is, and\r\n" );
document.write( "wait until last to substitute for r:\r\n" );
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document.write( "\"r=a%283%2Aexpr%28y%2Fr%29+-+4%2A%28y%2Fr%29%5E3%5E%22%22%29\"\r\n" );
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document.write( "\"r=a%28%283y%5E%22%22%29%2Fr%5E%22%22+-+4%28y%5E3%2Fr%5E3%29%5E%22%22%29%29\"\r\n" );
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document.write( "\"r=a%28%283y%5E%22%22%29%2Fr%5E%22%22+-+4y%5E3%2Fr%5E3%29%5E%22%22%29\"\r\n" );
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document.write( "\"r=%283ay%5E%22%22%29%2Fr%5E%22%22+-+4ay%5E3%2Fr%5E3%29%5E%22%22\"\r\n" );
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document.write( "Multiply through by r³\r\n" );
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document.write( "\"r%5E4=3ayr%5E2+-+4y%5E3%29\"\r\n" );
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document.write( "Finally substitute for r:\r\n" );
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document.write( "          \"r=sqrt%28x%5E2%2By%5E2%29\"\r\n" );
document.write( "          \"r%5E2=x%5E2%2By%5E2\"\r\n" );
document.write( "          \"r%5E4=x%5E4%2B2x%5E2y%5E2%2By%5E4\"\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "\"r%5E4=3ayr%5E2+-+4y%5E3%29\"\r\n" );
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document.write( "We get\r\n" );
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document.write( "\"x%5E4%2B2x%5E2y%5E2%2By%5E4=3ay%28x%5E2%2By%5E2%29+-+4y%5E3\"\r\n" );
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document.write( "Edwin
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