document.write( "Question 347139: 21cos(x)+20sin(x) can be written in the form Asin(x+b) where A>0 and -pi\n" ); document.write( "\n" ); document.write( "Cheers in advance\r
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Algebra.Com's Answer #248226 by Edwin McCravy(20081)\"\" \"About 
You can put this solution on YOUR website!
\"21cos%28x%29%2B20sin%28x%29\" can be written in the form \"Asin%28x%2Bb%29\" where A>0 and \"-pi%3Cb%3Cpi\"\r
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document.write( "Assume:\r\n" );
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document.write( "\"A%2Asin%28x%2Bb%29+=+21cos%28x%29%2B20sin%28x%29\"\r\n" );
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document.write( "\"A%28sin%28x%29cos%28b%29+%2B+cos%28x%29sin%28b%29%29+=+21cos%28x%29%2B20sin%28x%29\"\r\n" );
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document.write( "Let \"x+=+0\"\r\n" );
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document.write( "\"A%28sin%280%29cos%28b%29+%2B+cos%280%29sin%28b%29%29+=+21cos%280%29%2B20sin%280%29\"\r\n" );
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document.write( "\"A%280%2Acos%28b%29%2B1%2Asin%28b%29%29+=+21%281%29%2B20%280%29\"\r\n" );
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document.write( "\"A%280%2Bsin%28b%29%29=21\"\r\n" );
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document.write( "\"Asin%28b%29+=+21\"\r\n" );
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document.write( "Now go back to:\r\n" );
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document.write( "\"A%28sin%28x%29cos%28b%29+%2B+cos%28x%29sin%28b%29%29+=+21cos%28x%29%2B20sin%28x%29\"\r\n" );
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document.write( "And now let \"x+=+pi%2F2\"\r\n" );
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document.write( "\"A%281%2Acos%28b%29%2B0%2Asin%28b%29%29+=+21%280%29%2B20%281%29\"\r\n" );
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document.write( "\"A%28cos%28b%29%2B0%29=0%2B20\"\r\n" );
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document.write( "\"Acos%28b%29+=+20\"\r\n" );
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document.write( "-------------------------\r\n" );
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document.write( "So we have this system of equations:\r\n" );
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document.write( "\"system%28Asin%28b%29+=+21%2C+Acos%28b%29+=+20%29\"\r\n" );
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document.write( "Dividing equals by equals:\r\n" );
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document.write( "\"%28A%28sin%28b%29%29%29%2F%28A%28cos%28b%29%29%29=21%2F20\"\r\n" );
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document.write( "\"%28cross%28A%29%28sin%28b%29%29%29%2F%28cross%28A%29%28cos%28b%29%29%29=21%2F20\"\r\n" );
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document.write( "\"sin%28b%29%2Fcos%28b%29=21%2F20\"\r\n" );
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document.write( "\"tan%28b%29=21%2F20\"\r\n" );
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document.write( "\"b=%22tan%22%5E%28-1%29\"\"%2821%2F20%29\" \r\n" );
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document.write( "Draw a right triangle with horizontal leg 20 and vertical leg 21,\r\n" );
document.write( "so the angle at the bottom left will have tangent \"21%2F20\", and\r\n" );
document.write( "therefore be equal to b. \r\n" );
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document.write( " and by the Pythagorean theorem \r\n" );
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document.write( "Since\r\n" );
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document.write( "\"Asin%28b%29+=+21\"\r\n" );
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document.write( "\"A%2821%2F29%29=21\"\r\n" );
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document.write( "\"21A+=+21%2A29\"\r\n" );
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document.write( "\"A+=+%2821%2A29%29%2F21\"\r\n" );
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document.write( "\"A+=+%28cross%2821%29%2A29%29%2Fcross%2821%29\"\r\n" );
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document.write( "\"A+=+29\"\r\n" );
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document.write( "Therefore  \"A=29\" and \"b=%22tan%22%5E%28-1%29\"\"%2821%2F20%29\"\r\n" );
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document.write( "and:\r\n" );
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document.write( "\"21cos%28x%29%2B20sin%28x%29=A%2Asin%28x%2Bb%29\"\r\n" );
document.write( "\"21cos%28x%29%2B20sin%28x%29\"\"%22%22=%22%22\"\"29%2A%22sin%28x%22\"\"%22%22%2B%22%22\"\"%22tan%22%5E%28-1%29%29\"\"21%2F20\"\"%22%29%22\"\r\n" );
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document.write( "Edwin
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