document.write( "Question 247773: I need to write the following as an algebraic expression: sin(2arccos x) \n" ); document.write( "
Algebra.Com's Answer #180630 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
I need to write the following as an algebraic expression: sin(2arccos x)\r
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document.write( "let \"arccos%28x%29+=+alpha\"\r\n" );
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document.write( "then \"cos%28alpha%29=x\"\r\n" );
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document.write( "So draw a right triangle and label one of the acute angles \"alpha\"\r\n" );
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document.write( "Now we need to put values on the sides of the\r\n" );
document.write( "triangle to cause \"cos%28alpha%29\" to equal \"x\".\r\n" );
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document.write( "Since the cosine is the adjacent side divided by the\r\n" );
document.write( "hypotenuse, if we label the adjacent side as x and the\r\n" );
document.write( "hypotenuse as 1. then \"cos%28alpha%29\" will equal \"x\".\r\n" );
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document.write( "Now we use the Pythagorean theorem to find the opposite\r\n" );
document.write( "side:\r\n" );
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document.write( "\"adjacent%5E2+%2B+opposite%5E2=+hypotenuse%5E2\"\r\n" );
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document.write( "\"x%5E2%2Bopposite%5E2=1%5E2\"\r\n" );
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document.write( "\"x%5E2%2Bopposite%5E2=1\"\r\n" );
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document.write( "\"opposite%5E2=1-x%5E2\"\r\n" );
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document.write( "\"opposite+=+sqrt%281-x%5E2%29\"\r\n" );
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document.write( "So we write this on the opposite side:\r\n" );
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document.write( "Now we go back to the original problem:\r\n" );
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document.write( "\"sin%282arccos%28x%29%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"sin%282alpha%29\"\r\n" );
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document.write( "and we use the identity\r\n" );
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document.write( "\"sin%282alpha%29=2sin%28alpha%29cos%28alpha%29\"  \r\n" );
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document.write( "Then using the fact that the sine is the opposite over\r\n" );
document.write( "the hypotenuse and the cosine is the adjacent over the\r\n" );
document.write( "hypotenuse, we use the right triangle above:\r\n" );
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document.write( "\"sin%282alpha%29=2sin%28alpha%29cos%28alpha%29\"\r\n" );
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document.write( "\"sin%282alpha%29=2%28%28sqrt%281-x%5E2%29%29%2F1%29%28x%2F1%29\"\r\n" );
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document.write( "\"sin%282alpha%29=2%28sqrt%281-x%5E2%29%29%28x%29\"\r\n" );
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document.write( "\"sin%282alpha%29=2x%2Asqrt%281-x%5E2%29\"\r\n" );
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document.write( "Edwin
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