document.write( "Question 958763: how do you find the exact value of the expression\r
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Algebra.Com's Answer #854577 by ikleyn(53937)\"\" \"About 
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\n" ); document.write( "how do you find the exact value of the expression
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by @lwsshar3 is incorrect due to arithmetic error.\r
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document.write( "So, they want you find cos(arcsin(1/3) - arctan(1/2)).\r\n" );
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document.write( "An average school student will be shocked, I think.\r\n" );
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document.write( "Therefore, let's move forward accurately.\r\n" );
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document.write( "Let x = arcsin(1/3) and y = arctan(1/2).\r\n" );
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document.write( "So, sin(x) = 1/3 and x is in QI;\r\n" );
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document.write( "    tan(y) = 1/2 and y is in QI, too.\r\n" );
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document.write( "Then cos(x) = \"sqrt%281-sin%5E2%28x%29%29\" = \"sqrt%281-%281%2F3%29%5E2%29\" = \"sqrt%281-1%2F9%29\" = \"sqrt%288%2F9%29\" = \"sqrt%288%29%2F3\".\r\n" );
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document.write( "For 'y', tan(y) = 1/2  is the same as to say  sin(y) = \"1%2Fsqrt%285%29\",  cos(y) = \"2%2Fsqrt%285%29\".\r\n" );
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document.write( "Therefore,\r\n" );
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document.write( "      cos(arcsin(1/3)-acrtan(1/2)) = cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y) = \r\n" );
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document.write( "    = \"%28sqrt%288%29%2F3%29%2A%282%2Fsqrt%285%29%29\" + \"%281%2F3%29%2A%281%2Fsqrt%285%29%29\" = \"%282%2Asqrt%288%29%29%2F%283%2Asqrt%285%29%29\" + \"1%2F%283%2Asqrt%285%29%29\" = \"%282%2Asqrt%288%29%2B1%29%2F%283%2Asqrt%285%29%29\" = \"%284%2Asqrt%282%29%2B1%29%2F%283%2Asqrt%285%29%29\" = \"%284%2Asqrt%2810%29%2Bsqrt%285%29%29%2F15\".\r\n" );
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document.write( "ANSWER.  cos(arcsin(1/3) - arctan(1/2)) = \"%284%2Asqrt%2810%29%2Bsqrt%285%29%29%2F15\". \r\n" );
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