document.write( "Question 1132312: If a^2 is greater than (b^2 + c^2) in the triangle ABC, show that the value of the expression
\n" ); document.write( " [(1+ cos2B+isin2B)(1+cos2C+isin2C)]/(1+cos2A-isin2A) is real and positive. \n" ); document.write( "
Algebra.Com's Answer #749512 by Edwin McCravy(20081)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We REALize the denominator by multiplying by the conjugate of the\r\n" );
document.write( "denominator over itself:\r\n" );
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document.write( "The denominator could be simplified but we don't need to because it\r\n" );
document.write( "is the sum of two squares which is real and positive.  So we only \r\n" );
document.write( "need to show that the numerator is real and positive as well.\r\n" );
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document.write( "Numerator = \r\n" );
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document.write( "We use the double-angle identities:\r\n" );
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document.write( "      \"cos%282theta%29=2cos%5E2%28theta%29-1\"\r\n" );
document.write( "      \"sin%282theta%29=2sin%28theta%29cos%28theta%29\"\r\n" );
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document.write( "Numerator = \r\n" );
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document.write( "These are complex numbers in polar or trig form, so we multiply the magnitudes and add the angles:\r\n" );
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document.write( "Since B, C, and A are angles of a triangle, B+C+A = 180° or \"pi\"\r\n" );
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document.write( "\"8%2Acos%28B%29%2Acos%28C%29%2Acos%28A%29%2A%28cos%28pi%29%5E%22%22%2Bi%2Asin%28pi%29%29\"\r\n" );
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document.write( "Since \"cos%28pi%29=-1\" and \"sin%28pi%29=0\"\r\n" );
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document.write( "\"8%2Acos%28B%29%2Acos%28C%29%2Acos%28A%29%2A%28-1%2Bi%2A0%29\"\r\n" );
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document.write( "\"-8%2Acos%28B%29%2Acos%28C%29%2Acos%28A%29\", which is real.\r\n" );
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document.write( "So we only need to show that this is positive.\r\n" );
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document.write( "Since we are told that a² > b²+c², This makes the longest side\r\n" );
document.write( "greater than the hypotenuse of a right triangle with sides b\r\n" );
document.write( "and c, which means that angle A is greater than 90°, an obtuse\r\n" );
document.write( "angle.  An obtuse angle is a QII angle which has a negative\r\n" );
document.write( "cosine.\r\n" );
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document.write( "Therefore cos(A) is negative, and since a triangle with one\r\n" );
document.write( "obtuse angle has the other two angles acute.\r\n" );
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document.write( "So cos(A) < 0, cos(B) > 0, cos(C) > 0 \r\n" );
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document.write( "Therefore\r\n" );
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document.write( "\"-8%2Acos%28B%29%2Acos%28C%29%2Acos%28A%29\"\r\n" );
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document.write( "is the product of 4 factors, two of which are negative and two\r\n" );
document.write( "positive, which makes the product positive.\r\n" );
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document.write( "Edwin
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