document.write( "Question 1184170: Let a, b, c be three sides opposite to the angles A, B, C respectively in a triangle ABC. Find 2ac.Sin(A + C - B)/2? \n" ); document.write( "
Algebra.Com's Answer #814705 by ikleyn(52834)\"\" \"About 
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\n" ); document.write( "Let a, b, c be three sides opposite to the angles A, B, C respectively in a triangle ABC.
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document.write( "First,  \"%28%28A+%2B+C+-+B%29%2F2%29\" = \"%28%28A%2BB%2BC%29-2B%29%2F2\" = \"%28pi-2B%29%2F2\" = \"pi%2F2+-+B\".\r\n" );
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document.write( "THEREFORE,  \"sin%28%28A+%2B+C+-+B%29%2F2%29\" = \"sin%28pi%2F2-B%29\" = cos(B).\r\n" );
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document.write( "Next, due to the cosine law for triangle ABC\r\n" );
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document.write( "    \"b%5E2\" = \"a%5E2\" + \"c%5E2\" - \"2ac%2Acos%28B%29\".\r\n" );
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document.write( "It implies\r\n" );
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document.write( "    2ac*cos(B) = a^2 + c^2 - b^2\r\n" );
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document.write( "Now we can write the final identity\r\n" );
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document.write( "    \"2ac%2Asin%28%28A%2BC-B%29%2F2%29\" = 2ac*cos(B) = \"a%5E2+%2B+c%5E2+-+b%5E2\".\r\n" );
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document.write( "ANSWER.  \"2ac%2Asin%28%28A%2BC-B%29%2F2%29\" = \"a%5E2+%2B+c%5E2+-+b%5E2\".\r\n" );
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