document.write( "Question 1105327: Given Tan(A) = 5 in Quadrant III and Sin(B) = 2/3 in Quadrant II, find Sin(A-B),find Cos(A-B),and find the quadrant of A-B. \n" ); document.write( "
Algebra.Com's Answer #720128 by ikleyn(52788)\"\" \"About 
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document.write( "1.  tan(A) = 5  implies  \"tan%5E2%28A%29\" = 25  implies  \"sin%5E2%28A%29\" = \"1%2F%281+%2B+1%2Ftan%5E2%28A%29%29\" = \"1%2F%281+%2B+1%2F25%29\" = \"25%2F26\"  implies  sin(A) = \"-sqrt%2825%2F26%29\" = \"-5%2Fsqrt%2826%29\".\r\n" );
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document.write( "    The sign is  \"-\"  (minus)  at the square root, since A is in QIII.\r\n" );
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document.write( "    Then  cos(A) = \"-sqrt%281-sin%5E2%28A%29%29\" = \"-sqrt%281-25%2F26%29\" = \"-1%2Fsqrt%2826%29\".\r\n" );
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document.write( "    The sign is  \"-\"  (minus)  at the square root, since A is in QIII.\r\n" );
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document.write( "2.  sin(B) = \"2%2F3\"  implies  cos(B) = \"-sqrt%281-sin%5E2%28B%29%29\" = \"-sqrt%281-%282%2F3%29%5E2%29\" = \"-sqrt%281-4%2F9%29\" = \"-sqrt%285%2F9%29\" = \"-sqrt%285%29%2F3\".\r\n" );
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document.write( "    The sign is  \"-\"  (minus)  at the square root, since B is in QII.\r\n" );
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document.write( "3.  Now \r\n" );
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document.write( "    sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B) = \"%28-5%2Fsqrt%2826%29%29%2A%28-sqrt%285%29%2F3%29\" - \"%28-1%2Fsqrt%2826%29%29%2A%282%2F3%29\" = \"%285%2Asqrt%285%29%29%2F%283%2Asqrt%2826%29%29\" + \"2%2F%283%2Asqrt%2826%29%29\" = \"%285%2Asqrt%285%29%2B2%29%2F%283%2Asqrt%2826%29%29\" = \"%285%2Asqrt%285%29%2B2%29%2F%283%2Asqrt%2826%29%29\".\"sqrt%2826%29%2Fsqrt%2826%29\" = \"%285%2Asqrt%28130%29%2B2%2Asqrt%2826%29%29%2F%283%2A26%29\" = \r\n" );
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document.write( "             = \"%285%2Asqrt%28130%29%2B2%2Asqrt%2826%29%29%2F78\" = 0.861626  (approximately).\r\n" );
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document.write( "    cos(A-B) = cos(A)*cos(B) + sin(A)*sin(B) = \"%28-1%2Fsqrt%2826%29%29%2A%28-sqrt%285%29%2F3%29\" + \"%28-5%2Fsqrt%2826%29%29%2A%282%2F3%29\" = \"%28sqrt%285%29%29%2F%283%2Asqrt%2826%29%29\" - \"10%2F%283%2Asqrt%2826%29%29\" = \"%28sqrt%285%29-10%29%2F%283%2Asqrt%2826%29%29\" = \"%28sqrt%285%29-10%29%2F%283%2Asqrt%2826%29%29\".\"sqrt%2826%29%2Fsqrt%2826%29\" = \"%28sqrt%28130%29-10%2Asqrt%2826%29%29%2F%283%2A26%29\" = \r\n" );
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document.write( "             = \"%28sqrt%28130%29-10%2Asqrt%2826%29%29%2F78\" = -0.50754 (approximately).\r\n" );
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document.write( "    Since  sin(A-B) is positive, while cos(A-B) is negative,  the angle  A-B lies in QII.\r\n" );
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\n" ); document.write( "To see more solved similar problems on calculating trig functions,  look into the lessons \r
\n" ); document.write( "\n" ); document.write( "    - Calculating trigonometric functions of angles\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problems on calculating trigonometric functions of angles\r
\n" ); document.write( "\n" ); document.write( "    - Evaluating trigonometric expressions \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Trigonometry: Solved problems\". \r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
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