document.write( "Question 1052901: Find the exact value of the​ following, without using a calculator.
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Algebra.Com's Answer #668198 by ikleyn(52797)\"\" \"About 
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document.write( "0.  Let \"alpha\" be \"sin%5E%28-1%29%281%2F2%29\" = \"arcsin%281%2F2%29\", and\r\n" );
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document.write( "    let \"beta\" be \"tan%28-1%29%28-5%29\" = \"arctan%28-5%29\".\r\n" );
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document.write( "    Then we have \"sin%28alpha%29\" = \"1%2F2\" with \"alpha\" in QI and \"tan%28beta%29\" = -5 with \"beta\" in QIV, and we need to find \"sin%28alpha+%2B+beta%29\".\r\n" );
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document.write( "    Surely, we will use the formula\r\n" );
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document.write( "    \"sin%28alpha+%2B+beta%29\" = \"sin%28alpha%29%2Acos%28beta%29+%2B+cos%28alpha%29%2Asin%28beta%29\",                 (1)\r\n" );
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document.write( "    but first we shall calculate \"cos%28beta%29\", \"cos%28alpha%29\" and \"sin%28beta%29\".\r\n" );
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document.write( "1.  Since \"sin%28alpha%29\" = \"1%2F2\", we have \"alpha\" = \"pi%2F6\" and \"cos%28alpha%29\" = \"sqrt%283%29%2F2\".    (2)\r\n" );
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document.write( "2.  \"sin%5E2%28beta%29\" = \"1%2F%281%2B1%2Ftan%5E2%28beta%29%29\" = \"1%2F%281+%2B+1%2F%28-5%29%5E2%29\" = \"25%2F%2825%2B1%29\" = \"25%2F26\".\r\n" );
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document.write( "    Hence, \"sin%28beta%29\" = \"-sqrt%2825%2F26%29\" = \"-5%2Fsqrt%2826%29\".                            (3)\r\n" );
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document.write( "    We chose the sign \"-\" at the square root, since the angle \"beta\" lies in QIV (due to the sign of tan).\r\n" );
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document.write( "3.  Since \"sin%28beta%29\" = \"-5%2Fsqrt%2826%29\", we have \r\n" );
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document.write( "    \"cos%28beta%29\" = \"sqrt%281-sin%5E2%28beta%29%29\" = \"sqrt%281-%28-5%2Fsqrt%2826%29%29%5E2%29\" = \"sqrt%281-25%2F26%29\" = \"sqrt%28%2826-25%29%2F26%29%29\" = \"sqrt%281%2F26%29\" = \"1%2Fsqrt%2826%29\".    (4)\r\n" );
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document.write( "4.  Now you have everything to use the formula (1):\r\n" );
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document.write( "    \"sin%28alpha%2Bbeta%29\" = \"%281%2F2%29%2A%281%2Fsqrt%2826%29%29+%2B+%28sqrt%283%29%2F2%29%2A%28-5%2Fsqrt%2826%29%29\" = \"%281-5%2Asqrt%283%29%29%2F%282%2Asqrt%2826%29%29\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "For other similar solved problems see the lesson \r
\n" ); document.write( "\n" ); document.write( "    - Advanced problems on calculating trigonometric functions of angles\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 lesson is the part of this online textbook under the topic \"Trigonometry: Solved problems\".\r
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