document.write( "Question 1071949: Solve the following Trigonometric quadratic equations for all angles from 0<= X < 360 to the nearest degree
\n" ); document.write( "1) 4tan^2(x) +3tan(x ) -1=0 and 2) 6sin(x) +3 =2csc(x)\r
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Algebra.Com's Answer #686848 by ikleyn(52873)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Solve the following Trigonometric quadratic equations for all angles from 0<= X < 360 to the nearest degree
\n" ); document.write( "1) 4tan^2(x) +3tan(x ) -1=0 and 2) 6sin(x) +3 =2csc(x)\r
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\n" ); document.write( "\n" ); document.write( "Please listen me very attentively . . . \r
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document.write( "Introduce new variable y = tan(x). Then your original equation takes the form\r\n" );
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document.write( "\"4y%5E2+%2B+3y+-+1\" = 0.     (1)\r\n" );
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document.write( "Solve it by applying the quadratic formula:\r\n" );
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document.write( "\"y%5B1%2C2%5D\" = \"%28-3+%2B-+sqrt%289+%2B+4%2A4%29%29%2F%282%2A4%29\" = \"%28-3+%2B-+sqrt%2825%29%29%2F8\" = \"%28-3+%2B-+5%29%2F8\".\r\n" );
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document.write( "So, the equation (1) has two solution.\r\n" );
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document.write( "1.  \"y%5B1%5D\" = \"%28-3+%2B5%29%2F8\" = \"2%2F8\" = \"1%2F4\"  ====>  tan(x) = \"1%2F4\"  ====>  x = \"arctan%281%2F4%29\"   AND/OR  x = \"arctan%281%2F4%29+%2B+pi\".\r\n" );
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document.write( "2.  \"y%5B2%5D\" = \"%28-3+-+5%29%2F8\" = \"-1\"  ====>  tan(x) = -1  ====>  x = \"arctan%28-1%29\" = \"-pi%2F4\"  AND/OR   x = \"-pi%2F4%2Bpi\" = \"3pi%2F4\".\r\n" );
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document.write( "Answer.  There are 4 solutions:   x = \"arctan%281%2F4%29\",  x = \"arctan%281%2F4%29+%2B+pi\",  x = \"-pi%2F4\",   x = \"3pi%2F4\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "Introducing new variable is the standard way to solve such problems.\r
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\n" ); document.write( "\n" ); document.write( "For more examples see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Solving simple problems on trigonometric equations\r
\n" ); document.write( "\n" ); document.write( "    - Solving typical problems on trigonometric equations\r
\n" ); document.write( "\n" ); document.write( "    - Solving more complicated problems on trigonometric equations\r
\n" ); document.write( "\n" ); document.write( "    - Solving advanced problems on trigonometric equations\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on calculating trig functions and solving trig equations\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( "I know that in  5 minutes other tutors will come explaining that you can factor the equation (1) or even factor the original equation\r
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\n" ); document.write( "\n" ); document.write( "It is possible,  of course. But regarding factoring, you must know this\r
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document.write( "     It works smoothly in two cases:  a)  you can easily factor it mentally, \r\n" );
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document.write( "                                      b)  you know the answer in advance  (may be,  based on your previous experience).\r\n" );
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document.write( "     This is why I prefer to use quadratic function in such cases:\r\n" );
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document.write( "     It is universal and works ALWAYS.\r\n" );
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\n" ); document.write( "\n" ); document.write( "H A P P Y       L E A R N I N G !!\r
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\n" ); document.write( "One more advise: do not put more than one problem/one question to your post.\r
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\n" ); document.write( "\n" ); document.write( "By violating this rule, you work against yourself . . . \r
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