document.write( "Question 1204440: Hi, I am struggling with this problem. Thanks.
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Algebra.Com's Answer #840696 by ikleyn(52909) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( " $\"sin%5E2%28x%29\"$ + $\"1%2Fsin%5E2%28x%29\"$ + sin(x) + $\"1%2Fsin%28x%29\"$ = 4.
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document.write( "Introduce new variable\r\n" );
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document.write( "    y = sin(x) + .\r\n" );
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document.write( "Then y^2 = sin^2(x) + 2 + 1/sin^2(x),  and the original equation takes the form\r\n" );
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document.write( "    y^2 - 2 + y = 4,\r\n" );
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document.write( "or\r\n" );
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document.write( "    y^2 + y - 6 = 0.\r\n" );
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document.write( "Its roots are (using the quadratic formula)\r\n" );
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document.write( "     =  =  = ,\r\n" );
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document.write( "    or   =  = 2,   =  = -3.\r\n" );
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document.write( "It means that  sin(x) +   is  EITHER  2  OR  -3.\r\n" );
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document.write( "So, from this point, we have two cases for x.\r\n" );
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document.write( "(1)  sin(x) +  = 2,  which implies\r\n" );
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document.write( "     sin^2(x) - 2sin(x) + 1 = 0,\r\n" );
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document.write( "     (sin(x) - 1)^2 = 0\r\n" );
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document.write( "      sin(x) = 1\r\n" );
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document.write( "          x  = 90 degrees (or  x = ).\r\n" );
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document.write( "      Thus this case is complete.\r\n" );
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document.write( "(2)  sin(x) +  = -3,  which implies\r\n" );
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document.write( "     sin^2(x) + 3*sin(x) + 1 = 0,\r\n" );
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document.write( "     sin(x) = use the quadratic formula =  = . \r\n" );
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document.write( "      =  = ~ -0.382  --->  from it, you may find two values for x between 0 and 360 degrees.\r\n" );
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document.write( "      =  = ~ -2.618  --->  there is no solution for real x. \r\n" );
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document.write( "      Thus this case is complete, too.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Comparing with the solution by tutor @math_tutor2020, the substitution which I used, leads
\n" ); document.write( "to equation/equations of degree 2, ONLY. There is no need to work with equation of degree 4.\r
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