document.write( "Question 1208089: Find x,
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Algebra.Com's Answer #846242 by ikleyn(52835) $\"\"$ $\"About$

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\n" ); document.write( "Find x, $\"4%5Esin%5E2%28x%29\"$ + $\"4%5Ecos%5E2%28x%29\"$ = $\"3%2Asqrt%282%29\"$ .
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document.write( "The original equation is\r\n" );
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document.write( "     +  = .     (1)\r\n" );
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document.write( "Use  = .  \r\n" );
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document.write( "Then equation (1) becomes\r\n" );
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document.write( "     +  = .    (2)\r\n" );
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document.write( "Introduce new variable t = .  Then equation (2) takes the form\r\n" );
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document.write( "    t +  = .    (3)\r\n" );
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document.write( "Reduce it to the standard form quadratic equation\r\n" );
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document.write( "    t^2 + 4 = ,\r\n" );
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document.write( "    t^2 -  + 4 = 0.    (4)\r\n" );
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document.write( "Use the quadratic formula to find the roots\r\n" );
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document.write( "     =  =  = .\r\n" );
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document.write( "So, equation (4) has two roots.  \r\n" );
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document.write( "One root is       =  =  = .\r\n" );
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document.write( "Another root is   =  =  = .\r\n" );
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document.write( "So, further we consider two cases.\r\n" );
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document.write( "Case 1.   = .  \r\n" );
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document.write( "         Then    =  =  =  = .\r\n" );
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document.write( "         It implies  sin(x) = +/-  = +/- .\r\n" );
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document.write( "         Hence,  x = ,  k = 0, +/-1, +/-2, . . .  or  x = ,  k = 0, +/-1, +/-2, . . .\r\n" );
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document.write( "Case 2.   = .  \r\n" );
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document.write( "         Then    =  =  =  = .\r\n" );
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document.write( "         It implies  sin(x) = +/-  = +/- .\r\n" );
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document.write( "         Hence,  x = ,  k = 0, +/-1, +/-2, . . .  or  x = ,  k = 0, +/-1, +/-2, . . .\r\n" );
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document.write( "ANSWER.  The solutions to given equation are  these four infinite sets of real numbers\r\n" );
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document.write( "         x = ,  k = 0, +/-1, +/-2, . . .  or  x = ,  k = 0, +/-1, +/-2, . . .\r\n" );
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document.write( "     or  x = ,  k = 0, +/-1, +/-2, . . .  or  x = ,  k = 0, +/-1, +/-2, . . .\r\n" );
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