document.write( "Question 1197640: Solve for θ. 0° ≤ θ ≤ 360°.
\n" ); document.write( "1. (sin^2)θ + (1/(sin^2)θ) + sinθ + (1/sinθ) = 4
\n" ); document.write( "I tried substituting m for sinθ but after simplifying I get $\"m%5E4%2Bm%5E3%2Bm%2B1-4m%5E2=0\"$ . I'm not sure what to do next? \n" ); document.write( "

Algebra.Com's Answer #831006 by ikleyn(53762) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Solve for θ. 0° ≤ θ ≤ 360°.
\n" ); document.write( "(sin^2)θ+(1/(sin^2)θ)+sinθ+(1/sinθ)=4
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document.write( "Introduce new variable x =  + .\r\n" );
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document.write( "Notice that  =  +  + 2;\r\n" );
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document.write( "so           +  = .\r\n" );
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document.write( "THEREFORE, the original equation takes the form\r\n" );
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document.write( "     + x = 4,\r\n" );
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document.write( "or\r\n" );
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document.write( "     = 0.\r\n" );
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document.write( "Factor left side\r\n" );
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document.write( "    {x+3)*(x-2) = 0.\r\n" );
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document.write( "The roots are x= -3  and  x= 2.\r\n" );
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document.write( "Next we consider two cases.\r\n" );
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document.write( "    (a)  if x= -3, it means   +  = -3,  which implies\r\n" );
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document.write( "          = 0,  and then, due to the quadratic formula\r\n" );
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document.write( "          =  = .\r\n" );
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document.write( "         It gives only one root   =  = -0.382  (rounded),\r\n" );
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document.write( "         so   = -arcsin(0.382) = 360° - 22.457° = 337.543° \r\n" );
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document.write( "         or   = 180° + arcsin(0.382) = 180° + 22.457° = 202.457°.\r\n" );
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document.write( "    (b)  if x= 2, it means   +  = 2,  which implies\r\n" );
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document.write( "          = 0,  and then\r\n" );
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document.write( "          = 0.\r\n" );
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document.write( "         It gives one root   = 1 of multiplicity 2,  so   = 90° of multiplicity 2.\r\n" );
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document.write( "ANSWER.  The solutions are   = 90° of multiplicity 2,   = 202.457°  and   = 337.543°.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "The key to the solution is the substitution made at the very beginning of my post.\r
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\n" ); document.write( "\n" ); document.write( "It is a standard way to solve such equations, but far not everyone knows it.\r
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\n" ); document.write( "\n" ); document.write( "It is what you need to learn from my solution: how it works.\r
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