document.write( "Question 1183618: Need help with solving this system of Trig equations:
\n" ); document.write( "r = a sin(x)
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Algebra.Com's Answer #814023 by ikleyn(52909) $\"\"$ $\"About$

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\n" ); document.write( "Need help with solving this system of Trig equations:
\n" ); document.write( "r = a sin(x)
\n" ); document.write( "r = a cos(2x)
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\n" ); document.write( "\n" ); document.write( " In this system, \" a \" is the parameter.\r
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document.write( "Left sides of the two equations are identical - hence, their right sides are equal\r\n" );
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document.write( "    a*sin(x) = a*cos(2x)\r\n" );
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document.write( "Cancel the common factor \"a\" in both sides (assuming a =/= 0).  Then\r\n" );
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document.write( "    sin(x) = cos(2x)      (1)\r\n" );
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document.write( "    +-------------------------------------------------------+\r\n" );
document.write( "    |    From this point, there are two ways to proceed.    |\r\n" );
document.write( "    +-------------------------------------------------------+\r\n" );
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document.write( "One way is to notice that equation (1) implies  \r\n" );
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document.write( "    x + 2x =   for any integer  k = 0, +/-1, +/-2, . . . \r\n" );
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document.write( "It gives further\r\n" );
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document.write( "      3x  = ,\r\n" );
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document.write( "       x  = ,\r\n" );
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document.write( "which gives 3 (three) geometrically different answers for x\r\n" );
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document.write( "       x  =  = 30°  (at k=0);   x =  = 150°  (at k=1)   and   x =   = 270°  (at k=2).   (2)\r\n" );
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document.write( "At this point, first way solution is complete.\r\n" );
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document.write( "Another way to solve equation (1) is to replace the right side cos(2x) by , using well known trigonometric identity.\r\n" );
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document.write( "You will get then the equation\r\n" );
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document.write( "    sin(x) = ,\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( "It is a quadratic equation for sin(x). You can solve it  EITHER  using the quadratic formula  OR  factoring\r\n" );
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document.write( "     = (2sin(x)-1)*(sin(x)+1).\r\n" );
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document.write( "Doing this way, you will obtain the solutions\r\n" );
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document.write( "    sin(x) =   or  sin(x) = -1,\r\n" );
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document.write( "which lead to the same answers as the found above in (2).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Finally, if a =/= 0, then x=30°, r = 0.5a; or x=150°, r=0.5a; or x=270°, r= - a.\r
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\n" ); document.write( "\n" ); document.write( " If a = 0, then r=0 and x may have any value.\r
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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