document.write( "Question 1101623: Determine the values of
\n" ); document.write( "2θ (not θ) on [0,2π) that satisfy the following equation. (Separate multiple solutions with a comma. Give exact answers.)
\n" ); document.write( " 3sin(2θ)=−3/√2
\n" ); document.write( "2θ=\r
\n" ); document.write( "\n" ); document.write( "You now have two equations representing all possible solutions for 2θ. Solve each of those equations for θ. (Let θ1 and θ2 represent the solutions on [0,2π), where θ1is less than θ2.)
\n" ); document.write( " θ1=
\n" ); document.write( "
\n" ); document.write( " θ2=\r
\n" ); document.write( "\n" ); document.write( "Use these general solutions for θ to find the four solutions to 3sin(2θ)=−3/√2 on the intervall
\n" ); document.write( "[0,2π). (Separate multiple solutions with a comma. Give exact answers.)\r
\n" ); document.write( "\n" ); document.write( " θ=
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #716265 by ikleyn(52831) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "3*sin(2*a) = -3/sqrt(2)
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " Due to \"typography\" issues, I will replace $\"theta\"$ in my post by simple \"a\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( " =       (1)       ====>  (divide both sides by 3)  ====>\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " = ,   or, which is the same,\r\n" );
document.write( "\r\n" );
document.write( " = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It implies   =    or    = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Everything was simple to this point. \r\n" );
document.write( "\r\n" );
document.write( "    But in reality, accurate analysis only  STARTS  from this point.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "1)  It is obvious that   =   implies   = . \r\n" );
document.write( "\r\n" );
document.write( "    But if you stop here, you will loose another existing solution of the same family.\r\n" );
document.write( "\r\n" );
document.write( "    It is   =  = .\r\n" );
document.write( "\r\n" );
document.write( "    Indeed,   =  =  is GEOMETRICALLY the same angle as   and has the same value of sine,\r\n" );
document.write( "\r\n" );
document.write( "    so  is the solution to the original equation  (1), too.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Thus the relation   =   creates and generates not one solution , but TWO solutions    and    \r\n" );
document.write( "\r\n" );
document.write( "    of the same family.     Notice, that they BOTH belong to the interval  [0,).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2)  The same or the similar story is with the solution   = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    It is obvious that   =   implies   = . \r\n" );
document.write( "\r\n" );
document.write( "    But if you stop here, you will loose another existing solution of the same family.\r\n" );
document.write( "\r\n" );
document.write( "    It is   =  = .\r\n" );
document.write( "\r\n" );
document.write( "    Indeed,   =  =   is GEOMETRICALLY the same angle as   and has the same value of sine,\r\n" );
document.write( "\r\n" );
document.write( "    so  is the solution to the original equation  (1), too.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Thus the relation   =   creates and generates not one solution , but TWO solutions    and    \r\n" );
document.write( "\r\n" );
document.write( "    of the same family.     Notice, that they BOTH belong to the interval  [0,).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3.  Thus the original equation (1) has 4 (four, FOUR) solutions in the interval  [0,):\r\n" );
document.write( "\r\n" );
document.write( "    ,  ,    and  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "4.  The plot below visually confirms existing of 4 solutions to the given equality:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Plot y =   (red)  and y =  (green)\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );