document.write( "Question 1102804: An equation is given. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
\n" ); document.write( "2 cos(2θ) + 1 = 0\r
\n" ); document.write( "\n" ); document.write( "Find all solutions within the interval (0, 2pi).
\n" ); document.write( "

Algebra.Com's Answer #717548 by ikleyn(52839)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "\"2%2Acos%282theta%29\" + 1 = 0  ====>\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"cos%282theta%29\" = \"-1%2F2\"  ====>\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"2%2Atheta\"  may have one of the two values:  \"2%2Atheta\" = \"2pi%2F3\"  or  \"2%2Atheta\" = \"4pi%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Case 1.  If  \"2%2Atheta\" = \"2pi%2F3\",  then the solutions for \"theta\" in the interval  [0,\"2pi\")  are  these TWO values:\r\n" );
document.write( "\r\n" );
document.write( "         \"theta\" = \"pi%2F3\"   and/or   \"theta\" = \"pi%2F3+%2B+pi\" = \"4pi%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Case 2.  If  \"2%2Atheta\" = \"4pi%2F3\",  then the solutions for \"theta\" in the interval  [0,\"2pi\")  are  these TWO values:\r\n" );
document.write( "\r\n" );
document.write( "         \"theta\" = \"2pi%2F3\"   and/or   \"theta\" = \"2pi%2F3+%2B+pi\" = \"5pi%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Answer.  The given equation has 4 (four, FOUR) solutions in the interval  [0,\"2pi\"):\r\n" );
document.write( "\r\n" );
document.write( "                \"pi%2F3\",  \"4pi%2F3\",  \"2pi%2F3\",  and  \"5pi%2F3\".\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The plot below visually confirms existing of 4 solutions:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Plot y = \"2%2Acos%282%2Atheta%29+%2B+1\" \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        SOLVED.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----------------
\n" ); document.write( "N O T E. All the part of the condition  IN PARENTHESES  makes no sense  and  IS NOT RELEVANT  to the problem at all.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        When I see such a  NONSENSE  in the condition,  I become sad,
\n" ); document.write( "        because I clearly see that the person who prepared this post  does not understand WHAT HE IS WRITING.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "
\n" );