document.write( "Question 960547: Find all solutions if
\n" ); document.write( "0 ≤ x < 2π.
\n" ); document.write( " Use exact values only. (Enter your answers as a comma-separated list.)\r
\n" ); document.write( "\n" ); document.write( "cos 2x cos x − sin 2x sin x = square root of 2 over 2\r
\n" ); document.write( "\n" ); document.write( "x=?
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #854566 by ikleyn(53937) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Find all solutions in 0 ≤ x < 2π.
\n" ); document.write( "Use exact values only. (Enter your answers as a comma-separated list.)
\n" ); document.write( "cos 2x cos x − sin 2x sin x = square root of 2 over 2
\n" ); document.write( "x=?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        In his post, @lwsshar3 found two solutions, $\"pi%2F12\"$ and $\"7pi%2F12\"$ .\r
\n" ); document.write( "\n" ); document.write( "        Actually, this equation has 6 (six) solutions, but @lwsshar3 missed/lost most of them.\r
\n" ); document.write( "\n" ); document.write( "        See my correct and complete solution below.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Use identity:  cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)\r\n" );
document.write( "\r\n" );
document.write( "cos(2x)* cos(x) − sin(2x)* sin( x) = square root of 2 over 2\r\n" );
document.write( "\r\n" );
document.write( "cos(2x+x) = √2/2\r\n" );
document.write( "\r\n" );
document.write( "2x+x = π/4 + 2k*π;     7π/4 + 2k*π.\r\n" );
document.write( "\r\n" );
document.write( "  3x = π/4 + 2k*π;     7π/4 + 2k*π.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    For 3x, you must add the periods 2π and 4π for cosine.\r\n" );
document.write( "    For 3x, it will produce geometrically the same angle; \r\n" );
document.write( "    but for 'x' it will produce new angles that you will miss otherwise.\r\n" );
document.write( "\r\n" );
document.write( "    You should no add more hire periods for 3x, since it will not produce \r\n" );
document.write( "    new angles for x and will lead you out of the given interval.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now consider two cases.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Case (a).  3x = ,  ,  .\r\n" );
document.write( "\r\n" );
document.write( "            then  x = ,   =  = ,   = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Case (b).  3x = ,  ,  .\r\n" );
document.write( "\r\n" );
document.write( "            then  x = ,   =  = ,   = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The given equation has 6 solutions in the given interval\r\n" );
document.write( "\r\n" );
document.write( "         ,  ,  ,  ,  ,  .\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved completely and correctly - no one root is missed.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This analysis in my post is typical in trigonometry problems, when you work with a multiple of an angle.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If you miss this analysis, you will miss many solutions to your original equation.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );