document.write( "Question 1088217: If cosA-sinA=√2sin A then cosA+sinA equals \n" ); document.write( "

Algebra.Com's Answer #702532 by ikleyn(52874) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "If cos(A) - sin(A) = sqrt(2)*sin(A) then cos(A) + sin(A) equals
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "You are given\r\n" );
document.write( "\r\n" );
document.write( "cos(A) - sin(A) = sqrt(2)*sin(A).    (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides by . You will get\r\n" );
document.write( "\r\n" );
document.write( " -  = sin(A).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is the same as\r\n" );
document.write( "\r\n" );
document.write( " -  = sin(A).      (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now recall that  =  = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, you can re-write (2) in the form\r\n" );
document.write( "\r\n" );
document.write( " -  = sin(A).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Using the adding/subtracting formula for sine, it is the same as\r\n" );
document.write( "\r\n" );
document.write( " = ,                         (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "which implies EITHER\r\n" );
document.write( "\r\n" );
document.write( "     =  + ,                     (4)    \r\n" );
document.write( "\r\n" );
document.write( "OR\r\n" );
document.write( "\r\n" );
document.write( "     +  =                    (5)\r\n" );
document.write( "\r\n" );
document.write( "where k is any integer.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Equation (5) has no solution, obviously.\r\n" );
document.write( "\r\n" );
document.write( "Equation (4) has the solution\r\n" );
document.write( "\r\n" );
document.write( "    2A = ,   or   A = .      (6)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Actually, we have two cases:  A =   and  A = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is well known fact that \r\n" );
document.write( "\r\n" );
document.write( " = ,   = .\r\n" );
document.write( "\r\n" );
document.write( "    (see the lesson Miscellaneous Trigonometry problems in this site).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, if A = ,   then  cos(A) + sin(A) =  + .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    If A = ,  then  cos(A) + sin(A) = -(  +  ).\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Answer. If cos(A) - sin(A) = sqrt(2)*sin(A)   then \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                    a) A = $\"pi%2F8\"$ or A = $\"9pi%2F8\"$ ,   and\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                    b) cos(A) + sin(A) equals    $\"sqrt%282%2Bsqrt%282%29%29%2F2\"$ + $\"sqrt%282-sqrt%282%29%29%2F2\"$   or   -( $\"sqrt%282%2Bsqrt%282%29%29%2F2\"$ + $\"sqrt%282-sqrt%282%29%29%2F2\"$ ).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );