document.write( "Question 1063391: Solve the following equations for 0deg <= x <= 360deg:
\n" ); document.write( "(i) cos 2x cos x = sin 4x sin x
\n" ); document.write( "(ii) cos x + cos 2x + cos 3x + cos 4x = 0 \n" ); document.write( "

Algebra.Com's Answer #678492 by ikleyn(52787) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Solve the following equations for 0deg <= x <= 360deg:
\n" ); document.write( "(i) cos 2x cos x = sin 4x sin x
\n" ); document.write( "(ii) cos x + cos 2x + cos 3x + cos 4x = 0
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I will solve (ii).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "cos(x) + cos(2x) + cos(3x) + cos(4x) = 0.        (1)\r\n" );
document.write( "\r\n" );
document.write( "Use the general formula of Trigonometry\r\n" );
document.write( "\r\n" );
document.write( " = .                (2)\r\n" );
document.write( "\r\n" );
document.write( "You have \r\n" );
document.write( "\r\n" );
document.write( "cos(x) + cos(4x) =  = ,\r\n" );
document.write( "\r\n" );
document.write( "cos(2x) + cos(3x) =  = .\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the left side of the original equation is\r\n" );
document.write( "\r\n" );
document.write( "cos(x) + cos(2x) + cos(3x) + cos(4x) = 2*cos(2.5x)*cos(1.5x) + 2*cos(2.5x)*cos(0.5x) = 2*cos(2.5x)*(cos(1.5x) + cos(0.5x)).\r\n" );
document.write( "\r\n" );
document.write( "Hence, the original equation is equivalent to\r\n" );
document.write( "\r\n" );
document.write( "2*cos(2.5x)*(cos(1.5x) + cos(0.5x)) = 0,    or,  canceling  the factor  2*cos(2.5x),\r\n" );
document.write( "\r\n" );
document.write( "cos(1.5x) + cos(0.5x) = 0.                       (3)\r\n" );
document.write( "\r\n" );
document.write( "Again, apply the formula (2) to the left side of (3). You will get an equivalent equation\r\n" );
document.write( "\r\n" );
document.write( " = 0.                                (4)\r\n" );
document.write( "\r\n" );
document.write( "Equation (4) deploys in two independent separate equations:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "1.  cos(x) = 0  --->  x = ,  k = 0, +/-1, +/-2, . . . \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.  cos(x/2) = 0  --->  x = ,  k = 0, +/-1, +/-2, . . . \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, in the given interval the original equation has the roots 0, , ,   or  0°,  90°,  180°,  270°.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "But these are not ALL the roots.\r\n" );
document.write( "\r\n" );
document.write( "There is one more family of roots.\r\n" );
document.write( "\r\n" );
document.write( "Do you remember I canceled the factor 2*cos(2.5x) ?\r\n" );
document.write( "\r\n" );
document.write( "Of course, I must consider (and add !) all the solutions of the equation\r\n" );
document.write( "\r\n" );
document.write( "cos(2.5x) = 0.\r\n" );
document.write( "\r\n" );
document.write( "They are  2.5x = ,  k = 0, +/-1, +/-2, . . . \r\n" );
document.write( "\r\n" );
document.write( "or, which is the same,\r\n" );
document.write( "\r\n" );
document.write( " =  + ,  k = 0, +/-1, +/-2, . . . \r\n" );
document.write( "\r\n" );
document.write( "So, these additional solutions are x = , , ,   PLUS  , k = 0, +/-1, +/-2, . . . \r\n" );
document.write( "\r\n" );
document.write( "The final answer is:  There are two families of solutions. \r\n" );
document.write( "\r\n" );
document.write( "                      One family is 0°, 90°, 180° and 270°.\r\n" );
document.write( "\r\n" );
document.write( "                      The other family is 36°, 72°, 108°, 144°, 180°, 216°, 252°, 288°, 324°.\r\n" );
document.write( "

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

\n" );