document.write( "Question 1195997: cos(2\alpha +(\pi )/(4))=3tan(2\alpha +(\pi )/(4)) \n" ); document.write( "

Algebra.Com's Answer #828651 by ikleyn(52802) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "I read this equation in this way\r\n" );
document.write( "\r\n" );
document.write( "     = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Notice that the argument under cosine function is the same as under the tangent function.\r\n" );
document.write( "\r\n" );
document.write( "So, I introduce new variable\r\n" );
document.write( "\r\n" );
document.write( "    x = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the given equation takes the form\r\n" );
document.write( "\r\n" );
document.write( "    cos(x) = 3*tan(x).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It implies\r\n" );
document.write( "\r\n" );
document.write( "    cos^2(x) = 3*sin(x),\r\n" );
document.write( "\r\n" );
document.write( "    1-sin^2(x) = 3*sin(x)\r\n" );
document.write( "\r\n" );
document.write( "    sin^2(x) + 3*sin(x) - 1 = 0\r\n" );
document.write( "\r\n" );
document.write( "It is a quadratic equation relative sin(x).  Use the quadratic formula\r\n" );
document.write( "\r\n" );
document.write( "    sin(x) =  = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Since sin(x) must be between -1 and 1, only the root sin(x) =  = 0.30278  really fits.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence,  there are two solutions for x:\r\n" );
document.write( "\r\n" );
document.write( "    x = arcsin(0.30278) = 0.3076  OR  x =  = 3.14159 - 0.3076 = 2.834 (rounded).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It gives two solutions for :\r\n" );
document.write( "\r\n" );
document.write( "    (a)   =  =  = -0.2389  (rounded);\r\n" );
document.write( "\r\n" );
document.write( "    (b)   =   =   =  1.0243  (rounded).\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" ); document.write( "

\n" );