document.write( "Question 635392: Hi, I'm terribly stuck with this question:\r
\n" ); document.write( "\n" ); document.write( "The function f is defined on the domain [0,pi] by f(x)=4cos(x)+3sin(x)
\n" ); document.write( "a) express in the form Rcos(x-y), where 0 < y < (pi)/2
\n" ); document.write( "b) Hence or otherwise, write down the value of x for which f(x) takes its maximum value\r
\n" ); document.write( "\n" ); document.write( "Thankyou in advance \n" ); document.write( "

Algebra.Com's Answer #400286 by AnlytcPhil(1806) $\"\"$ $\"About$

You can put this solution on YOUR website!
Hi, I'm terribly stuck with this question:\r
\n" ); document.write( "\n" ); document.write( "The function f is defined on the domain [0,pi] by f(x)=4cos(x)+3sin(x)
\n" ); document.write( "a) express in the form Rcos(x-y), where 0 < y < (pi)/2\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Thankyou in advance
\n" ); document.write( "

 \r\n" );
document.write( "4cos(x)+3sin(x) = R()[4cos(x)+3sin(x)] = R[(cos(x)+sin(x)] =\r\n" );
document.write( "\r\n" );
document.write( " R·cos(x-y) = R·[cos(x)cos(y)+sin(x)sin(y)]\r\n" );
document.write( "\r\n" );
document.write( "So\r\n" );
document.write( "\r\n" );
document.write( "R[(cos(x)+sin(x)] = R·[cos(x)cos(y)+sin(x)sin(y)]\r\n" );
document.write( "\r\n" );
document.write( "These will be equal if we choose y and R such that \r\n" );
document.write( "\r\n" );
document.write( " = cos(y) and  = sin(y)\r\n" );
document.write( "\r\n" );
document.write( "So we draw a right triangle with angle y, adjacent side 4, opposite side 3,\r\n" );
document.write( "and hypotenuse R:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So we choose y = arctan() = 0.6435\r\n" );
document.write( "and calculate R =  =  =  = 5\r\n" );
document.write( "\r\n" );
document.write( "Therefore \r\n" );
document.write( "\r\n" );
document.write( "f(x) = 4·cos(x)+3·sin(x) = R·cos(x-y) = 5·cos(x-0.6435)\r\n" );
document.write( "

\n" ); document.write( "b) Hence or otherwise, write down the value of x for which f(x) takes its maximum value
\n" ); document.write( "

\r\n" );
document.write( "f(x) will have maximum value when cos(x-0.6435) has maximum value.\r\n" );
document.write( "cos(q) has maximum value of 1 when q = 0, so we set\r\n" );
document.write( "\r\n" );
document.write( "              x-0.6435 = 0\r\n" );
document.write( "                     x = 0.6435\r\n" );
document.write( "Edwin

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

\n" );