document.write( "Question 388621: 1. Express 3cos(x°)+5sin(x°) in the form kcos(x°-a°), where k>0 and 0≤a≤360\r
\n" ); document.write( "\n" ); document.write( "Can you provide full working with the solutions please.\r
\n" ); document.write( "\n" ); document.write( "Many Thanks,
\n" ); document.write( "Andrew. \r
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "

Algebra.Com's Answer #275021 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "\"3cos%28x%29%2B5sin%28x%29\"
\n" ); document.write( "
\r\n" );
document.write( "Draw a right triangle with those coefficients as legs:\r\n" );
document.write( "\r\n" );
document.write( "horizontal leg = 3 and vertical leg = 5\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Calculate the hypotenuse using the Pythagorean theorem\r\n" );
document.write( "\r\n" );
document.write( "\"c%5E2=a%5E2%2Bb%5E2\"\r\n" );
document.write( "\"c%5E2=3%5E2%2B5%5E2\"\r\n" );
document.write( "\"c%5E2=9%2B25\"\r\n" );
document.write( "\"c%5E2=34\"\r\n" );
document.write( "\"c=sqrt%2834%29\"\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "\"sin%28theta%29+=+5%2Fsqrt%2834%29\" so \"5=sqrt%2834%29sin%28theta%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"cos%28theta%29+=+3%2Fsqrt%2834%29\" so \"3=sqrt%2834%29cos%28theta%29\"\r\n" );
document.write( "\r\n" );
document.write( "Substitute \"sqrt%2834%29cos%28theta%29\" for 3 and\r\n" );
document.write( "substitute \"sqrt%2834%29sin%28theta%29\" for 5 in\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"3cos%28x%29%2B5sin%28x%29\" =\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%2834%29cos%28theta%29cos%28x%29%2Bsqrt%2834%29sin%28theta%29sin%28x%29\" =\r\n" );
document.write( "\r\n" );
document.write( "factor out \"sqrt%2834%29\" on the left side:\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%2834%29%28cos%28theta%29cos%28x%29%2Bsin%28theta%29sin%28x%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "Now we remember the formula \"cos%28alpha-beta%29=cos%28alpha%29cos%28beta%29%2Bsin%28alpha%29sin%28beta%29\", and the above can be written:\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%2834%29%28cos%28theta%29cos%28x%29%2Bsin%28theta%29sin%28x%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%2834%29cos%28theta-x%29\"\r\n" );
document.write( "\r\n" );
document.write( "Since \"cos%28alpha%29+=+cos%28-alpha%29\", the above becomes:\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%2834%29cos%28-%28theta-x%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%2834%29cos%28-theta%2Bx%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%2834%29cos%28x-theta%29\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This is in the form  \"k%2Acos%28%22x%B0%22-%22a%B0%22%29\"\r\n" );
document.write( "\r\n" );
document.write( "where \"k=sqrt%2834%29\" and \"theta=arctan%285%2F3%29+=+%2259.03624347%B0%22\"\r\n" );
document.write( " \r\n" );
document.write( "So we have\r\n" );
document.write( "\r\n" );
document.write( "\"3cos%28x%29%2B5sin%28x%29\" = \"k%2Acos%28%22x%B0%22-%22a%B0%22%29\" = \"sqrt%2834%29%2Acos%28%22x%B0%22-%2259.03624347%B0%22%29\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "
\n" );