document.write( "Question 1156563: \cos\theta-\sqrt{3}\sin\theta=1 \n" ); document.write( "
Algebra.Com's Answer #779303 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            I will show you  TRULY  elegant  method of solution . . . \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "    \"cos%28theta%29\" - \"sqrt%283%29%2Asin%28theta%29\" = 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply both sides by \"1%2F2\".\r\n" );
document.write( "\r\n" );
document.write( "    \"%281%2F2%29%2Acos%28theta%29\" - \"%28sqrt%283%29%2F2%29%2Asin%28theta%29\" = \"1%2F2\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It can be written in this form\r\n" );
document.write( "\r\n" );
document.write( "    \"sin%28pi%2F6%29%2Acos%28theta%29\" - \"cos%28pi%2F6%29%2Asin%28theta%29\" = \"sin%28pi%2F3%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Use the formula of adding arguments for sine\r\n" );
document.write( "\r\n" );
document.write( "    \"sin%28%28pi%2F6%29+-+theta%29\" = \"sin%28pi%2F6%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It may happen in one of the two cases\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Case 1.  \"%28pi%2F6%29+-+theta\" = \"pi%2F6\".\r\n" );
document.write( "\r\n" );
document.write( "          which implies \"theta\" = 0,\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "Case 2.  \"%28pi%2F6%29+-+theta\" = \"pi\" - \"pi%2F6\",\r\n" );
document.write( "\r\n" );
document.write( "         which implies  \"theta\" = \"%282pi%29%2F6+-+pi\" = \"pi%2F3+-+pi\" = -\"%282pi%29%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "         Since we consider the angles in the interval [0,2pi), it is equivalent to  \"theta\" = \"%284pi%29%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we get the ANSWER :  The given equation has two solutions in the interval [0,2pi)\r\n" );
document.write( "\r\n" );
document.write( "                         a)  \"theta\" = 0,   and  b)  \"theta\" = \"%284pi%29%2F3\".\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "@greenestamps,  double check your answer:  it  DOES  NOT  correspond to your plot  (and should be corrected (!) )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "
\n" );