document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=488238>Question 488238</A>: <I><SPAN STYLE=\"word-break: break-all;\"> sin 2x + cos 60 = 0, 0 < x < 360. <BR>\n" );
document.write( "i have been able to find out that sin inverse of -1/2 is -30, but how do i find answers in the 3rd and fourth quadrant for 2x?? then i can find x. my answers are coming wrong, and i can apply no suitable rule. please help, and show working if possible. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #333324 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=333324\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your equation is:<BR>\n" );
document.write( "sin (2x) + cos(60) = 0<BR>\n" );
document.write( "since cos(60) = 1/2, this equation becomes:<BR>\n" );
document.write( "sin (2x) + 1/2 = 0<BR>\n" );
document.write( "subtract 1/2 from both sides of this equation to get:<BR>\n" );
document.write( "sin (2x) = -(1/2)<BR>\n" );
document.write( "the sin is negative in the third and fourth quadrants only.<BR>\n" );
document.write( "solve the angle as if it was in the first quadrant.<BR>\n" );
document.write( "you get sin(2x) = 1/2<BR>\n" );
document.write( "this means that 2x = sin^-1(1/2) which becomes 2x = 30 degrees.<BR>\n" );
document.write( "your answer would be 2x = 30 degrees if the angle was in the first quadrant.<BR>\n" );
document.write( "the equivalent angle in the third quadrant would be 180 + 2x which would make the angle 210 degrees.<BR>\n" );
document.write( "the equivalent in the fourth quadrant would be 360 - 2x which would make the angle 330 degrees.<BR>\n" );
document.write( "your original equation becomes:<BR>\n" );
document.write( "sin(210) + cos(60) = 0<BR>\n" );
document.write( "sin(330) + cos(60) = 0<BR>\n" );
document.write( "solve both of these equations using your calculator and you'll see that they are both true.<BR>\n" );
document.write( "a picture of what we just did is shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2011/aug302.jpg\" alt = \"$$$$$\"/ ><BR>\n" );
document.write( "we first solved for the reference angle.<BR>\n" );
document.write( "that was the angle in the first quadrant.<BR>\n" );
document.write( "we got a positive value because we were in the first quadrant.<BR>\n" );
document.write( "we then determined that our value had to be negative.<BR>\n" );
document.write( "we had to know that the sine is negative in the third and fourth quadrant only.<BR>\n" );
document.write( "we then had to find an equivalent angle in the third and fourth quadrant.<BR>\n" );
document.write( "the formula to derive that is as follows:<BR>\n" );
document.write( "the reference angle is always in the first quadrant.<BR>\n" );
document.write( "if the angle is in the second quadrant, then the angle is equal to 180 minus the reference angle.<BR>\n" );
document.write( "if the angle is in the third quadrant, then the angle is equal to 180 plus the reference angle.<BR>\n" );
document.write( "if the angle is in the fourth quadrant, then the angle is equal to 360 minus the reference angle.<BR>\n" );
document.write( "we found the angle in the third and fourth quadrant because the sine is negative in the third and fourth quadrant.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );