document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=959041>Question 959041</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1-sinx = square root 3cosx <BR>\n" );
document.write( "On the interval (0, 360)<BR>\n" );
document.write( "Find all solutions  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #586231 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=586231\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-sin%28x%29+=+sqrt%283%29cos%28x%29 ALIGN=MIDDLE  ALT=\"1-sin%28x%29+=+sqrt%283%29cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We start by transforming this equation into one or more equations of the form:<BR>\n" );
document.write( "trigfunction(argument) = number\r<BR>\n" );
document.write( "\n" );
document.write( "One way to do this is to start by squaring both sides. This will allow us to \"convert\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"1-sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then, with all sin's we will be able to get the desired forms.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-sin%28x%29%29%5E2+=+%28sqrt%283%29cos%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%281-sin%28x%29%29%5E2+=+%28sqrt%283%29cos%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2sin%28x%29%2Bsin%5E2%28x%29+=+3cos%5E2%28x%29 ALIGN=MIDDLE  ALT=\"1-2sin%28x%29%2Bsin%5E2%28x%29+=+3cos%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Replacing the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2sin%28x%29%2Bsin%5E2%28x%29+=+3%281-sin%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"1-2sin%28x%29%2Bsin%5E2%28x%29+=+3%281-sin%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2sin%28x%29%2Bsin%5E2%28x%29+=+3-3sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"1-2sin%28x%29%2Bsin%5E2%28x%29+=+3-3sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since this is a quadratic equation for sin(x) we will solve it by getting one side to be zero:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4sin%5E2%28x%29-2sin%28x%29+-2+=+0 ALIGN=MIDDLE  ALT=\"4sin%5E2%28x%29-2sin%28x%29+-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And factoring:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282sin%5E2%28x%29-sin%28x%29+-1%29+=+0 ALIGN=MIDDLE  ALT=\"2%282sin%5E2%28x%29-sin%28x%29+-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282sin%28x%29%2B1%29%28sin%28x%29-1%29+=+0 ALIGN=MIDDLE  ALT=\"2%282sin%28x%29%2B1%29%28sin%28x%29-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the Zero Product Property:<BR>\n" );
document.write( "2sin(x)+1 = 0 or sin(x)-1 = 0<BR>\n" );
document.write( "Solving these for sin(x):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+-1%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or sin(x) = 1<BR>\n" );
document.write( "And we finally have the desired forms. (This is often the most difficult part of these problems.)<br><BR>\n" );
document.write( "Next we find the general solution for these equations. For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+-1%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we should recognize that 1/2 is a special angle value for sin and that the reference angle whose sin is 1/2 is 30 degrees. Since our sin is negative, we know that x terminates in either the 3rd or 4th quadrant. This gives us general solution equations of:<BR>\n" );
document.write( "x = 180+30 + 360n (for the 3rd quadrant)<BR>\n" );
document.write( "x = 360-30 + 360n (for the 4th quadrant)<BR>\n" );
document.write( "These simplify to:<BR>\n" );
document.write( "x = 210 + 360n (for the 3rd quadrant)<BR>\n" );
document.write( "x = 330 + 360n (for the 4th quadrant)<br><BR>\n" );
document.write( "For sin(x) = 1 we should recognize that this also is a special angle value. And we should know that only angles which terminate on the positive part of the y-axis, like 90 degrees, will have a sin of 1. So we get only one general solution for sin(x) = 1:<BR>\n" );
document.write( "x = 90 + 360n<br><BR>\n" );
document.write( "Before we leap into finding the solutions in the given interval, we must check them. We must check because we squared both sides of the equation and whenever that is done, solutions must be checked.<br><BR>\n" );
document.write( "Checking 210:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-sin%28210%29+=+sqrt%283%29cos%28210%29 ALIGN=MIDDLE  ALT=\"1-sin%28210%29+=+sqrt%283%29cos%28210%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-%28-1%2F2%29+=+sqrt%283%29%28-sqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"1-%28-1%2F2%29+=+sqrt%283%29%28-sqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2+=+-3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2+=+-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This does not check. So we reject 210 (and all coterminal angles from that general solution.)<br><BR>\n" );
document.write( "Checking 330:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-sin%28330%29+=+sqrt%283%29cos%28330%29 ALIGN=MIDDLE  ALT=\"1-sin%28330%29+=+sqrt%283%29cos%28330%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-%28-1%2F2%29+=+sqrt%283%29%28sqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"1-%28-1%2F2%29+=+sqrt%283%29%28sqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2+=+3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2+=+3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Check!<br><BR>\n" );
document.write( "Checking 90:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-sin%2890%29+=+sqrt%283%29cos%2890%29 ALIGN=MIDDLE  ALT=\"1-sin%2890%29+=+sqrt%283%29cos%2890%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-%281%29+=+sqrt%283%29%280%29 ALIGN=MIDDLE  ALT=\"1-%281%29+=+sqrt%283%29%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+0 ALIGN=MIDDLE  ALT=\"0+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Check!<br><BR>\n" );
document.write( "So our true general solution is:<BR>\n" );
document.write( "x = 330 + 360n<BR>\n" );
document.write( "x = 90 + 360n<br><BR>\n" );
document.write( "TO find specific solutions within the (0, 360) interval we replace the n's with various integers until we have found them all.<BR>\n" );
document.write( "From<BR>\n" );
document.write( "x = 330 + 360n<BR>\n" );
document.write( "when n = 0 we get 330 which is in the desired interval<BR>\n" );
document.write( "when n = 1 (or higher) we get an angle more than 360<BR>\n" );
document.write( "when n = -1 (or lower) we get an angle less than 0<BR>\n" );
document.write( "From<BR>\n" );
document.write( "x = 90 + 360n<BR>\n" );
document.write( "when n = 0 we get 90 which is in the desired interval<BR>\n" );
document.write( "when n = 1 (or higher) we get an angle more than 360<BR>\n" );
document.write( "when n = -1 (or lower) we get an angle less than 0<BR>\n" );
document.write( "So there are only two solutions in the given interval: 90 and 330. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );