document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=214062>Question 214062</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions of tan x = 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #161682 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=161682\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> tan x = 1<BR>\n" );
document.write( "tan 45 degrees = 1 *****<BR>\n" );
document.write( "add 90 degrees to this and you get 135 degrees<BR>\n" );
document.write( "tan 135 degrees = -1<BR>\n" );
document.write( "add 90 degrees to this and you get 225 degrees<BR>\n" );
document.write( "tan 225 degrees = 1 *****<BR>\n" );
document.write( "add 90 degrees to this and you get 315 degrees<BR>\n" );
document.write( "tan 315 degrees = -1<BR>\n" );
document.write( "add 90 degrees to this and you get 405 degrees<BR>\n" );
document.write( "now:<BR>\n" );
document.write( "405 degrees - 360 degrees = 45 degrees and you're back where you started.<BR>\n" );
document.write( "every 360 degrees you will be back where you started.<BR>\n" );
document.write( "135 degrees + 360 = 495 degrees.<BR>\n" );
document.write( "this is the same place on the graph.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your question was:<BR>\n" );
document.write( "find all solutions of tan(x) = 1<BR>\n" );
document.write( "looking at the above, we see that tan(x) = 1 when:<BR>\n" );
document.write( "x = 45 degrees<BR>\n" );
document.write( "x = (180 + 45) = 225 degrees<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "If you subtract 180 from 45 degrees you will get:<BR>\n" );
document.write( "-135 degrees.<BR>\n" );
document.write( "tan -135 degrees = 1<BR>\n" );
document.write( "this is not to be unexpected because -135 degrees is equivalent to 225 degrees.<BR>\n" );
document.write( "just add 360 to it and you get:<BR>\n" );
document.write( "-135 + 360 = 225 degrees.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "whether you add 180 to 45 degrees and keep adding 180 to the result you will get either the equivalent of 45 degrees or 225 degrees.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "whether you subtract 180 from 45 degrees and keep subtracting 180 from the result you will get either the equivalent of 45 degrees or 225 degrees.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "why is tan (x) equal to 1 at 45 and 225 degrees and equal to -1 at the others?<BR>\n" );
document.write( "it has to do with the sign of the result.<BR>\n" );
document.write( "tan (x) = opposite divided by hypotenuse = y/x on the graph because the angle being measured is between the x axis and the line from the origin to the point (x,y).<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "on the graph, x is the horizontal direction and y is the vertical direction.<BR>\n" );
document.write( "in the first quadrant (45 degrees), x and y are both positive.  divide a + by a + and you get a +.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "in the second quadrant (135 degrees), x is negative and y is positive.  divide a - by a + and you get a -.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "in the third quadrant (225 degrees), x is negative and y is negative.  divide a - by a - and you get a +.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "in the fourth quadrant (315 degrees), x is positive and y is negative.  divide a + by a - and you get a -.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "i have some pictures that will show you what is happening.<BR>\n" );
document.write( "go to the following website and click on problem number 214062.  There may be more than one picture up there.  You will see them as 214062.1, 214062.2, etc.<BR>\n" );
document.write( "click on each in turn if that is the case.  if the pictures aren't there when you look, wait 30 minutes and try again.  they will be there.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the website to go to is:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "http://theo.x10hosting.com/<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your answer is:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "45 + (180*n) for all real integer values of n<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "you could probably get away with all integer values of n since I suppose they are all real.  I can't think of an instance where they wouldn't.  <BR>\n" );
document.write( "-----<BR>\n" );
document.write( "here's how that would work:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "45 + (180*0) = 45 which is good.<BR>\n" );
document.write( "45 + (180*1) = 45 + 180 = 225 which is good.<BR>\n" );
document.write( "45 + (180*2) = 45 + 360 = 405 which is equivalent to 45 so that is good.<BR>\n" );
document.write( "45 + (180*3) = 45 + 540 = 585 which is equivalent to 225 so that is good.<BR>\n" );
document.write( "45 - (180*1) = 45 - 180 = -135 which is equivalent to 225 so that is good.<BR>\n" );
document.write( "45 - (180*2) = 45 - 360 = -315 which is equivalent to 45 so that is good.<BR>\n" );
document.write( "etc.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "pictures to follow:<BR>\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );