document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=966873>Question 966873</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of each of the following. Enter an exact answer; decimal approximations will be marked incorrect.\r<BR>\n" );
document.write( "\n" );
document.write( "tan(arctan(41/15))\r<BR>\n" );
document.write( "\n" );
document.write( "arctan(tan(70&#960;/56))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #590982 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=590982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> by definition, tan(a) = b if and only if arctan(b) = a\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "therefore:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if tan(a) = b, then arctan(b) = a\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "tan(arctan(b)) is therefore equal to tan(a) which is equal to b.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this gets you tan(arctan(b)) = b.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you let b = 41/15, then you get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "tan(arctan(b)) = b becomes tan(arctan(41/15)) = 41/15.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that takes care of your first statement.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "now to move on to your second statement.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "by definition, arctan(c) = d if and only if tan(d) = c.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "therefore, if arctan(c) = d, then tan(d) = c.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "arctan(tan(d) is therefore equal to arctan(c) which is equal to d.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this gets you arctan(tan(d)) = d.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you let d = 70pi/56, then you get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "arctan(tan(70pi/56)) = 70pi/56.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can confirm these properties are correct by using your calculator.<BR>\n" );
document.write( "set your calculator to degree mode.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "tan(45) = 1<BR>\n" );
document.write( "arctan(1) = 45\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this leads to:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "tan(arctan(1)) = 1<BR>\n" );
document.write( "arctan(tan(45) = 45\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );