document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1115576>Question 1115576</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve the following problem <BR>\n" );
document.write( "Find all the values of z such that <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=exp%282z-1%29=1+ ALIGN=MIDDLE  ALT=\"exp%282z-1%29=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "My attempt at a solution.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282%28x%2Biy%29-1%29=1 ALIGN=MIDDLE  ALT=\"e%5E%282%28x%2Biy%29-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x-1%29+%2A+e%5E%28i2y%29=1 ALIGN=MIDDLE  ALT=\"e%5E%282x-1%29+%2A+e%5E%28i2y%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x-1%29+%2A%28cos2y+%2Bi+sin2y%29=+1 ALIGN=MIDDLE  ALT=\"e%5E%282x-1%29+%2A%28cos2y+%2Bi+sin2y%29=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x-1%29+%2Acos2y+%2B+e%5E%282x-1%29+%2Asin2y=+1 ALIGN=MIDDLE  ALT=\"e%5E%282x-1%29+%2Acos2y+%2B+e%5E%282x-1%29+%2Asin2y=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Pairing the real and imaginary parts  gives \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x-1%29+%2Acos2y+=+1+ ALIGN=MIDDLE  ALT=\"e%5E%282x-1%29+%2Acos2y+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ----------- (1)\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x-1%29+%2Asin2y+=+0+ ALIGN=MIDDLE  ALT=\"e%5E%282x-1%29+%2Asin2y+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ----------- (2) \r<BR>\n" );
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document.write( "Then \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin2y=0 ALIGN=MIDDLE  ALT=\"sin2y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=+pi++ ALIGN=MIDDLE  ALT=\"+y=+pi++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "since Im trying to find a value of of y such that <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sin2y=0+ ALIGN=MIDDLE  ALT=\"+sin2y=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+1%2F2++ ALIGN=MIDDLE  ALT=\"x=+1%2F2++\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+z=+1%2F2+%2B+i%28pi%29+%2B+i%282pi%29n ALIGN=MIDDLE  ALT=\"+z=+1%2F2+%2B+i%28pi%29+%2B+i%282pi%29n\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "but the book gives the answer as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+z=+1%2F2+%2B+i%28pi%29n ALIGN=MIDDLE  ALT=\"+z=+1%2F2+%2B+i%28pi%29n\"   DISABLED_style= \"max-width:90%; height:auto;\" > so I know I'm messing up the imaginary part I'm just not sure how .\r<BR>\n" );
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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 #730413 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=730413\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> At the step  sin(2y) = 0,  you wrote <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=pi+ ALIGN=MIDDLE  ALT=\"+y=pi+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  but  it is probably better to write  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%28n%2Api%29%2F2+ ALIGN=MIDDLE  ALT=\"+y+=+%28n%2Api%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <em>potential</em> solutions.   That solves eq (2), but we need to check this solution in eq (1):\r<BR>\n" );
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document.write( "In solving eq (1),  let n=0:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5E%282x-1%29%2Acos%282%2A%280%2Api%2F2%29%29+=+1+ ALIGN=MIDDLE  ALT=\"+e%5E%282x-1%29%2Acos%282%2A%280%2Api%2F2%29%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —>   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=1%2F2+ ALIGN=MIDDLE  ALT=\"+x=1%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "However, eq (1) must be checked for other values of n:<BR>\n" );
document.write( "n=1:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos%282%2A%281%2Api%2F2%29%29+=++cos%28pi%29+=+-1+ ALIGN=MIDDLE  ALT=\"+cos%282%2A%281%2Api%2F2%29%29+=++cos%28pi%29+=+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  thus n=1 is NOT a solution for eq 1<BR>\n" );
document.write( "n=2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos%282%2A%282%2Api%2F2%29%29+=+cos%282pi%29+=+1+ ALIGN=MIDDLE  ALT=\"+cos%282%2A%282%2Api%2F2%29%29+=+cos%282pi%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  n=2 is a solution<BR>\n" );
document.write( "n=3:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos%282%2A%283%2Api%2F2%29%29+=+cos%283pi%29+=+-1+ ALIGN=MIDDLE  ALT=\"+cos%282%2A%283%2Api%2F2%29%29+=+cos%283pi%29+=+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  n=3 is NOT a solution<BR>\n" );
document.write( "n=4:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos%282%2A%284%2Api%2F2%29%29+=+cos%284pi%29+=+1+ ALIGN=MIDDLE  ALT=\"+cos%282%2A%284%2Api%2F2%29%29+=+cos%284pi%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   n=4 is a solution<br>\r<BR>\n" );
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document.write( "The pattern is the solution to eq (1) with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=n%2Api%2F2+ ALIGN=MIDDLE  ALT=\"+y=n%2Api%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is only valid for  even n:<BR>\n" );
document.write( "Thus,  the overall solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+z+=+%281%2F2%29+%2B+i%2An%2Api%2F2+ ALIGN=MIDDLE  ALT=\"+z+=+%281%2F2%29+%2B+i%2An%2Api%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   n = 0, 2, 4, 6, ….   <br>\r<BR>\n" );
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document.write( "Which reduces to the equivalent (letting m = n/2 just to highlight this step):<BR>\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+z+=+%281%2F2%29+%2B+i%2Am%2Api+ ALIGN=MIDDLE  ALT=\"+z+=+%281%2F2%29+%2B+i%2Am%2Api+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   m = 0,1,2,3,….  <br>\r<BR>\n" );
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