document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102550>Question 1102550</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I've forgotten how to do this.<BR>\n" );
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document.write( "cos(x) = 2<BR>\n" );
document.write( "Solve for x.<BR>\n" );
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document.write( "As an example from my book:<BR>\n" );
document.write( "cos(+/-arctan(2sqrt(6)/5) - i*ln(7) + n*2pi) = 10<BR>\n" );
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document.write( "--> cos(x) = 10<BR>\n" );
document.write( "x = +/-arctan(2sqrt(6)/5) - i*ln(7) + n*2pi<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717251 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=717251\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There must be an error or omission in the stated problem, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1+%3C=++cos%28x%29+++%3C=+1++ ALIGN=MIDDLE  ALT=\"+-1+%3C=++cos%28x%29+++%3C=+1++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "cos(x) = 2  has no solution,  real or complex  x.  <BR>\n" );
document.write( "—<BR>\n" );
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document.write( "Here is what happens if you try to solve cos(z) = 2  for  z complex:<BR>\n" );
document.write( "—<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos%28z%29+=+%281%2F2%29%28e%5E%28iz%29+%2B+e%5E%28-iz%29+%29+ ALIGN=MIDDLE  ALT=\"+cos%28z%29+=+%281%2F2%29%28e%5E%28iz%29+%2B+e%5E%28-iz%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   <— definition <BR>\n" );
document.write( "Let z=a+bi  where a,b are real and i is the imaginary unit\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos%28a%2Bbi%29+=+%281%2F2%29%28e%5E%28i%28a%2Bbi%29%29+%2B+e%5E%28-i%28a%2Bbi%29%29%29+ ALIGN=MIDDLE  ALT=\"+cos%28a%2Bbi%29+=+%281%2F2%29%28e%5E%28i%28a%2Bbi%29%29+%2B+e%5E%28-i%28a%2Bbi%29%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=+%281%2F2%29%28e%5E%28-b%29%29%28e%5E%28ai%29%29+%2B+%281%2F2%29%28e%5Eb%29%28e%5E%28-ai%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "—<BR>\n" );
document.write( "using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5E%28i%2Atheta%29+=+cos%28theta%29+%2B+i%2Asin%28theta%29+ ALIGN=MIDDLE  ALT=\"+e%5E%28i%2Atheta%29+=+cos%28theta%29+%2B+i%2Asin%28theta%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <— Euler's equation<BR>\n" );
document.write( "—<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281%2F2%29%28e%5E%28-b%29%29%28cos%28a%29+%2B+i%2Asin%28a%29%29+%2B+%281%2F2%29%28e%5Eb%29%28cos%28-a%29+%2B+i%2Asin%28-a%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now setting this last equation to 2  you get (real part is 2, imaginary part is 0) .  Start by setting the imaginary part to 0:<BR>\n" );
document.write( "Noting sin(-a) = -sin(a): <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281%2F2%29e%5E%28-b%29sin%28a%29+-+%281%2F2%29%28e%5Eb%29sin%28a%29++=+0+ ALIGN=MIDDLE  ALT=\"+%281%2F2%29e%5E%28-b%29sin%28a%29+-+%281%2F2%29%28e%5Eb%29sin%28a%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=++++e%5E%28-b%29+-+e%5Eb+=+0+ ALIGN=MIDDLE  ALT=\"++++e%5E%28-b%29+-+e%5Eb+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==>  b = 0\r<BR>\n" );
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document.write( "and for the real part:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++%281%2F2%29e%5E%28-b%29cos%28a%29+%2B+%281%2F2%29%28e%5Eb%29cos%28-a%29+=+2+ ALIGN=MIDDLE  ALT=\"++%281%2F2%29e%5E%28-b%29cos%28a%29+%2B+%281%2F2%29%28e%5Eb%29cos%28-a%29+=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "plugging in b=0, and noting cos(a) = cos(-a):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+++%281%2F2%29+cos%28a%29+%2B+%281%2F2%29cos%28a%29+=+2+ ALIGN=MIDDLE  ALT=\"+++%281%2F2%29+cos%28a%29+%2B+%281%2F2%29cos%28a%29+=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+++cos%28a%29+=+2+ ALIGN=MIDDLE  ALT=\"+++cos%28a%29+=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   which looks like we are back to the original problem….  EXCEPT now 'a' is REAL and we know this has no solution.<BR>\n" );
document.write( "—<BR>\n" );
document.write( "  \r<BR>\n" );
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document.write( " </SPAN>\n" );
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