document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=14373>Question 14373</A>: <I><SPAN STYLE=\"word-break: break-all;\"> log (1 + cos x + i sin x)/(cos x – 1 – i sin x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #7224 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=7224\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  log <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281+%2B+cos+x+%2B+i%2A+sin+x%29%2F%28cos+x+-+1+-+i%2A+sin+x%29++ ALIGN=MIDDLE  ALT=\"+%281+%2B+cos+x+%2B+i%2A+sin+x%29%2F%28cos+x+-+1+-+i%2A+sin+x%29++\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " Multiplying the complex conjugate of the denominator.\r<BR>\n" );
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document.write( " Since z  = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281+%2B+cos+x+%2B+i%2A+sin+x%29%2F%28cos+x+-+1+-+i%2A+sin+x%29++ ALIGN=MIDDLE  ALT=\"+%281+%2B+cos+x+%2B+i%2A+sin+x%29%2F%28cos+x+-+1+-+i%2A+sin+x%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+%2B+cos+x+%2B+i%2A+sin+x%29%2A+%28cos+x+-+1+%2B+i%2A+sin+x%29%2F%0D%0A++%28%28cos+x+-+1+-+i%2A+sin+x%29%2A%28cos+x+-+1+%2B+i%2A+sin+x%29%29++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+%28cos+x%29%5E2+-1+-+%28sin+x%29%5E2+%2B+i+%2A+sin+x+%2A2%2Acos+x+%29%2F%0D%0A+++%28+%28cos+x+-+1%29%5E2+-+%28sin+x%29%5E2+%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+-2%2A%28sin+x%29%5E2+%2B+i+%2A+2+%2Asin+x+%2Acos+x+%29%2F%0D%0A+++%28+2%2A%28cos+x%29%5E2+-+2%2A+cosx+%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin+x+%28-+sin+x+%2B+i+cos+x+%29%29%2F+++%28cos+x+%28cos+x+-+1%29+%29+ ALIGN=MIDDLE  ALT=\"+%28sin+x+%28-+sin+x+%2B+i+cos+x+%29%29%2F+++%28cos+x+%28cos+x+-+1%29+%29+\"   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=sin+x%2F%28cos+x+%28cos+x+-+1%29+%29+ ALIGN=MIDDLE  ALT=\"sin+x%2F%28cos+x+%28cos+x+-+1%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+sin+x+%2B+i+cos+x%29+ ALIGN=MIDDLE  ALT=\"%28-+sin+x+%2B+i+cos+x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sin+x%2F%28cos+x+%281-cos+x%29+%29+ ALIGN=MIDDLE  ALT=\"+sin+x%2F%28cos+x+%281-cos+x%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+sin+x+-+i+cos+x+%29+ ALIGN=MIDDLE  ALT=\"%28+sin+x+-+i+cos+x+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin+x%2F%28cos+x+%281-cos+x+%29+%29 ALIGN=MIDDLE  ALT=\"sin+x%2F%28cos+x+%281-cos+x+%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cos%28x-+pi%2F2%29+%2B+i+sin+%28x-pi%2F2%29%29+ ALIGN=MIDDLE  ALT=\"+%28cos%28x-+pi%2F2%29+%2B+i+sin+%28x-pi%2F2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " (polar coordinates) ...(***)\r<BR>\n" );
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document.write( " The definition of log z for complex number may be beyond your level.\r<BR>\n" );
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document.write( " Anyway,I give it here.<BR>\n" );
document.write( " If the polar form for z = r <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28i%2Atheta%29+ ALIGN=MIDDLE  ALT=\"e%5E%28i%2Atheta%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( " Here, by (***) the given z =  <BR>\n" );
document.write( "(where the radius r = |sin x/(cos x (1-cos x ) )|, <BR>\n" );
document.write( " the argument <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+x-pi%2F2+ ALIGN=MIDDLE  ALT=\"theta+=+x-pi%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Hence, the natural logarithm ln(z).<BR>\n" );
document.write( " ln(z) = ln(r) + i * theta = ln|sin x| - ln|cosx (1-cos x) + i <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-pi%2F2%29 ALIGN=MIDDLE  ALT=\"%28x-pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " ... Answer<BR>\n" );
document.write( "  <BR>\n" );
document.write( " This is a very boring question about calculations, try your best to understand the details.\r<BR>\n" );
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document.write( " Kenny <BR>\n" );
document.write( " </SPAN>\n" );
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