document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138879>Question 1138879</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Express 2 - 2i in polar form using principal argument </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756728 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=756728\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Express <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+-+2i ALIGN=MIDDLE  ALT=\"2+-+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" > in polar form using principal argument \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2+-+2i ALIGN=MIDDLE  ALT=\"z=2+-+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=x+-+yi ALIGN=MIDDLE  ALT=\"z=x+-+yi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The modulus (or length) of the complex number is simply given by using Pythagoras' theorem to find the hypotenuse, ie\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=r%2Acos%28theta%29%2Br%2Ai%2Asin%28theta%29 ALIGN=MIDDLE  ALT=\"z=r%2Acos%28theta%29%2Br%2Ai%2Asin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=r%28cos%28theta%29%2Bi%2Asin%28theta%29%29 ALIGN=MIDDLE  ALT=\"z=r%28cos%28theta%29%2Bi%2Asin%28theta%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=z=2+-+2i ALIGN=MIDDLE  ALT=\"z=2+-+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2Acos%28theta%29=2 ALIGN=MIDDLE  ALT=\"r%2Acos%28theta%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=r%2Asin%28theta%29=-2 ALIGN=MIDDLE  ALT=\"r%2Asin%28theta%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=tan%28theta%29=-2%2F2=-1 ALIGN=MIDDLE  ALT=\"tan%28theta%29=-2%2F2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "will be easier to find acute angle in quadrant I whose tan is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"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=alpha=tan%5E%28-1%29%281%29=pi%2F4 ALIGN=MIDDLE  ALT=\"alpha=tan%5E%28-1%29%281%29=pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=45 ALIGN=MIDDLE  ALT=\"alpha=45\"   DISABLED_style= \"max-width:90%; height:auto;\" >°\r<BR>\n" );
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document.write( "now find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=sqrt%282%5E2%2B%28-2%29%5E2%29 ALIGN=MIDDLE  ALT=\"r=sqrt%282%5E2%2B%28-2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=sqrt%288%29 ALIGN=MIDDLE  ALT=\"r=sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=sqrt%284%2A2%29 ALIGN=MIDDLE  ALT=\"r=sqrt%284%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=2sqrt%282%29 ALIGN=MIDDLE  ALT=\"r=2sqrt%282%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=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in quadrant IV <BR>\n" );
document.write( "=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=360-alpha=360-45=315 ALIGN=MIDDLE  ALT=\"theta=360-alpha=360-45=315\"   DISABLED_style= \"max-width:90%; height:auto;\" >° or <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=2pi-alpha=2pi-pi%2F4=7pi%2F4 ALIGN=MIDDLE  ALT=\"theta=2pi-alpha=2pi-pi%2F4=7pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> we can also choose to use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=-alpha ALIGN=MIDDLE  ALT=\"theta=-alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is also in quadrant IV\r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=45 ALIGN=MIDDLE  ALT=\"theta=45\"   DISABLED_style= \"max-width:90%; height:auto;\" >° or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=-pi%2F4 ALIGN=MIDDLE  ALT=\"theta=-pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2sqrt%282%29%28cos%28-pi%2F4%29%2Bi%2Asin%28-pi%2F4%29%29 ALIGN=MIDDLE  ALT=\"z=2sqrt%282%29%28cos%28-pi%2F4%29%2Bi%2Asin%28-pi%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >......since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28-pi%2F4%29=cos%28pi%2F4%29 ALIGN=MIDDLE  ALT=\"cos%28-pi%2F4%29=cos%28pi%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-pi%2F4%29=+-sin%28pi%2F4%29 ALIGN=MIDDLE  ALT=\"sin%28-pi%2F4%29=+-sin%28pi%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2sqrt%282%29%28cos%28pi%2F4%29-i%2Asin%28pi%2F4%29%29 ALIGN=MIDDLE  ALT=\"z=2sqrt%282%29%28cos%28pi%2F4%29-i%2Asin%28pi%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "or using<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=45 ALIGN=MIDDLE  ALT=\"theta=45\"   DISABLED_style= \"max-width:90%; height:auto;\" >° , we have  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2sqrt%282%29%28cos%2845%29-i%2Asin%2845%29%29 ALIGN=MIDDLE  ALT=\"z=2sqrt%282%29%28cos%2845%29-i%2Asin%2845%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "also, using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=7pi%2F4 ALIGN=MIDDLE  ALT=\"theta=7pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=315 ALIGN=MIDDLE  ALT=\"theta=315\"   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=z=2sqrt%282%29%28cos%287pi%2F4%29%2Bi%2Asin%287pi%2F4%29%29 ALIGN=MIDDLE  ALT=\"z=2sqrt%282%29%28cos%287pi%2F4%29%2Bi%2Asin%287pi%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2sqrt%282%29%28cos%28315%29%2Bi%2Asin%28315%29%29 ALIGN=MIDDLE  ALT=\"z=2sqrt%282%29%28cos%28315%29%2Bi%2Asin%28315%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );