document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=177505>Question 177505</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Give the polar and exponential forms of each of the complex numbers:<BR>\n" );
document.write( "a)6-2j<BR>\n" );
document.write( "b)-4j </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #132537 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=132537\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large r=\left| z \right|=\sqrt{a^2+b^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \varphi =\arg (z)=\arctan (\frac {b}{a})\">\r<BR>\n" );
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document.write( "So for 6 - 2j, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+6 ALIGN=MIDDLE  ALT=\"a+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-2 ALIGN=MIDDLE  ALT=\"b+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large r=\left| z \right|=\sqrt{6^2+(-2)^2}=sqrt {40}=2sqrt{10}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \varphi =\arg (z)=\arctan (\frac {-2}{6})=\arctan(-\frac{1}{3})\">\r<BR>\n" );
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document.write( "Polar form notation:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large z = r(\cos{\varphi} + i\sin{\varphi})\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large z = 2sqrt{10}(\cos{(\arctan(-\frac{1}{3}))} + j\sin{{(\arctan(-\frac{1}{3})))} \">\r<BR>\n" );
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document.write( "Exponential notation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \cos{\varphi} + i\sin{\varphi}=e^{i\varphi}\">, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large z = re^{i\varphi}\">\r<BR>\n" );
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document.write( "So for 6 - 2j, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large z = (2sqrt{10})e^{j\arctan(-\frac {1}{3})\">\r<BR>\n" );
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document.write( "For your other problem:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+0 ALIGN=MIDDLE  ALT=\"a+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-4 ALIGN=MIDDLE  ALT=\"b+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Do it the same way, except that arctan(b/a) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > because b/a is undefined. </SPAN>\n" );
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