document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=616995>Question 616995</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write the following numbers in the polar form r(cos(theta)+isin(theta)) 0<(theta)<2pi. Thank you!\r<BR>\n" );
document.write( "\n" );
document.write( "1) 4i<BR>\n" );
document.write( "2) -4+2i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #388082 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=388082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In general<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%28a%5E2%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"r+=+sqrt%28a%5E2%2Bb%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=cos%28theta%29+=+a%2Fsqrt%28a%5E2%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+a%2Fsqrt%28a%5E2%2Bb%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=sin%28theta%29+=+b%2Fsqrt%28a%5E2%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+b%2Fsqrt%28a%5E2%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where the \"a\" and \"b\" come from the \"a + bi\" form of complex numbers.<br><BR>\n" );
document.write( "1) 4i<BR>\n" );
document.write( "First we write it in \"a + bi\" form:<BR>\n" );
document.write( "0 + 4i<BR>\n" );
document.write( "So the \"a\" = 0 and the \"b\" = 4<BR>\n" );
document.write( "Putting these values into the formulas:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%280%5E2%2B4%5E2%29+=+sqrt%280+%2B+16%29+=+sqrt%2816%29+=+4 ALIGN=MIDDLE  ALT=\"r+=+sqrt%280%5E2%2B4%5E2%29+=+sqrt%280+%2B+16%29+=+sqrt%2816%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29+=+0%2F4+=+0 ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+0%2F4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+4%2F4+=+1 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+4%2F4+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From our knowledge of special angles we should know what <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is. The angle between 0 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a cos of 0 and a sin of 1 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;\" >. So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4i+=+4%28cos%28pi%2F2%29%2Bi%2Asin%28pi%2F2%29%29 ALIGN=MIDDLE  ALT=\"4i+=+4%28cos%28pi%2F2%29%2Bi%2Asin%28pi%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "2) -4+2i<BR>\n" );
document.write( "This makes \"a\" = -4 and \"b\" = 2. Putting these values into the formulas:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%28%28-4%29%5E2%2B2%5E2%29+=+sqrt%2816%2B+4%29+=+sqrt%2820%29+=+sqrt%284%2A5%29+=+sqrt%284%29%2Asqrt%285%29+=+2%2Asqrt%285%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=cos%28theta%29+=+%28-4%29%2F2%2Asqrt%285%29+=+%28-2%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+%28-4%29%2F2%2Asqrt%285%29+=+%28-2%29%2Fsqrt%285%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%28theta%29+=+2%2F2%2Asqrt%285%29+=+1%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+2%2F2%2Asqrt%285%29+=+1%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From our knowledge of special angles we should know that none of the special angle values have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in them. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not a special angle. So we will need our calculators to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >. First we get decimals for cos and sin:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29+=+%28-4%29%2F2%2Asqrt%285%29+=+%28-2%29%2Fsqrt%285%29+=+-0.89442719 ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+%28-4%29%2F2%2Asqrt%285%29+=+%28-2%29%2Fsqrt%285%29+=+-0.89442719\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+2%2F2%2Asqrt%285%29+=+1%2Fsqrt%285%29+=+0.44721360 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+2%2F2%2Asqrt%285%29+=+1%2Fsqrt%285%29+=+0.44721360\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we use the inverse sin button on our calculator to find the reference angle. (Be sure your calculator is set for radian angles. If you don't know how to set the mode then find the angle in degrees and then multiply your answer by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F180 ALIGN=MIDDLE  ALT=\"pi%2F180\"   DISABLED_style= \"max-width:90%; height:auto;\" > to convert it to radians.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E%28-1%29%280.44721360%29+=+0.46364761 ALIGN=MIDDLE  ALT=\"sin%5E%28-1%29%280.44721360%29+=+0.46364761\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So our reference angle is 0.46364761 radians. With a cos that is negative and a sin that is positive, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > must terminate in the 2nd quadrant. To find an angle in the 2nd quadrant, you subtract the reference angle from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+pi+-+0.46364761+=+3.14159265+-+0.46364761+=+2.67794504 ALIGN=MIDDLE  ALT=\"theta+=+pi+-+0.46364761+=+3.14159265+-+0.46364761+=+2.67794504\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+%2B+2i+=+2%2Asqrt%285%29%28cos%282.67794504%29+%2B+i%2Asin%282.67794504%29%29 ALIGN=MIDDLE  ALT=\"-4+%2B+2i+=+2%2Asqrt%285%29%28cos%282.67794504%29+%2B+i%2Asin%282.67794504%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );