document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086747>Question 1086747</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Perform the indicated operations using trigonometric form.  Please leave your answer in trigonometric form. (1+i)^4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #701015 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=701015\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">z = 1+i = 1+1*i is in the form a+b*i where a = 1 and b = 1\r<BR>\n" );
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document.write( "r = sqrt(a^2+b^2)<BR>\n" );
document.write( "r = sqrt(1^2+1^2)<BR>\n" );
document.write( "r=sqrt(2)\r<BR>\n" );
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document.write( "theta = arctan(b/a)<BR>\n" );
document.write( "theta = arctan(1/1)<BR>\n" );
document.write( "theta = arctan(1)<BR>\n" );
document.write( "theta = pi/4 radians (45 degrees)\r<BR>\n" );
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document.write( "If z = 1+i, then z = sqrt(2)*(cos(pi/4) + i*sin(pi/4)) is the trig form of the given complex number.<BR>\n" );
document.write( "Recall that z = r*(cos(theta)+i*sin(theta)) is the general trig form. \r<BR>\n" );
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document.write( "De Moivre's Theorem is the idea of raising a trig complex number to some integer power n\r<BR>\n" );
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document.write( "z = r*(cos(theta)+i*sin(theta))\r<BR>\n" );
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document.write( "z^n = [r*(cos(theta)+i*sin(theta))]^n\r<BR>\n" );
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document.write( "z^n = r^n*(cos(n*theta)+i*sin(n*theta))\r<BR>\n" );
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document.write( "Using De Moivre's Theorem, we can say\r<BR>\n" );
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document.write( "z^n = r^n*(cos(n*theta)+i*sin(n*theta))\r<BR>\n" );
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document.write( "z^4 = r^4*(cos(4*theta)+i*sin(4*theta))\r<BR>\n" );
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document.write( "z^4 = (sqrt(2))^4*(cos(4*pi/4)+i*sin(4*pi/4))\r<BR>\n" );
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document.write( "z^4 = 4*(cos(pi)+i*sin(pi))\r<BR>\n" );
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document.write( "z^4 = 4*(-1+i*0)\r<BR>\n" );
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document.write( "z^4 = 4*(-1+0)\r<BR>\n" );
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document.write( "z^4 = 4*(-1)\r<BR>\n" );
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document.write( "z^4 = -4\r<BR>\n" );
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document.write( "Earlier we defined z = 1+i and we just found out z^4 = -4\r<BR>\n" );
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document.write( "So in conclusion, (1+i)^4 = -4\r<BR>\n" );
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document.write( "If you wish to leave the answer in trig form, then writing it as 4*(cos(pi)+i*sin(pi)) should be fine (though it won't be fully simplified)</font> </SPAN>\n" );
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