document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208325>Question 1208325</A>: <I><SPAN STYLE=\"word-break: break-all;\"> True or False: If S, P, and A are the cube roots of a complex number, then Arg(S)+Arg(P)=2×Arg(A). </SPAN>\n" );
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document.write( "Answer: <font color=red>False</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Let's look at a counterexample. Consider the cube roots of 16.<BR>\n" );
document.write( "We wish to solve z^3 = 16 or z^3-16 = 0.<BR>\n" );
document.write( "According to the <a  rel=nofollow HREF=\"https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/06%3A_The_Polar_System/6.05%3A_De_Moivre's_and_the_nth_Root_Theorem\">Nth Root Theorem</a> we find the 3 cube roots are<BR>\n" );
document.write( "S = 16^(1/3)*cis(0)<BR>\n" );
document.write( "P = 16^(1/3)*cis(120)<BR>\n" );
document.write( "A = 16^(1/3)*cis(240)<BR>\n" );
document.write( "where cis(x) = cos(x)+i*sin(x) and the angle mode is in degrees.<BR>\n" );
document.write( "If you want to convert to radians, then,<BR>\n" );
document.write( "120 degrees = 2pi/3 radians<BR>\n" );
document.write( "240 degrees = 4pi/3 radians<BR>\n" );
document.write( "The order of the cube roots doesn't matter. Try out different permutations to see why. \r<BR>\n" );
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document.write( "Arg(S)+Arg(P) = 2*Arg(A)<BR>\n" );
document.write( "0+120 = 2*240<BR>\n" );
document.write( "120 = 480<BR>\n" );
document.write( "Reaching a false statement proves that the original equation is <font color=red>false</font>.\r<BR>\n" );
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document.write( "A bit of extra info:<BR>\n" );
document.write( "Notice how angles 120 and 480 are coterminal. This is because 120+360=480. <BR>\n" );
document.write( "If you were to involve mod 360 then the two sides would agree on the same number and the equation would be true. <BR>\n" );
document.write( "To learn more, search \"modular arithmetic\".<BR>\n" );
document.write( "</font> </SPAN>\n" );
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