document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205952>Question 1205952</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, re-asking this question as my coding didn't work last time. Use De Moivres Theorem to simplify (cos(5pi/6)-sin(5pi/6))^7. I have tried a few examples and can't seem to figure out this one. Thank you </SPAN>\n" );
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document.write( "De Moivre's Theorem:<BR>\n" );
document.write( "If z = r*cis(theta), then z^n = r^n*cis(n*theta)<BR>\n" );
document.write( "where cis(theta) = cos(theta)+i*sin(theta) and n is an integer.\r<BR>\n" );
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document.write( "I'll assume that your expression is <font color=blue>(cos(5pi/6)-i*sin(5pi/6))^7 </font><BR>\n" );
document.write( "I introduced the imaginary number 'i' just before the \"sin\"\r<BR>\n" );
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document.write( "The minus sign between those trig functions means we cannot immediately use De Moivre's Theorem just yet. We need to replace the minus sign with a plus sign.\r<BR>\n" );
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document.write( "We can use the idea that sine is an odd function. Odd as in \"not even\", though I suppose someone could argue that it is a strange function.<BR>\n" );
document.write( "Since sine is odd, we know that -sin(x) = sin(-x) for all real numbers x.\r<BR>\n" );
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document.write( "Cosine being even means cos(-x) = cos(x)<BR>\n" );
document.write( "It has y-axis symmetry.\r<BR>\n" );
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document.write( "z = cos(5pi/6) - i*sin(5pi/6)<BR>\n" );
document.write( "z = cos(5pi/6) + i*sin(-5pi/6) ... since sine is odd<BR>\n" );
document.write( "z = cos(-5pi/6) + i*sin(-5pi/6) ... since cosine is even<BR>\n" );
document.write( "z = cis(-5pi/6)\r<BR>\n" );
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document.write( "Now we can use De Moivre's Theorem<BR>\n" );
document.write( "z = cis(-5pi/6)<BR>\n" );
document.write( "z^7 = cis( 7*(-5pi/6) )<BR>\n" );
document.write( "z^7 = cis( -35pi/6 )<BR>\n" );
document.write( "z^7 = cis( pi/6 )\r<BR>\n" );
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document.write( "How did we go from -35pi/6 to pi/6?<BR>\n" );
document.write( "I'm using the fact that both sine and cosine have period 2pi.<BR>\n" );
document.write( "sin(x+2pi) = sin(x)<BR>\n" );
document.write( "cos(x+2pi) = sin(x)<BR>\n" );
document.write( "We can add and subtract multiples of 2pi to find coterminal angles.\r<BR>\n" );
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document.write( "Add 3 multiples of 2pi onto angle -35pi/6 radians to end up at pi/6<BR>\n" );
document.write( "-35pi/6 + 3*2pi = pi/6<BR>\n" );
document.write( "This is the same as rotating a full 360 degrees exactly three times. We end up pointing at the same direction we started at (somewhere in the northeast).\r<BR>\n" );
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document.write( "From here, we can use the <a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Unit_circle\">unit circle</a> to wrap things up.<BR>\n" );
document.write( "z^7 = cis( pi/6 )<BR>\n" );
document.write( "z^7 = cos( pi/6 ) + i*sin( pi/6 )<BR>\n" );
document.write( "z^7 = sqrt(3)/2 + i*(1/2)<BR>\n" );
document.write( "z^7 = sqrt(3)/2 + (1/2)*i<BR>\n" );
document.write( "z^7 = (1/2)*( sqrt(3) + i )\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "(cos(5pi/6)-i*sin(5pi/6))^7 = <font color=red>sqrt(3)/2 + (1/2)*i</font><BR>\n" );
document.write( "or<BR>\n" );
document.write( "(cos(5pi/6)-i*sin(5pi/6))^7 = <font color=red>(1/2)*( sqrt(3) + i )</font><BR>\n" );
document.write( "depending on how you prefer to write the answer.\r<BR>\n" );
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document.write( "Various computer algebra systems (CAS) can be used to verify the answer.<BR>\n" );
document.write( "Some examples include: WolframAlpha and GeoGebra.<BR>\n" );
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document.write( " </SPAN>\n" );
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