Question 1112816
.
using de'Moivres {{{highlight(cross(method))}}} formula evaluate sin(5theta)/sin(theta) and leave your answer in {{{highlight(terms_of)}}} cos(theta)
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This problem is for those students &nbsp;<U>who know complex numbers and Trigonometry very well</U>&nbsp; and &nbsp;<U>want to rise their level from good to perfect</U>.


I will show you how to solve it for more simple &nbsp;(but still very close) &nbsp;formulation:


<pre>
    Using de'Moivres formula evaluate  {{{sin(3a)/sin(a)}}}  and leave your answer in terms of cos(a).
</pre>

<U>Solution</U>


<pre>
Consider  complex number  z = cos(a) + i*sin(a).


According to de'Moivre formula,  


{{{z^3}}} = {{{(cos(a) + i*sin(a))^3}}} = {{{cos(3a) + i*sin(3a)}}}.    (1)


By applying Newton's binomial formula to  {{{(cos(a) + i*sin(a))^3}}}  and accounting that  {{{i^2}}} = -1,  you get


{{{(cos(a) + i*sin(a))^3}}} = {{{cos^3(a) + 3i*cos^2(a)*sin(a) + 3i^2*cos(a)*sin^2(a) + i^3sin^3(a)}}} = {{{cos^3(a) + 3i*cos^2(a)*sin(a) - 3*cos(a)*sin^2(a) - i*sin^3(a)}}}.    (2)


From (1) and (2),  you get for the imaginary part coefficients


sin(3a) = {{{3*cos^2(a)*sin(a)}}} - {{{sin^3(a)}}}.    (3)


Next step divide both sides of (3)  by sin(a).  You will get


{{{sin(3a)/sin(a)}}} = {{{3*cos^2(a) - sin^2(a)}}}.    (4)


Your last step is to replace {{{sin^2(a)}}} in the RHS of (4) by  {{{1-cos^2(a)}}} :


{{{sin(3a)/sin(a)}}} = {{{3*cos^2(a) - (1-cos^2(a))}}} = {{{4*cos^2(a)-1}}}.


<U>Answer</U>.  Using de'Moivre formula, we get  {{{sin(3a)/sin(a)}}} = {{{4*cos^2(a)-1}}}.
</pre>

Doing by the same way, &nbsp;you can solve the problem for &nbsp;&nbsp;{{{sin(5a)/sin(a)}}}.


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There is a bunch of my lessons on complex numbers in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetical operations on them</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson>Addition and subtraction of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Multiplication-and-division-of-complex-numbers-in-complex-plane-.lesson>Multiplication and division of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Raising-a-complex-number-to-an-integer-power.lesson>Raising a complex number to an integer power</A> (*)

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-root-of-a-complex-number.lesson>How to take a root of a complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-real-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with real coefficients on complex domain</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-square-root-of-a-complex-number.lesson>How to take a square root of a complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-complex-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with complex coefficients on complex domain</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-taking-roots-of-complex-numbers.lesson>Solved problems on taking roots of complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-arithmetic-operations-on-complex-numbers.lesson>Solved problems on arithmetic operations on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problem-on-taking-square-roots-of-complex-numbers.lesson>Solved problem on taking square root of complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Miscellaneous-problems-on-complex-numbers.lesson>Miscellaneous problems on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Advanced-problem-in-complex-numbers.lesson>Advanced problem on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/An-equation-in-complex-numbers-which-HAS-NO-a-solution.lesson>A curious example of an equation in complex numbers which HAS NO a solution</A>



On de'Moivre formula, &nbsp;see the lesson &nbsp;marked &nbsp;(*)&nbsp; in the list.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Complex numbers</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.