Polynomial question (1 Viewer)

RealiseNothing · Apr 2, 2012

Given that:

$z^n + z^{-n} = 2cos{n\theta}$

Where $z = cos{\theta} + sin{\theta}$

Show that:

$4cos{\theta}cos{2\theta}cos{3\theta} = 1 + cos{2\theta} + cos{4\theta} + cos{6\theta}$

This was from a past half yearly paper from my school.

deswa1 · Apr 2, 2012

$2cos\theta=z+\frac{1}{z}\\ 2cos2\theta=z^2+\frac{1}{z^2}\\ 2cos3\theta=z^3+\frac{1}{z^3}\\ \therefore 4cos\theta cos2\theta cos3\theta=\frac{1}{2}(z+\frac{1}{z})(z^2+\frac{1}{z^2})(z^3+\frac{1}{z^3})\\ =\frac{1}{2}(z^3+z+\frac{1}{z}+\frac{1}{z^3})(z^3+\frac{1}{z^3})\\ =\frac{1}{2}(z^6+z^4+1+z^2+1+\frac{1}{z^2}+\frac{1}{z^6}+\frac{1}{z^4})\\ =\frac{1}{2}(2cos6\theta+2cos4\theta+2cos2\theta+2)\\$
<img src="http://latex.codecogs.com/gif.latex?=cos6\theta+cos4\theta+cos2\theta+1" title="=cos6\theta+cos4\theta+cos2\theta+1" />

BTW, the question is wrong. It should be where z=cos(theta)+isin(theta)

RealiseNothing · Apr 2, 2012

Thanks deswa, repped.

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Polynomial question (1 Viewer)

RealiseNothing

what is that?It is Cowpea

deswa1

Well-Known Member

RealiseNothing

what is that?It is Cowpea

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