Payton Benson

2022-03-12

Why can I cancel this $\sum _{i}2{\mathrm{cos}}^{3}\left({\alpha}_{i}\right){\mathrm{sin}\alpha}_{i}$ term for pairs 180degrees apart?

I have the equation,

$K}_{11}\sum _{i}{\mathrm{cos}}^{4}{\alpha}_{i}+{K}_{12}\sum _{i}2{\mathrm{cos}}^{3}{\alpha}_{i}{\mathrm{sin}\alpha}_{i}+{K}_{22}\sum _{i}{\mathrm{sin}}^{2}{\alpha}_{i}{\mathrm{cos}}^{2}{\alpha}_{i}=\sum _{i}{\rho}_{i}{\mathrm{cos}}^{2}{\alpha}_{i$

In the paper it states, "For equal intervals in 180 degree segments the sums of odd powers are zero; therefore,"

$K}_{11}\sum _{i}{\mathrm{cos}}^{4}{\alpha}_{i}+{K}_{22}\sum _{i}{\mathrm{sin}}^{2}{\alpha}_{i}{\mathrm{cos}}^{2}{\alpha}_{i}=\sum _{i}{\rho}_{i}{\mathrm{cos}}^{2}{\alpha}_{i$

The way I interpret what they mean is that, if I sum the values of$\mathrm{cos}}^{3}{\alpha}_{i$ for 0 degree and 180 degree I get a value of 0. Similarly, if I sum the values of $\mathrm{cos}}^{3}{\alpha}_{i$ for 1 degrees and 181 degree I get a value of 0. And so on and so forth all the way up to 179 degree and 359 degree, the sum of the values of $\mathrm{cos}}^{3}{\alpha}_{i$ will be 0. Therefore, we can cancel out the $K}_{12}\sum _{i}2{\mathrm{cos}}^{3}{\alpha}_{i}{\mathrm{sin}\alpha}_{i$ term of Eqn(1). However, if I were to make the same sums with the term $\sum _{i}2{\mathrm{cos}}^{3}{\alpha}_{i}{\mathrm{sin}\alpha}_{i}$ I see that I do not get values of zero.

I have the equation,

In the paper it states, "For equal intervals in 180 degree segments the sums of odd powers are zero; therefore,"

The way I interpret what they mean is that, if I sum the values of

Veronica Riddle

Beginner2022-03-13Added 9 answers

Note that

$\sum _{\alpha =x}^{{180}^{\circ}+x}\mathrm{cos}\left(\alpha \right)=\sum _{\alpha -x=0}^{{180}^{\circ}}\mathrm{cos}(\alpha -x)=\sum _{y=0}^{{180}^{\circ}}\mathrm{cos}y$

and this last sum telescopes to 0 by the identity$\mathrm{cos}({180}^{\circ}-y)=-\mathrm{cos}y$

The same happens with$\sum {\mathrm{cos}}^{3}\alpha \mathrm{sin}\alpha$ since the sign stays the same because of the identity $\mathrm{sin}({180}^{\circ}-y)=\mathrm{sin}y$

and this last sum telescopes to 0 by the identity

The same happens with

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