Why can I cancel this \(\displaystyle\sum_{{i}}{2}{{\cos}^{{3}}{\left(\alpha_{{i}}\right)}}{{\sin{\alpha}}_{{i}}}\) term

Payton Benson

Payton Benson

Answered question


Why can I cancel this i2cos3(αi)sinαi term for pairs 180degrees apart?
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 cos3αi for 0 degree and 180 degree I get a value of 0. Similarly, if I sum the values of cos3αi 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 cos3αi will be 0. Therefore, we can cancel out the K12i2cos3αisinαi term of Eqn(1). However, if I were to make the same sums with the term i2cos3αisinαi I see that I do not get values of zero.

Answer & Explanation

Veronica Riddle

Veronica Riddle

Beginner2022-03-13Added 9 answers

Note that
and this last sum telescopes to 0 by the identity cos(180y)=cosy
The same happens with cos3αsinα since the sign stays the same because of the identity sin(180y)=siny

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