In another question one user helped me prove that the sum of three angles was a multiple of 360 degrees with formulas for sine and cosine sums of three angles. The sine formula was:
sin(α+β+γ)=sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ−sinαsinβsinγ
I infer that the pattern for five angles is as shown below? For brevity, I'm using a shorthand, e.g. sin(α+β+γ):s(a1+a2+a3) and \sin α\cos \beta\cos γ∶s_1 c_2 c_3. So, is the following the proper pattern for summing five angles?
s(a_1+a_2+a_3+a_4+a_5)=s_1 c_2 c_3 c_4 c_5+c_1 s_2 c_3 c_4 c_5+c_1 c_2 s_3 c_4 c_5+c_1 c_2 c_3 s_4 c_5+c_1 c_2 c_3 c_4 s_5-s_1 s_2 s_3 s_4 s_5
If so, I can also infer the pattern for cosine and use the patterns for any number of angles.
Answer
s(a_1+a_2+a_3) = s_1c_2c_3+c_1s_2c_3+c_1c_2s_3-s_1s_2s_3
c(a_1+a_2+a_3) = s(\frac \pi 2 -(a_1+a_2+a_3)) \\ =s((\frac \pi 2 -a_1)+(-a_2)+(-a_3)) \\= c_1c_2c_3 -s_1s_2c_3- s_1c_2s_3-c_1s_2s_3
And we also know ...
s(a_4+a_5) = s_4c_5+ c_4s_5
c(a_4+a_5) = c_4c_5- s_4s_5
So
s(a_1+a_2+a_3+a_4+a_5) \\ =( s_1c_2c_3+c_1s_2c_3+c_1c_2s_3-s_1s_2s_3) ( c_4c_5- s_4s_5) \\+ ( c_1c_2c_3 -s_1s_2c_3- s_1c_2s_3-c_1s_2s_3)(s_4c_5+ c_4s_5 )
There will be no cancellations so you will get 16 terms ( they will be the half of the 2^5=32 possible arrangements of sines and cosines having an odd number of sines )
The coefficient in front of each term having (2k+1) sines will be (-1)^k
So for 5 angles you would expect...
\binom 51=5 terms having 1 sine , coefficient = +1
\binom 53=10 terms having 3 sines , coefficient = -1
\binom 55=1 term having 5 sines , coefficient = +1
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