Tuesday, January 24, 2017

trigonometry - Simplify a quick sum of sines





Simplify sin2+sin4+sin6++sin88



I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.


Answer



The angles are in arithmetic progression. Use the formula




n1k=0sin(a+kb)=sinnb2sinb2sin(a+(n1)b2)



See here for two proofs (using trigonometry, or using complex numbers).



In your case, a=b=2 and n=44.


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