Sum of Sine Series
Date: 07/17/2008 at 23:15:35 From: Lauren Subject: Sum of Sines and Cosines I am completely lost with how to even get started on this question! Show that for any integer n >= 1, the sum from k = 1 to k = n of sin(kt) = [cos(1/2t) - cos(n + 1/2)t]/[2 sin(1/2t)] The hint says to write sin(kt) sin(1/2t) as a linear combination of cosine functions, but I am unsure how to even do that. Is there a formula for sin kt or sin 1/2t that involves cosines? Will the half angle formula for cosines help? It involves a square root so that doesn't seem right. I do know the half angle formula. Also, I know that there is a formula for sin 2t, but is there a general formula for sin(kt)?
Date: 07/18/2008 at 10:19:48 From: Doctor Ali Subject: Re: Sum of Sines and Cosines Hi Lauren! Thanks for writing to Dr. Math. So you want to find, Sin(t) + Sin(2t) + Sin(3t) + ... + Sin(nt) Does that make sense? The trick is to multiply and divide the series by 2 Sin(t/2). That is, 2 Sin(t/2) [ Sin(t) + Sin(2t) + Sin(3t) + ... + Sin(nt) ] --------------------------------------------------------- 2 Sin(t/2) If you distribute that 2 Sin(t/2) to all terms in the numerator, you'll be able to write the expression in terms of cosines. For example, the first term will be: 2 Sin(t/2) Sin(t) We know that: 2 Sin(a) Sin(b) = Cos(a-b) - Cos(a+b) Are you familiar with the formula? So, by assuming a = t/2 b = t we can write 2 Sin(t/2) Sin(t) = Cos(t/2) - Cos(3t/2). For the second term we have: 2 Sin(t/2) Sin(2t) = Cos(3t/2) - Cos(5t/2) If you continue like this, you'll have: 2 Sin(t/2) Sin(t) = Cos(t/2) - Cos(3t/2) 2 Sin(t/2) Sin(2t) = Cos(3t/2) - Cos(5t/2) 2 Sin(t/2) Sin(3t) = Cos(5t/2) - Cos(7t/2) 2 Sin(t/2) Sin(4t) = Cos(7t/2) - Cos(9t/2) . . . 2 Sin(t/2) Sin(nt) = Cos((2n-1)t/2) - Cos((2n+1)t/2) Adding all of the above equalities together, you'll see that the terms on the right-hand sides will all cancel out, except Cos(t/2) and -Cos((2n+1)t/2). That is: n --- 2 Sin(t/2) ) Sin(kt) = Cos(t/2) - Cos((2n+1)t/2) --- k=1 Does that make sense? Please write back if you still have any difficulties. - Doctor Ali, The Math Forum http://mathforum.org/dr.math/
Date: 07/18/2008 at 15:44:04 From: Lauren Subject: Thank you (Sum of Sines and Cosines) Thank you so much for your help! This helps a lot! I am a middle school math teacher taking a graduate class and Ask Dr. Math is a huge help. Is there any way I can pay back by volunteering to answer some lower level questions?
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