Computing the trigonometric sum $ \sum_{j=1}^{n} \cos(j) $

Question

I have a task to compute such a sum:

$$ \sum_{j=1}^{n} \cos(j) $$

Of course I know that the answer is $$ \frac{1}{2} (\cos(n)+\cot(\frac{1}{2}) \sin(n)-1) = \frac{\cos(n)}{2}+\frac{1}{2} \cot(\frac{1}{2}) \sin(n)-\frac{1}{2} $$ but I don't have any idea how to start proving it.

Answer 1

You could also use $\sin(n+\frac{1}{2})-\sin(n-\frac{1}{2})=2\cos n\sin\frac{1}{2}$; $\;\;$ so if we let $a=2\sin\frac{1}{2}$,

$\cos n =\frac{1}{a}\left[\sin(n+\frac{1}{2})-\sin(n-\frac{1}{2})\right]\implies$

$\cos 1+\cos2+\cos 3+\cdots+\cos(n-1)+\cos n=$

$\frac{1}{a}\left[(\sin\frac{3}{2}-\sin\frac{1}{2})+(\sin\frac{5}{2}-\sin\frac{3}{2})+(\sin\frac{7}{2}-\sin\frac{5}{2})+\cdots+(\sin(n+\frac{1}{2})-\sin(n-\frac{1}{2}))\right]=$

$\frac{1}{a}\left[\sin(n+\frac{1}{2})-\sin\frac{1}{2}\right]=\frac{1}{2\sin\frac{1}{2}}\left[\sin n\cos\frac{1}{2}+\cos n\sin\frac{1}{2}-\sin\frac{1}{2}\right]$

$=\frac{1}{2}\left[\sin n\cot\frac{1}{2}+\cos n-1\right]$

Answer 2

Using the formula for a geometric series: $$\sum_{k=1}^Ne^{ik}=\frac{e^{i(N+1)}-1}{e^{i}-1}-1=\frac{e^{i(N+1)}-1}{e^{i}-1}\frac{e^{-i}-1}{e^{-i}-1}-1$$

Taking real parts : \begin{align} \sum_{k=1}^N\cos(k)&=\Re\left[\frac{\left(e^{i(N+1)}-1\right)\left(e^{-i}-1\right)}{2-2\cos(1)}\right]-1\\ &=\Re\left[\frac{e^{iN}-e^{i(N+1)}-e^{-i}+1}{2-2\cos(1)}\right]-1\\ &=\frac{\cos(N)-\cos(N+1)-\cos(1)+1}{2-2\cos(1)}-1\\ &=\frac{\cos(N)-\cos(N+1)}{2-2\cos(1)}-\frac{1}{2}\\ &=\frac{\cos(N)-\cos(N)\cos(1)+\sin(N)\sin(1)}{2-2\cos(1)}-\frac{1}{2}\\ &=\frac{\sin(N)\sin(1)}{2-2\cos(1)}-\frac{1}{2}+\frac{1}{2}\cos(N)\\ &=\frac{2\sin(N)\sin\left(\frac{1}{2}\right)\cos\left(\frac{1}{2}\right)}{4\sin^2\left(\frac{1}{2}\right)}-\frac{1}{2}+\frac{1}{2}\cos(N)\\ &=\frac{1}{2}\cot\left(\frac{1}{2}\right)\sin(N)-\frac{1}{2}+\frac{1}{2}\cos(N)\\ \end{align}

Answer 3

Let $$S_n = \sum_{j=1}^{n}\cos j.$$ Then: $$S_n\cdot \sin\frac{1}{2}=\frac{1}{2}\sum_{j=1}^{n}\left(\sin(j+1/2)-\sin(j-1/2)\right)=\frac{1}{2}\left(\sin(n+1/2)-\sin(1/2)\right).$$