How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

Question

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:

$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$

There is a slight difference in case of $\sin$, which is: $$\sum_{k=0}^{n-1}\sin (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin\biggl( \frac{2 a + (n-1)\cdot d}{2}\biggr)$$

How do we prove the above two identities?

Answer 1

Let $$ S = \sin{(a)} + \sin{(a+d)} + \cdots + \sin{(a+nd)}$$ Now multiply both sides by $\sin\frac{d}{2}$. Then you have $$S \times \sin\Bigl(\frac{d}{2}\Bigr) = \sin{(a)}\sin\Bigl(\frac{d}{2}\Bigr) + \sin{(a+d)}\cdot\sin\Bigl(\frac{d}{2}\Bigr) + \cdots + \sin{(a+nd)}\cdot\sin\Bigl(\frac{d}{2}\Bigr)$$

Now, note that $$\sin(a)\sin\Bigl(\frac{d}{2}\Bigr) = \frac{1}{2} \cdot \biggl[ \cos\Bigl(a-\frac{d}{2}\Bigr) - \cos\Bigl(a+\frac{d}{2}\Bigr)\biggr]$$ and $$\sin(a+d) \cdot \sin\Bigl(\frac{d}{2}\Bigr) = \frac{1}{2} \cdot \biggl[ \cos\Bigl(a + d -\frac{d}{2}\Bigr) - \cos\Bigl(a+d+\frac{d}{2}\Bigr) \biggr]$$

Then by doing the same thing you will have some terms cancelled out. You can easily see which terms are going to get Cancelled. Proceed and you should be able to get the formula.

I tried this by seeing this post. This has been worked for the case when $d=a$. Just take a look here:

• https://web.archive.org/web/20180927043531/http://mathforum.org/library/drmath/view/72777.html

Answer 2

Here's a trigonograph for $a = 0$ and $d = 2\theta$:

Answer 3

Writing $\cos x = \frac12 (e^{ix} + e^{-ix})$ will reduce the problem to computing two geometric sums.

Answer 4

From Euler's Identity we know that $\cos (a+kd) = \text{Re}\{e^{i(a+kd)}\}$ and $\sin (a+kd) = \text{Im}\{e^{i(a+kd)}\}$.$\,$ Thus,

$$\begin{align} \sum_{k=0}^{n-1} \cos (a+kd) &= \sum_{k=0}^{n-1} \text{Re}\{e^{i(a+kd)}\}\\\\ &=\text{Re}\left(\sum_{k=0}^{n-1} e^{i(a+kd)}\right)\\\\ &=\text{Re}\left(e^{ia} \sum_{k=0}^{n-1} (e^{id})^{k} \right)\\\\ &=\text{Re} \left( e^{ia} \frac{1-e^{idn}}{1-e^{id}}\right) \\\\ &=\text{Re} \left( e^{ia} \frac{e^{idn/2}(e^{-idn/2}-e^{idn/2})}{e^{id/2}(e^{-id/2}-e^{id/2})}\right) \\\\ &=\frac{\cos(a+(n-1)d/2)\sin(nd/2)}{\sin(d/2)} \end{align}$$

as was to be shown. Likewise for the sine function identity, follow the same procedure and take the imaginary part of the sum rather than the real part.

Answer 5

This is similar to the currently accepted answer, but more straightforward. You can use the trig identity \begin{equation*} \sin(\alpha + \beta) - \sin(\alpha - \beta) = 2\sin \beta \cos \alpha. \end{equation*}

Let $a_n = a + 2dk$ be an arithmetic sequence of difference $2d$, and set $b_n = a_n - d = a + d(2k - 1)$. Note that $\{b_n\}$ is also an arithmetic sequence of difference $2d$, hence $a_n + d = b_n + 2d = b_{n + 1}$. Therefore

\begin{equation*} 2 \sin d \cos a_n = \sin(a_n + d) - \sin(a_n - d) = \sin b_{n + 1} - \sin b_n. \end{equation*}

Summing both sides from $0$ to $n$ yields

\begin{align*} 2 \sin d \sum_{k = 0}^n \cos a_k &= \sin b_{n + 1} - \sin b_0 \\ &= \sin(a + d(2n + 1)) - \sin(a - d). \end{align*}

From our original trig identity, \begin{equation*} 2\sin((n + 1)d) \cos(a + nd) = \sin(a + d(2n + 1)) - \sin(a - d). \end{equation*} Thus, if $\sin d \neq 0$, we can rewrite our result as \begin{equation*} \sum_{k = 0}^n \cos (a + 2dk) = \frac{\sin((n + 1)d) \cos(a + nd)}{\sin d}. \end{equation*} This is OP's formula with $2d$ and $n$ instead of $d$ and $n - 1$. A similar process will yield the formula for $\sum_{k = 0}^n \sin(a + 2dk)$.

Answer 6

Small comment:

If we have one of the identities then we can derive the other!

Consider:

$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$

Take the derivative on both sides with $a$ while keeping everything else constant:

$$\sum_{k=0}^{n-1}\sin (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$

Answer 7

I did this by using $\Large{\frac{e^{i\theta}+e^{-i\theta}}{2}}$ formula to their corresponding cos terms

L.H.S

=$\Large{cos{(\theta)}+cos{(\theta+\alpha)}+......+cos(\theta+n\alpha)}$

=$\Large{\frac{e^{i\theta}}{2}(1+e^{i\alpha}+e^{i2\alpha}+......+e^{in\alpha})+\frac{e^{-i\theta}}{2}(1+e^{-i\alpha}+e^{-i2\alpha}+......+e^{-in\alpha})}$

using geometric progression formula

=$\Large{\frac{e^{i\theta}}{2}(\frac{e^{i\alpha(n+1)}-1}{e^{i\alpha}-1})+\frac{e^{i\theta}}{2}(\frac{1-e^{-i\alpha(n+1)}}{1-e^{-i\alpha}})}$

=$\Large{\frac{e^{i\theta}}{2}(\frac{e^{i\alpha(n+1)}-1}{e^{i\alpha}-1})+\frac{1}{2e^{(\theta+n\alpha)}}(\frac{e^{i\alpha(n+1)}-1}{e^{i\alpha}-1})}$

=$\Large{(\frac{e^{i\alpha(n+1)}-1}{e^{i\alpha}-1})(\frac{e^{i(2\theta+n\alpha)}+1}{2e^{i(\theta+n\alpha)}})}$ ........(1)

Now $\Large{(\frac{e^{i\alpha(n+1)}-1}{e^{i\alpha}-1})*\frac{(e^{-i\alpha}-1)}{(e^{-i\alpha}-1)}}$

=$\Large{\frac{e^{i\alpha n}-e^{-i\alpha}+1-e^{-i\alpha(n+1)}}{2-2cos(\alpha)}}$

and $\Large{(\frac{e^{i(2\theta+n\alpha)}+1}{2e^{i(\theta+n\alpha)}})*\frac{(2e^{-i(\theta+n\alpha)})}{(2e^{-i(\theta+n\alpha)})}}$

=$\Large{\frac{e^{i\theta}+e^{-i(\theta+\alpha n)}}{2}}$

So from (1)

$\Large{(\frac{e^{i\alpha n}-e^{-i\alpha}+1-e^{-i\alpha(n+1)}}{2-2cos(\alpha)})}*\Large{(\frac{e^{i\theta}+e^{-i(\theta+\alpha n)}}{2})}$

=$\Large{\frac{e^{i(\alpha n + \theta)}-e^{-i(\alpha n + \theta)}-e^{i(\theta-\alpha)}-e^{-i(\theta-\alpha)}+e^{i\theta}+e^{-i\theta}-e^{i(\theta+\alpha(n+1))}-e^{-i(\theta+\alpha(n+1))}}{4(1-cos(\alpha))}}$

=$\Large{\frac{cos(\alpha n+\theta)-cos(\theta-\alpha)+cos(\theta)-cos(\theta+\alpha(n+1))}{2(1-cos(\alpha))}}$ ..........(2)

Now

$cos(\theta)-cos(\theta-\alpha)=2sin(\frac{\alpha}{2}-\theta)(sin(\frac{\alpha}{2}))$

and

$cos(\alpha n+\theta)-cos(\theta+\alpha(n+1))=2sin(\frac{2\alpha n +\alpha+2\theta}{2})(sin(\frac{\alpha}{2}))$

also $(1-cos(\frac{\alpha}{2})=2sin^{2}(\frac{\alpha}{2}))$

from (2)

$\Large{\frac{sin(\frac{\alpha}{2}-\theta)+sin(\theta+\alpha n+\frac{\alpha}{2})}{2sin(\frac{\alpha}{2})}}$

=$\Large{\frac{sin(\frac{\alpha(n+1)}{2})cos(\theta+\frac{\alpha n}{2})}{sin(\frac{\alpha}{2})}}$

The sine series can also be done in this way

Answer 8

There's also a nice proof of the cosine formula by Michael Knapp in Mathematics Magazine (vol 82, number 5, December 2009, p. 371-372). Download a free version here: https://www.maa.org/sites/default/files/Knapp200941575.pdf

Knapp mentions in his article that a proof for the sine formula was given by Samuel Greitzer in the obscure student-oriented math journal Arbelos from 1986.