All Questions
41
questions
1
vote
1
answer
148
views
Geometry formulas, how to show identities.
Given $d$ is integer:
How do I show:
$$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$
How do I rewrite and show, for $k$ is an integer:
$$ \sum_{p=1}^{d-1}\frac{\...
- 593
10
votes
3
answers
12k
views
Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$
State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$.
By letting $z=\cos\theta+i\sin\theta$, show that
$$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\...
- 1,927
2
votes
3
answers
14k
views
Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+....+\cos (n-1)x$ [duplicate]
By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ${1-\cos(\theta)+\cos(n-1)...
- 741
0
votes
1
answer
120
views
Prove that $\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$ if $n=1\mod 2$
I found out that this equality holds by accident,$$\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$$ if $n=1\mod 2$. However, I am not able to prove this directly with rules ...
- 2,996
3
votes
1
answer
1k
views
Understanding a step in applying deMoivre's Theorem to $\sum_{k=0}^n \cos(k\theta)$
I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far:
\begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} \...
- 231
1
vote
1
answer
5k
views
Summation of $\cos (2n-1) \theta$
By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that
$$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$
where $\sin\theta\neq0$
I ...
- 1,927
1
vote
4
answers
6k
views
Show $1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac{1}{2}+\frac{\sin[(n+1/2)θ]}{2\sin(θ/2)}$ [duplicate]
Show
$$1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac12+\frac{\sin\left(\left(n+\frac12\right)θ\right)}{2\sin\left(\frac\theta2\right)}$$
I want to use De Moivre's formula and $$1+z+z^2+\cdots+z^n=\frac{z^{n+...
- 47
25
votes
5
answers
2k
views
Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$
In old popular science magazine for school students I've seen problem
Prove that $\quad $
$\dfrac{1}{\cos^2 20^\circ} +
\dfrac{1}{\cos^2 40^\circ} +
\dfrac{1}{\cos^2 60^\circ} +
\dfrac{1}{\cos^...
- 17.4k
1
vote
2
answers
1k
views
Summing $\sum_{k=1}^{n-1} |1- e^{{2\pi ik}\over {n}}| $
I need to sum$$\sum_{k=1}^{n-1} |1- e^{{2\pi ik}\over {n}}| $$ which finally reduces to
$$\sum_{k=1}^{n-1} 2\sin\ {{\pi k} \over {n}}.$$
But I'm stuck here.The final answer is supposed to be $n$ .
- 744
2
votes
1
answer
399
views
A finite sum of trigonometric functions
By taking real and imaginary parts in a suitable exponential equation, prove that
$$\begin{align*}
\frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases}
1&\text{if } k \...
- 33
21
votes
7
answers
18k
views
Finite Sum $\sum\limits_{k=0}^{n}\cos(kx)$
I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$
I have some progress made, but I am stuck and could use some help.
What I did:
It ...
- 23.2k