All Questions
41
questions
2
votes
2
answers
97
views
How do I prove that $\sum_{i=1}^m \cos^2\left(\frac{2\pi i}{m}\right) = \sum_{i=1}^m \sin^2\left(\frac{2\pi i}{m}\right) = \frac{m}{2}$? [duplicate]
I haven't had any good ideas nor found any helpful identities so far, so I'd appreciate some help.
Also, here $m > 2$.
Update: Thanks to the hints and to this previous post I managed to get to ...
- 6,833
0
votes
1
answer
74
views
Trigonometry and Complex Numbers with Series [duplicate]
The number
$$\text{cis}75^\circ + \text{cis}83^\circ + \text{cis}91^\circ + \dots + \text{cis}147^\circ$$ is expressed in the form $r \, \text{cis } \theta,$ where $0 \le \theta < 360^\circ$. ...
- 61
6
votes
2
answers
2k
views
Prove that $\cos \frac{2\pi}{2n+1}+\cos \frac{4\pi}{2n+1}+\cos \frac{6\pi}{2n+1}+...+\cos \frac{2n\pi}{2n+1}=\frac{-1}{2}$
Prove that $$\cos \frac{2\pi}{2n+1}+\cos \frac{4\pi}{2n+1}+\cos \frac{6\pi}{2n+1}+...+\cos \frac{2n\pi}{2n+1}=\frac{-1}{2}$$
My attempt,
Let an equation $x^{2n+1}-1=0$, which has roots $$\cos \frac{...
- 7,798
1
vote
1
answer
2k
views
Find the sum of a geometric progression involving cos using complex numbers and proof trigonometric formula
I got the following problem:
I need to prove - using complex numbers:
$\sum_{t=0}^n \cos(tb) = \frac{\cos\frac{nb}{2}\sin\frac{nb+b}{2}}{\sin\frac{b}{2}}$
Ok so what I came up with so far:
we know ...
- 537
1
vote
2
answers
46
views
Using Euler and polynomials
I want to show that $\sum_{k=-N}^{N}e^{ikx}=\frac{\sin((N+\frac{1}{2})x)}{\sin(\frac{x}{2})}$ for $N\in \mathbb{N}$
Any tips on how to proceed?
I tried doing it in two ways:
First using the sum of ...
- 1,394
1
vote
3
answers
106
views
If $n>3$ prove that $\sum_{k=0}^{n-1} (k-n)\cos\frac{2k\pi}{n}=\frac{n}{2}$.
Do you have any ideas on this IIT exercise?
If $n>3$ is an integer, prove that
$$\sum_{k=0}^{n-1} (k-n)\cos(2kπ /n) = n/2$$
In my attempt, I have considered
$$z=cis(2kπ/n), k=[1, 2,..., n-1]...
- 514
1
vote
0
answers
487
views
A generalised formula for $\cos {n\theta}$ in terms of powers of $\cosθ$ using De Moivre's Theroem
I am trying to generalise a formula for $\cos{nθ}$ in terms of powers of $\cosθ$ using De Moivre's Theorem for a high school assignment.
The equations are here:
I was wondering if letting $m= ⌊n/2⌋-j+...
- 101
4
votes
1
answer
263
views
How to calculate $\sum_{k=0}^n a^k\sin(kx)$?
I tried to evaluate
$$
\sum_{k=0}^n a^k\sin(kx)
$$
using complex numbers but it didn't work... Any hint?
$a$ and $x$ are real numbers.
- 145
0
votes
1
answer
104
views
How to prove this quasi-geometric trigonometric series identity without induction
$$\frac{2}{\sin{x}}\sum_{r=1}^{n-1} \sin{rx}\cos{[(n-r)y]} \equiv \frac{\cos{(nx)}-\cos{(ny)}}{\cos{x}-\cos{y}} - \frac{\sin{(nx)}}{\sin{x}}$$
The identity can be tediously proven using the Axiom of ...
- 4,394
7
votes
2
answers
305
views
Bounding a sum involving a $\Re((z\zeta)^N)$ term
This is a follow up to this question. Any help would be very much appreciated.
Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$.
Let $\...
- 7,789
2
votes
2
answers
185
views
Simplifying this (perhaps) real expression containing roots of unity
Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant.
Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$.
...
- 7,789
7
votes
3
answers
3k
views
Cotangent summation (proof)
How to sum up this thing, i tried it with complex number getting nowhere so please help me with this,$$\sum_{k=0}^{n-1}\cot\left(x+\frac{k\pi}{n}\right)=n\cot(nx)$$
- 164
1
vote
2
answers
217
views
Summing up trigonometric series [duplicate]
By considering $\sum_{r=1}^n z^{2r-1}$ where z= $\cos\theta + i\sin\theta$, show that if $\sin\theta$ $\neq$ 0, $$\sum_{r=1}^n \sin(2r-1)\theta=\frac{\sin^2n\theta}{\sin\theta}$$
I couldn't solve ...
- 346
1
vote
2
answers
1k
views
About a binomial expansion of complex numbers
Prove that $$1+{n \choose 1}\cos x + {n \choose 2}\cos 2x+... \cos nx=(2 \cos\frac{x}{2})^n(\cos\frac{nx}{2})$$ given that
$$(1+\cos x+i\sin x)^n=(2\cos\frac{x}{2})^n(\cos\frac{nx}{2}+i\sin\frac{nx}{2}...
- 37
0
votes
2
answers
78
views
find the coefficient
If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$.
I have tried this using trigonometric expansion but unable to find solution ...
- 97