All Questions
41
questions
7
votes
3
answers
157
views
History of the general formula for linearising $\cos^n(x)$
I was wondering where the formula:
$$\cos^n(x)=\sum_{k=0}^n\frac{n!}{k!(n-k)!}\cos(x(2k-n))$$
Was first originally published. I accidentally derived it a few days ago and was wondering where it was ...
- 474
18
votes
2
answers
496
views
Prove $\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \frac n4$
How do you prove that $\displaystyle\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \dfrac{n}{4}$, where $x\in\mathbb{R}$?
I tried mathematical induction, but it doesn't work. I also ...
- 405
0
votes
5
answers
199
views
Evaluating $\sum_{k=1}^{10}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$
Find the value of $$\sum_{k=1}^{10}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$$
I have heard somewhere that this question can be done by $nth$ roots of unity or by vector algebra. But I'...
- 1,253
1
vote
1
answer
117
views
If $\beta=e^{i\frac{2\pi}7}$, and $\left|\sum_{r=0}^{3n-1}\beta^{2^r}\right|=4\sqrt2$, find $n$ [duplicate]
If $\beta=e^{i\frac{2\pi}7}$, and $\left|\sum_{r=0}^{3n-1}\beta^{2^r}\right|=4\sqrt2$, find $n$.
My Attempt:
$\beta+\beta^2+\beta^4+\beta^8+\beta^{16}+...=4\sqrt2$
This is no GP. Can we do something ...
- 8,318
-1
votes
2
answers
99
views
What trigonometric identities was used to get this? [closed]
I have this solution for a math question, but I didn't get how did we get from the line $3$ to $4$ and also from $4$ to $5$. I can understand that this is done through trigonometric identities. but ...
- 1
3
votes
1
answer
359
views
Finding the sum of $\cos{x}+2\cos{2x}+...+n\cos{nx}$ [duplicate]
I'm struggling to find the sum of $S_n=\cos{x}+2\cos{2x}+3\cos{3x}...+n\cos{nx}$
I know that for $z=e^{ix}$, $2\cos{nx}=z^n+\frac{1}{z^n}$.
So I've tried $2S_n=(z+2z^2+3z^3+...+nz^n)+(\frac{1}{z}+\...
- 1,381
4
votes
0
answers
128
views
How to solve $\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}$?
I need to solve this sum:
$$\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}.$$
Do you have any ideas for how I could do this?
I know that this sum:
$$\sum_{n=-\infty}^\infty\frac{y}{(x-n\pi)^...
- 1,027
-1
votes
2
answers
106
views
Summation from 0 to n of cos((k/n)2 pi) [duplicate]
Recently I came across this summation:
$$\sum_{k=0}^n\cos\Bigl(k\frac{2\pi}{n}\Bigr)$$ when I was trying to evaluate the following summation $$\sum_{k=0}^{n} z^k+z^{-k}$$ where $z \in \mathbb{C}, z=\...
- 36
1
vote
1
answer
194
views
Trigonometric Identities Using De Moivre's Theorem
I am familiar with solving trigonometric identities using De Moivre's Theorem, where only $\sin(x)$ and $\cos(x)$ terms are involved. But could not use it to solve identities involving other ratios. ...
2
votes
1
answer
86
views
Find the sum of infinite series $\cos{\frac{\pi}{3}}+\frac{\cos{\frac{2\pi}{3}}}{2}+..$
Find the sum of infinite series $$\cos{\dfrac{\pi}{3}}+\dfrac{\cos{\dfrac{2\pi}{3}}}{2}+\dfrac{\cos{\dfrac{3\pi}{3}}}{3}+\dfrac{\cos{\dfrac{4\pi}{3}}}{4}+\dots$$
I tried to convert it to $\mathrm{cis}$...
- 1,157
3
votes
4
answers
174
views
Evaluate $\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}$, where $q^7=1$ and $q\neq 1$.
Let $q$ be a complex number such that $q^7=1$ and $q\neq 1$. Evaluate $$\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}.$$
The given answer is $\frac{3}{2}$ or $-2$. But my answer is $\pm 2$.
...
- 373
2
votes
1
answer
95
views
Proving $\sum_{r=0}^{n-1} (-1)^r \cos^n{\left(\frac{r\pi}{n}\right)}=\frac{n}{2^{n-1}}$
Prove that $$\sum_{r=0}^{n-1} (-1)^r \cos^n{\left(\dfrac{r\pi}{n}\right)}=\dfrac{n}{2^{n-1}}$$
I proved the result using induction, however am more interested in finding the sum using complex numbers....
- 1,157
0
votes
1
answer
78
views
Summation of an Infinite Series involving Trigonometry
I came across this summation problem the other day and I am not quite sure how to approach it
$$S=\sum_{n=0}^{n=\infty}\frac{2^{n-1}}{3^{2n-2}}\sin\left(\frac{\pi}{3.2^{n-1}}\right)$$
My approach ...
- 1,292
1
vote
1
answer
306
views
How to solve sum of cos(kx) for the case cos(x)=1
I have the solution for $\sum_{k=1}^n \cos(kx)$:
\begin{align}
\sum_{k=1}^n \cos(kx) & = \Re\left(\sum_{k=1}^n e^{ikx}\right)\\
& = \Re\left(e^{ix} {e^{inx}-1 \over e^{ix}-1}\right) \\
& ...
- 9
0
votes
2
answers
85
views
Exponential double angle formula
My question is whether someone could provide a proof for the following identity:
$$ \frac{1 - e^{int}}{1 - e^{it}} = e^{i(n-1)t/2} \frac{\sin(nt/2)}{\sin(t/2)} $$
Motivation:
The left hand side is ...
- 3