All Questions
39
questions
1
vote
0
answers
103
views
Restructuring Jacobi-Anger Expansion
In Jacobi-Anger expansion, $$e^{\iota z \sin(\theta)}$$ can be written as:
$$e^{\iota z \sin(\theta)} = \sum_{n=-\infty}^{\infty} J_n(z)e^{\iota n \theta}$$
where $J_n(z)$ is the Bessel function of ...
- 31
8
votes
1
answer
253
views
Compute $\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$
Question: Compute $$\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$$
I began by rearranging the sum as follows:
$$\...
- 690
-1
votes
1
answer
126
views
Find the value $\sum_{n=a}^b\frac1{\sin (2^{n+3})}$ [closed]
Find the value of: $$\sum_{n=0}^{10}\frac1{\sin (2^{n+3})}$$
I'm stuck on this problem, can someone please help?
- 596
0
votes
1
answer
113
views
Clean way to prove that $\sum_{k=0}^{N-1}\cos \left( \frac{2m \pi}{N}k\right)=0$
Here's the identity:
For $0 < m < N$, we have
$$
\sum_{k=0}^{N-1}\cos \left( \frac{2m \pi}{N}k\right) = 0
$$
I know I can solve this using a variety of methods, i.e. anything described here: ...
- 175
4
votes
1
answer
108
views
Clean proof for trigonometry identity? I know what the answer is, but I feel like there should be like a $1$-$2$ liner to compute this
Fix $j,k$ with $0 \leq j,k \leq N$. If $j+k$ is even, (i.e. if $j,k$ have same parity), then
$$
\sum_{n=1}^{N-1} \cos\left(\frac{j\pi}Nn\right)\sin\left(\frac{k \pi}Nn\right) = 0
$$
and
$$
\sum_{n=0}^...
- 175
5
votes
1
answer
189
views
Evaluating $\sum_{r=1}^{89} \frac{1}{1+\tan^3 r}$
$$\sum_{r=1}^{89} \frac{1}{1+\tan^3 r }$$
where $r$ is in degrees
I tried this a lot using the $a^3+b^3$ identity but I don't seem to be getting anything fruitful :(
Can someone please give me a hint?...
- 205
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. ...
0
votes
2
answers
141
views
Proving a formula for $\sum ^{n}_{k=1}\sin kt$ [duplicate]
Let $t$ be a real number such that $\sin \dfrac{t}{2}\neq 0$. Show that
$$\sum ^{n}_{k=1}\sin kt=\dfrac{\cos\dfrac{t}{2}-\cos \left( n+\dfrac{1}{2}\right) t}{2\sin \dfrac{t}{2}}$$
for every positive $...
- 33
0
votes
1
answer
63
views
How to get $A$ and $B$ from $A\csc 10^\circ+B=$ $\sin 10^\circ+\cos 60^\circ+\cos 40^\circ+\sin 70^\circ+\sin 90^\circ$?
The problem is as follows:
Find $A+B$ from:
$A\csc 10^\circ+B=\sin 10^\circ+\cos 60^\circ+\cos 40^\circ+\sin 70^\circ+\sin 90^\circ$
The alternatives given in my book are as follows:
$\begin{array}{ll}...
- 4,185
2
votes
2
answers
82
views
Using trigonometric power formulas to derive an identity for $\cos^3(x)$
I am practicing with manipulating sigma notation and binomial coefficients right now. I am using the formula given here to derive the identity for $\cos^3(x)$
The identity for $\cos^3(x)$ is
$$\cos^3(...
- 3,187
0
votes
0
answers
30
views
How to prove $\sum_{k=0}^{n-1} \cos\frac{2k\pi}{n}=0$ [duplicate]
My teacher told us that :
$$\sum_{k=0}^{n-1} \cos\frac{2k\pi}{n}=0$$
Without giving any proof
If someone could tell me how to approach this that would be great !
4
votes
0
answers
306
views
Closed form for Sum of Tangents with Angles in Arithmetic Progression
The formulae that can be used to evaluate series of sines and cosines of angles in arithmetic progressions are well known:
$$\sum_{k=0}^{n-1}\cos (a+k d) =\frac{\sin( \frac{nd}{2})}{\sin ( \frac{d}{2} ...
- 6,560
2
votes
3
answers
256
views
Evaluate $\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$
I would like to know how to evaluate $$\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$$
There are a couple of issues I have with this. Firstly, depending on the value of $x$, it seems, at ...
- 6,560
5
votes
2
answers
191
views
$\sum_{n=1}^{\infty} {\frac{1}{4^n \cos^2 (\frac{\pi}{2^{n+2}})}}$ [duplicate]
$\sum_{n=1}^{\infty} {\frac{1}{4^n \cos^2 (\frac{\pi}{2^{n+2}})}}$
How can I calculate this? Since there are $4^n$ and $\cos^2x$, I tried:
$$\sum_{n=1}^{\infty} {\frac{1}{4^n \cos^2 (\frac{\pi}{2^{n+2}...
- 596
4
votes
3
answers
867
views
Finding $\frac{\sum_{r=1}^8 \tan^2(r\pi/17)}{\prod_{r=1}^8 \tan^2(r\pi/17)}$
I have tried to wrap my head around this for some time now, and quite frankly I am stuck.
Given is that :
$$a=\sum_{r=1}^8 \tan^2\left(\frac{r\pi}{17}\right) \qquad\qquad b=\prod_{r=1}^8 \tan^2\left(\...
- 183