All Questions
Tagged with summation trigonometry
424
questions
8
votes
1
answer
642
views
Finite Series - reciprocals of sines
Find the sum of the finite series
$$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$
This problem was asked in a test in my school.
The answer seems to be $\dfrac{\cos1^{\circ}}{\sin^...
- 984
5
votes
1
answer
603
views
Sum : $\sum \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right)$.
Calculate : $$ \sum_{k=1}^{n-1} \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right).$$
- 4,928
1
vote
1
answer
2k
views
Solution for Summation of $\cos^2x$
Can you give me the solution for the summation
$$
\sum_{n=0}^{\infty} \cos^2(\pi n)
$$
Edit: Please give me the explanation of how it is calculated and also final answer in integers.
- 113
42
votes
3
answers
1k
views
Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
- 13.2k
3
votes
1
answer
139
views
Proof the following trig series
Prove that
$$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$
I am not necessarily looking for a ...
- 1,542
3
votes
1
answer
71
views
Trigonometric Integration + Series
I am doing an integration question:
$$\int \frac{1-\cos^{2m}x}{1-\cos^2x}$$
So I have to show that $$\frac{1-\cos^{2m}x}{1-\cos^2x}=1+\cos^2x+\cos^4x+...+\cos^{2(m-1)}$$
How can I do that?
- 1,542
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
3
votes
2
answers
237
views
Finding an infinite trigonometric sum
Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only ...
- 1,066
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
$$\begin{align*}
\frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases}
1&\text{if } k \...
- 33
0
votes
1
answer
113
views
trigonometric summation
Taking into consideration the functions
$$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$
and
$$\sum_{t=0}^{n}\cos{(\...
- 217
4
votes
3
answers
194
views
Help in manipulating Integrals
I try to express : $\displaystyle 1+2\sum _{ k=1 }^n \cos(2k\theta ) $
as : $\dfrac { \sin\left( \theta +2\theta n \right) }{ \sin\left( \theta \right) } $
I tried to use the exponential function :...
- 153
3
votes
1
answer
5k
views
Sum $\cos x + \cos 2x + \cdots + \cos (n-1)x.$ [duplicate]
Find the sum of the series $$\cos x + \cos 2x + \cdots + \cos (n-1)x.$$
You must calculate the sum of this series only by multiplying through by $2\sin\left(\frac{x}{2}\right)$.
Now I've heard of ...
- 522
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
9
votes
2
answers
2k
views
Prove that $\frac{1}{4-\sec^{2}(2\pi/7)} + \frac{1}{4-\sec^{2}(4\pi/7)} + \frac{1}{4-\sec^{2}(6\pi/7)} = 1$
How can I prove the fact $$\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$$
When asked somebody told me to use the ideas of ...
user47520
1
vote
1
answer
417
views
Expressing $\int \tan^n x\,dx$ with a sum
I was playing around with integrals of $\tan x$, because I knew that both $\int\tan x\,dx$ and $\int\tan^2x\,dx$ were solvable. I then came across the fact that
$$\begin{align}
\int \tan^n x\,dx &...
- 1,535
6
votes
1
answer
296
views
A curious identity on sums of secants
I was working on proving a variant of Markov's inequality, and in doing so I managed to come across an interesting (conjectured) identity for any $n\in\mathbb{N}$:
$$\sum_{m=0}^{n-1} \sec^2\left(\...
- 303
13
votes
2
answers
3k
views
Reference for a tangent squared sum identity
Can anyone help me find a formal reference for the following identity about the summation of squared tangent function:
$$
\sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+.
$$
I ...
- 131
0
votes
1
answer
230
views
find the multiplicative factor for get a specific amount of sum on sin
i am not a math guru so please sorry if this is a silly question. i'm not sure on how to latexize this question so i've done a spreadsheets with openoffice (and i'm interest also in the best way to ...
- 1,823
3
votes
2
answers
3k
views
Evaluation of $ \sum_{k=0}^n \cos k\theta $
I just wanted to evaluate
$$ \sum_{k=0}^n \cos k\theta $$
and I know that it should give
$$ \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)} $$
...
- 1,197
13
votes
3
answers
16k
views
How to prove Lagrange trigonometric identity [duplicate]
I would to prove that
$$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+
\frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$
given that
$$1+...
- 2,318
6
votes
1
answer
8k
views
Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]
For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
- 293
11
votes
2
answers
55k
views
Sum of $\cos(k x)$ [duplicate]
I'm trying to calculate the trigonometric sum : $$\sum\limits_{k=1}^{n}\cos(k x)$$
This is what I've tried so far : $$\renewcommand\Re{\operatorname{Re}}
\begin{align*}
\sum\limits_{k=1}^{n}\cos(k x) ...
- 891
18
votes
1
answer
11k
views
$\sum \cos$ when angles are in arithmetic progression [duplicate]
Possible Duplicate:
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\...
- 189
196
votes
8
answers
101k
views
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{...
- 22.5k