All Questions
Tagged with summation trigonometry
85
questions
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:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{...
- 22.5k
18
votes
1
answer
11k
views
$\sum \cos$ when angles are in arithmetic progression [duplicate]
Possible Duplicate:
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\...
- 189
35
votes
3
answers
4k
views
Trig sum: $\tan ^21^\circ+\tan ^22^\circ+\cdots+\tan^2 89^\circ = \text{?}$
As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+\cdots+\tan^2 89^\circ$$
I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch ...
- 3,153
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
51
votes
9
answers
6k
views
Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
Question : Is the following true for any $m\in\mathbb N$?
$$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$
Motivation : I reached $(\star)$ by ...
- 145k
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
7
votes
1
answer
5k
views
Bernoulli numbers, taylor series expansion of tan x
I found the following formula here: Taylor Series of $\tan x$.
Taylor series of $\tan x$:
$$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$.
I do not ...
- 1,948
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
$$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\...
- 1,927
13
votes
2
answers
3k
views
Reference for a tangent squared sum identity
Can anyone help me find a formal reference for the following identity about the summation of squared tangent function:
$$
\sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+.
$$
I ...
- 131
11
votes
2
answers
55k
views
Sum of $\cos(k x)$ [duplicate]
I'm trying to calculate the trigonometric sum : $$\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
10
votes
2
answers
889
views
Find $\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$
Find $$M:=\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$$
There's a solution here that uses complex numbers which I didn't understand and I was wondering if the following is also a correct method.
My ...
- 1,206
47
votes
1
answer
2k
views
Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?
Why is the following function $f(n)$ constant for any natural number $n$?
$$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt k}}}}{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}-{\sqrt{n+1+\...
- 145k
13
votes
3
answers
16k
views
How to prove Lagrange trigonometric identity [duplicate]
I would to prove that
$$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
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
6
votes
2
answers
360
views
Simplify $\prod_{k=1}^5\tan\frac{k\pi}{11}$ and $\sum_{k=1}^5\tan^2\frac{k\pi}{11}$
My question is:
If $\tan\frac{\pi}{11}\cdot \tan\frac{2\pi}{11}\cdot \tan\frac{3\pi}{11}\cdot \tan\frac{4\pi}{11}\cdot \tan\frac{5\pi}{11} = X$ and $\tan^2\frac{\pi}{11}+\tan^2\frac{2\pi}{11}+\tan^2\...
- 2,586
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
4
votes
3
answers
248
views
Proving $\sum_{x=0}^{n-1} \cos\left(k +x{2\pi\over n}\right) =\sum_{x=0}^{n-1} \sin\left(k +x{2\pi\over n}\right) =0. $
Is there anybody who can help me show the following?
$$
\sum_{x=0}^{n-1} \cos\left(k +x{2\pi\over n}\right) =0
\qquad\hbox{and}\qquad
\sum_{x=0}^{n-1} \sin\left(k +x{2\pi\over n}\right) =0
$$
I ...
user357524
3
votes
2
answers
6k
views
arccot limit: $\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$
I have to find the limit of this sum:
$$\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$$
I tried using sandwich theorem , observing:
$$\cot ^{-1}(r^3)\leq\cot ^{-1}(r^2+\frac{3}{4})\leq\cot ^{-1}(r^...
- 3,517
26
votes
1
answer
869
views
Finite sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$
I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum.
I ...
- 665
25
votes
5
answers
2k
views
Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$
In old popular science magazine for school students I've seen problem
Prove that $\quad $
$\dfrac{1}{\cos^2 20^\circ} +
\dfrac{1}{\cos^2 40^\circ} +
\dfrac{1}{\cos^2 60^\circ} +
\dfrac{1}{\cos^...
- 17.4k
9
votes
2
answers
2k
views
Prove that $\frac{1}{4-\sec^{2}(2\pi/7)} + \frac{1}{4-\sec^{2}(4\pi/7)} + \frac{1}{4-\sec^{2}(6\pi/7)} = 1$
How can I prove the fact $$\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.$$
When asked somebody told me to use the ideas of ...
user47520
3
votes
4
answers
174
views
Evaluate $\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}$, where $q^7=1$ and $q\neq 1$.
Let $q$ be a complex number such that $q^7=1$ and $q\neq 1$. Evaluate $$\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}.$$
The given answer is $\frac{3}{2}$ or $-2$. But my answer is $\pm 2$.
...
- 373
2
votes
2
answers
852
views
Find the $\frac mn$ if $T=\sin 5°+\sin10°+\sin 15°+\cdots+\sin175°=\tan \frac mn$
It's really embarrassing to be able to doesn't solve this simple-looking trigonometry question.
$$T=\sin(5^\circ) +\sin(10^\circ) + \sin(15^\circ) + \cdots +\sin(175^\circ) =\tan \frac mn$$
Find the ...
user548054
1
vote
4
answers
2k
views
Prove by induction: $\sum\limits_{k=1}^{n}sin(kx)=\frac{sin(\frac{n+1}{2}x)sin\frac{nx}{2}}{sin\frac{x}{2}}$
$\sum\limits_{k=1}^{n}sin(kx)=\frac{sin(\frac{n+1}{2}x)sin\frac{nx}{2}}{sin\frac{x}{2}}$
Base case: For $n=1$
$sinx=\frac{sinx\cdot sin\frac{x}{2}}{sin\frac{x}{2}}=sinx$
Induction hypothesis: For $...
- 3,479
16
votes
6
answers
596
views
If $x+y+z=xyz$, prove $\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$ [duplicate]
If $x+y+z=xyz$, prove
$\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$
given that $x^2~,~y^2~,~z^2\ne1$
I came across this question in ...
- 6,560
12
votes
3
answers
772
views
Sum of reciprocal sine function $\sum\limits_{k=1}^{n-1} \frac{1}{\sin(\frac{k\pi}{n})}=?$
The question comes to me when I find there are answers on summation of some forms of trigonometric functions, i.e.
$$
\sum\limits_{k=1}^{n-1} \frac{1}{\sin^2(\frac{k\pi}{n})}\\
\sum\limits_{k=0}^{n-1}...
- 591
11
votes
0
answers
449
views
Tricky Sum involving Binomial Coefficients and Sine
I am stumped by the sum
$$\sum_{x=0}^n \binom{n}{x}\sin\big(\frac{\pi x}{n}\big)$$
but I can't figure it out. I tried expanding the taylor series of sine and using Euler's identity, but to no avail. ...
11
votes
0
answers
483
views
Simplify: $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\ldots+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\ldots+\sqrt{10-\sqrt{99}}}$ [duplicate]
$\frac{ \sqrt{1+\sin(x)}+\sqrt{1+\cos(x)} }{ \sqrt{1-\sin(x)}+\sqrt{1-\cos(x)} }=\sqrt2+1\,\forall x\in [0,\pi/2]$
the fraction of square roots may be simplified using double angle formulas.. though i ...
- 4,266
9
votes
1
answer
543
views
Is there any identity for $\sum_{k=1}^{n}\tan\left(\theta+\frac{k\pi}{\color{red} {2n+1}}\right)$?
Is there any identity for $\sum_{k=1}^{n}\tan\left(\theta+\frac{k\pi}{\color{red} {2n+1}}\right)$ or $\sum_{k=1}^{n}\tan\left(\frac{k\pi}{\color{red} {2n+1}}\right)$ ?
I thought maybe wrongly that ...
- 1,379
8
votes
2
answers
685
views
Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$
I want to show that
$$\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})} = \frac{n^2-1}6$$
With induction I don't know how I could come back from $\frac{1}{1-\cos(\frac{2k\pi}{n+1})}$ to $\frac{1}{1-\...
- 81
7
votes
2
answers
288
views
Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?
What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$$
here it can be shown that where $x=y$,
$$n^2 \csc^2(nx) = \sum_{k=0}^...
- 1,379
6
votes
1
answer
338
views
Finding a closed form expression for $\sum_{i=1}^{n-1}\csc{\frac{i\pi}{n}}$
Consider
$\displaystyle{S =
\sum_{k = 1}^{n - 1}\csc\left(k\,{\pi \over n}\right) =
\frac{1}{\sin\left(\pi/n\right)} +
\frac{1}{\sin\left(2\pi/n\right)} +
\frac{1}{\sin\left(3\pi/n\right)} + \cdots +
\...
- 4,359
4
votes
1
answer
127
views
Evaluating $\frac{\sum_{k=0}^6 \csc^2(a+\frac{k\pi}{7})}{7\csc^2(7a)}$
The question is to evaluate $$\frac{\sum_{k=0}^{6}\csc^2(a+\frac{k\pi}{7})}{7\csc^2(7a)}$$ where $a=\pi/8$ without looking at the trigonometric table.
I tried to transform the $\csc^2$ term to $\cot^...
- 2,595
4
votes
3
answers
4k
views
Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]
I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$.
So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get:
$\sin(\theta)=\theta-\frac{\...
- 1,346
3
votes
2
answers
369
views
A series $10^{12} + 10^7 - 45\sum_{k=1}^{999}\csc^4\frac{k\pi}{1000}.$
There's a math clock with formulas for each of $1,\ldots,12$, most of which are easy. Number 11, however, intrigues me: $$10^{12} + 10^7 - 45\sum_{k=1}^{999}\csc^4\frac{k\pi}{1000}.$$
Wolfram Alpha ...
- 1,098
3
votes
2
answers
3k
views
Evaluation of $ \sum_{k=0}^n \cos k\theta $
I just wanted to evaluate
$$ \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)} $$
...
- 1,197
3
votes
1
answer
262
views
On $\mathrm{\sum_{x\in\Bbb Z}sech(x)=3.142242…}$
Inspired by
This question,
I started to wonder about simpler series. I have seen similar questions to the following, but none had this special case explicitly. It is related to the q-digamma ...
- 12.5k
1
vote
1
answer
311
views
reference for finit sum of cotangents
Hi I found a finite summation of cotangents as
$$\sum_{k=0}^{n-1}\cot^2(z + \frac{k\pi}{n}) = n^2 - n + n^2 \cot^2(n z), \quad n > 0$$
in the URL bellow
http://functions.wolfram.com/...
- 369
22
votes
0
answers
2k
views
How to find the approximate basic period or GCD of a list of numbers?
I want to tell the number which act as the best approximate basic period (or wavelenght as pointed out by Eric) of a list of real numbers: e.g for {14, 21, 35} we should obtain 7 as the basic period, ...
- 366
18
votes
2
answers
497
views
Prove $\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \frac n4$
How do you prove that $\displaystyle\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \dfrac{n}{4}$, where $x\in\mathbb{R}$?
I tried mathematical induction, but it doesn't work. I also ...
- 405
17
votes
2
answers
579
views
Is there a way to evaluate analytically the following infinite double sum?
Consider the following double sum
$$
S = \sum_{n=1}^\infty \sum_{m=1}^\infty
\frac{1}{a (2n-1)^2 - b (2m-1)^2} \, ,
$$
where $a$ and $b$ are both positive real numbers given by
\begin{align}
a &= ...
12
votes
1
answer
267
views
Could this conjecture be proved ? (sum of even powers of cotangents in arithmetic progression )
Having tried (in vain) to answer this question, I worked the explicit formulae of
$$\color{blue}{S_k=\sum _{n=1}^m \Big[\cot \left(\frac{n \,\pi }{2 m+1}\right)\Big]^k}$$ where $k$ is an even integer....
- 268k
11
votes
2
answers
268
views
Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$
$${\color\green{\sum_{x=2}^\infty (x^2 \operatorname{arcoth} (x) \operatorname{arccot} (x) -1)}}$$
This is an impressive sum that has bothered me for a while. Here are the major points behind the sum.....
user266519
10
votes
1
answer
717
views
Product of Sines and Sums of Squares of Tangents
There is a nice formula for products of cosines, found by multiplying by the complementary products of sines and using the double angle sine formula (as I asked in my question here): $$\prod_{k=1}^n \...
- 8,945
8
votes
2
answers
2k
views
Largest possible value of trigonometric functions
Find the largest possible value of
$$\sin(a_1)\cos(a_2) + \sin(a_2)\cos(a_3) + \cdots + \sin(a_{2014})\cos(a_1)$$
Since the range of the $\sin$ and $\cos$ function is between $1$ and $-1$, shouldn'...
- 2,506
7
votes
1
answer
272
views
Random Wolfram|Alpha identity related to $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$
I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series
$$\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{...
- 1,817
6
votes
1
answer
296
views
A curious identity on sums of secants
I was working on proving a variant of Markov's inequality, and in doing so I managed to come across an interesting (conjectured) identity for any $n\in\mathbb{N}$:
$$\sum_{m=0}^{n-1} \sec^2\left(\...
- 303
6
votes
1
answer
220
views
Prove |cos(x−1)|+|cos(x)|+|cos(x+1)|≥3/2
I'm working on an induction proof, but I keep coming up against a brick wall.
While working through the induction proof process I keep ending up with $$|\cos(m)|\ge\frac12$$ ,but clearly this isn't ...
- 1,250
6
votes
1
answer
346
views
Proving these trigonometric sums $\sum\limits_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$
Can someone help me to prove that:
$$ \sum_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$$
$$\sum_{k=0}^{n-1}\cos\frac{2k^2\pi}{n}=\frac{\...
- 61
6
votes
1
answer
8k
views
Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]
For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
- 293