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