All Questions
Tagged with summation trigonometry
424
questions
1
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2
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178
views
Evaluating $\sum\limits_{k=1}^{n-1} k\cos (k x)$
My intent is to find an expression for the following summation,
$$\sum_{k=1}^{n-1} k\cos (k x).$$
We know that (cfr. Sines and Cosines of Angles in Arithmetic Progression by M. P. Knapp)
$$\sum_{k=0}^{...
- 115
-2
votes
1
answer
189
views
Find the Value of $\sin^2A + \sin^2B + \sin^2C$ given the following data. [duplicate]
If $2\tan^2A\tan^2B\tan^2C + \tan^2A\tan^2B + \tan^2B\tan^2C + \tan^2C\tan^2A = 1$, then find the value of $\sin^2A + \sin^2B + \sin^2C$.
My attempt
1). I tried to multiply both sides by $\cos^2A\...
- 77
1
vote
1
answer
107
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Find $ \cot\left(\sum_{n=1}^{23} \cot^{-1} (1+\sum_{k=1}^n 2k )\right)$
Find the value of
$$ \cot\left(\sum_{n=1}^{23} \cot^{-1} (1+\sum_{k=1}^n 2k )\right)$$
I tried to simplify it but it became messy. How can I find $\sum_{n=1}^{23} \cot^{-1} (1+n+n^2)$? Please help ...
- 3,789
3
votes
1
answer
587
views
Prove that $\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$ converges absolutely.
Prove that $$\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$$ converges absolutely.
The answer here suggests to use the Limit Comparison Test but it works for $a_n \geq 0$ while $\ln(\cos (1/...
- 1,206
8
votes
2
answers
220
views
Proving that ${n+3\choose 3} =\frac{n+2}{2}\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$
Fancy physics predicts the equality
$${n+3\choose 3} =\frac{n+2}{2}\;\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$$
which I can check (numerically and symbolically) for small $n$, but cannot prove for every ...
- 300
4
votes
0
answers
127
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Products of trig functions and the Thue–Morse sequence
I was studying transformations of finite products of trig functions into sums, and empirically observed that the following curious identity appears to hold for all non-negative integer $m$:
$$\prod_{...
- 47.3k
0
votes
0
answers
151
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sum of the series: $\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+....+\cos^3 {(2n-1)\alpha}$. [duplicate]
Question: Find the sum of the series:
$$\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+\ldots+\cos^3 {(2n-1)\alpha}$$
The book from which this question was taken says that the answer is $$\...
user677025
0
votes
2
answers
157
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Trigonometric Summation $\sum_{r=0}^∞ \frac {\sin(r θ)}{3^r}$
$$\sum_{r=0}^∞ \frac {\sin(r θ)}{3^r} $$
Given: $\sinθ=1/3$
I recently came across this question, and I tried numerous ways of solving it, but I could reach nowhere.
My initial approach was to ...
- 535
12
votes
3
answers
772
views
Sum of reciprocal sine function $\sum\limits_{k=1}^{n-1} \frac{1}{\sin(\frac{k\pi}{n})}=?$
The question comes to me when I find there are answers on summation of some forms of trigonometric functions, i.e.
$$
\sum\limits_{k=1}^{n-1} \frac{1}{\sin^2(\frac{k\pi}{n})}\\
\sum\limits_{k=0}^{n-1}...
- 591
0
votes
0
answers
21
views
Proving $\sum_{j=0}^{N-1}\cos\frac{\left(2j+1\right)\pi}{2N}=0$ [duplicate]
Let $l\in\mathbb{Z}$ and $N\in\mathbb{N}$. I need to prove the following:
\begin{equation}
\sum_{j=0}^{N-1}\cos\left(l\frac{\left(2j+1\right)\pi}{2N} \right)=0
\end{equation}
I tried to use Euler ...
- 2,917
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
1
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4
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506
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Find the value of $\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$
Find the value of $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$
My apporach:-
$$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$
$$=\log_2 (\sin(36^{\circ}))+\log_2 (\sin(2*36^{\circ}))+\log_2 (\sin(3*...
- 1,131
1
vote
3
answers
83
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Sum of sinusoids
Let $f_k : = \sin(\omega t+ k \alpha)$ for $k \in \mathbb{N}_0$ and $\alpha, \omega \in \mathbb{R}$ fixed. The following formula is convenient in some engineering applications: $$\sum_{j=0}^N f_k = \...
0
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3
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146
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Trigonometric series sum with $\sin$ function
Sum of Trigonometric series
$\sin (x)-\sin(2x)+\sin(3x)-\sin(4x)+\cdots n$ terms
Try: I have take $2$ cases
$\bullet$ If $n$ is even natural number, Then
$S=\sin(x)-\sin(2x)+\sin(3x)+\cdots -\sin(nx)...
- 11.3k
2
votes
1
answer
83
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Simplifying $\frac{{\sum_{i=1}^{i=n}}1+\tan^2\theta_i}{{\sum_{i=1}^{i=n}}1+\cot^2\theta_i}$, where $\theta_i = \frac{2^{i-1}\pi}{2^n+1}$
How to simplify this expression?
$$\frac{\sum_{i=1}^{n}\left(1+\tan^2\theta_i\right)}{\sum_{i=1}^{n}\left(1+\cot^2\theta_i\right)}$$
where $$\theta_i = \frac{2^{i-1}\pi}{2^n+1}$$
- 2,188
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3
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If $\sum_{n=1}^\infty\tan^{-1}\left(\frac4{n^2+n+16}\right)=\tan^{-1}\left(\frac\alpha{ 10 }\right)$, then find $\alpha$.
$$\sum _ { n = 1 } ^ { \infty } \tan ^ { -1 } \left( \frac { 4 } { n ^ {
2 } + n + 16 } \right)= \tan ^ { -1 } \left( \frac { \alpha } { 10 }
\right)$$ Find $\alpha$.
I know I need to convert to $$\...
- 245
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1
answer
799
views
If $\sum_{i=1}^n \cos \theta_i$=n, then find the value of $\sum_{i=1}^n \sin \theta_i$ [closed]
If $\displaystyle\sum_{i=1}^{n} \cos(\theta_{i}) = n$, then find the value of $\displaystyle\sum_{i=1}^n \sin(\theta_i)$
user642405
2
votes
1
answer
189
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Evaluating $\sum_{n=1}^\infty \frac{\sin(nx)}{n}$ without integrating $\sum_{n=1}^\infty e^{nx}$
I am looking for alternative solutions for finding this sum
$$\sum_{n=1}^\infty \frac{\sin(nx)}{n} $$
My solution proceeds by integrating $$\sum_{n=1}^\infty e^{nx}=\frac{e^{ix}}{1-e^{ix}}$$
With ...
- 313
17
votes
2
answers
579
views
Is there a way to evaluate analytically the following infinite double sum?
Consider the following double sum
$$
S = \sum_{n=1}^\infty \sum_{m=1}^\infty
\frac{1}{a (2n-1)^2 - b (2m-1)^2} \, ,
$$
where $a$ and $b$ are both positive real numbers given by
\begin{align}
a &= ...
1
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3
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584
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What is the value of: $\lim_{n\rightarrow \infty}\sum_{r=1}^{n-1}\frac{\cot^2(r\pi/n)}{n^2}$
$$\lim_{n\rightarrow \infty}\sum_{r=1}^{n-1}\frac{\cot^2(r\pi/n)}{n^2}$$
How can I calculate the value of this trigonometric function where limits tends to infinity? I have thought and tried various ...
5
votes
2
answers
880
views
If $\sin 5°+\sin 10°+\sin15°+\cdots+\sin 40°=a$, then $\sin 5°+\sin 10°+\sin15°+\cdots+\sin 175°=?$
I'm stuck in this question
If $\sin 5°+\sin 10°+\sin15°+\cdots+\sin 40°=a$
$\sin 5°+\sin 10°+\sin15°+\cdots+\sin 175°=?$
I know that, (I asked before) $\sin 5°+\sin 10°+\sin15°+\cdots+\sin 175°=\tan\...
user548054
2
votes
2
answers
852
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Find the $\frac mn$ if $T=\sin 5°+\sin10°+\sin 15°+\cdots+\sin175°=\tan \frac mn$
It's really embarrassing to be able to doesn't solve this simple-looking trigonometry question.
$$T=\sin(5^\circ) +\sin(10^\circ) + \sin(15^\circ) + \cdots +\sin(175^\circ) =\tan \frac mn$$
Find the ...
user548054
10
votes
2
answers
889
views
Find $\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$
Find $$M:=\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$$
There's a solution here that uses complex numbers which I didn't understand and I was wondering if the following is also a correct method.
My ...
- 1,206
1
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1
answer
440
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IQ modulation, sum of I and Q sinewaves
In data transmission, IQ modulation is frequently used to generate FM, AM, PSK, and other modulated signals. I is a "in phase" signal. It's a sinewave with a given frequency. Q is "quadrature" and is ...
- 45
1
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3
answers
233
views
Calculate $\sum_{n=1}^{\infty} \arctan\bigl(\frac{2\sqrt2}{n^2+1}\bigr) $
$$ \lim_{n \to\infty} \sum_{k=1}^{n} \arctan\frac{2\sqrt2}{k^2+1}= \lim_{n \to\infty} \sum_{k=1}^{n} \arctan\frac{(\sqrt{k^2+2}+\sqrt2)-\sqrt{k^2+2}-\sqrt2)}{(\sqrt{k^2+2}+\sqrt2)(\sqrt{k^2+2}-\sqrt2)+...
- 1,219
3
votes
3
answers
131
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Maximizing the sum $\sum_{n=1}^m \sin n$
Consider the sum : $$\displaystyle \sum_{n=1}^m \sin n$$
For which value of $m,$ we will obtain the maximum sum?
Here's my approach :
$\displaystyle \sum_{n=1}^m \sin n=\dfrac{\sin 1}{4 \sin^2 \dfrac{...
- 41
4
votes
1
answer
216
views
Evaluating the sum $\sum\limits_{k=0}^{15} (-1)^k \cos^{560} (k\pi/16).$
To eliminate the pesky $(-1)^k$ term, I have rewritten this as
$ S = -\sum\limits_{k=0}^{15} \cos^{560} (k\pi/16) + 2\sum\limits_{k=0}^{7} \cos^{560} (k\pi/8).$
However, neither of these sums are ...
- 5,230
3
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1
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82
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Show that $\ln\left(1+3x + 2x^2\right) = 3x - \frac{5x^2}{2} + \frac{9x^3}{3} -\cdots + \left(-1\right)^{n-1}\frac{2^n+1}{n}x^n+\cdots$
Show that $$\ln\left(1+3x + 2x^2\right) = 3x - \frac{5x^2}{2} + \frac{9x^3}{3} -\cdots + \left(-1\right)^{n-1}\frac{2^n+1}{n}x^n+\cdots$$
I know that $$\ln(1+x) = \sum\limits_{n=0}^{\infty}\frac{\...
- 307
12
votes
4
answers
540
views
Prove that $1 + \frac{2}{3!} + \frac{3}{5!} + \frac{4}{7!} + \dotsb = \frac{e}{2}$
Prove that $1 + \frac{2}{3!} + \frac{3}{5!} + \frac{4}{7!} + \dotsb = \frac{e}{2}$.
This is problem 4 from page 303 of S.L.Loney's 'Plane Trigonometry'.
It seems fairly obvious that the series ...
- 307
2
votes
4
answers
436
views
Simplify $\sum_{k = 1}^n \tan(k) \tan(k - 1)$ by first proving $\tan(k)\tan(k - 1) = \frac{\tan(k) - \tan(k - 1)}{\tan(1)} - 1$
I have the following problem:
Use the formula
$$\tan(A - B) = \dfrac{\tan(A) - \tan(B)}{1 + \tan(A) \tan(B)}$$
to prove that
$$\tan(k)\tan(k - 1) = \dfrac{\tan(k) - \tan(k - 1)}{\tan(1)} - 1$$
Hence ...
- 4,322
0
votes
0
answers
38
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An approach for this question accompanied by a solution
This question is very elementary when compared to the level of problems asked on this site. However, I am preparing for the Joint Entrance Examination in India and I needed some help in finding an ...
- 125
8
votes
2
answers
2k
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Largest possible value of trigonometric functions
Find the largest possible value of
$$\sin(a_1)\cos(a_2) + \sin(a_2)\cos(a_3) + \cdots + \sin(a_{2014})\cos(a_1)$$
Since the range of the $\sin$ and $\cos$ function is between $1$ and $-1$, shouldn'...
- 2,506
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votes
1
answer
126
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Determine whether the given series is absolutely convergent or conditionally convergent
Consider the series
$$\sum_{n=1}^\infty \log\left(1+\frac{1}{|\sin(n)|}\right).$$
Determine whether it converges absolutely or conditionally.
I am trying to apply Cauchy condensation test, but I ...
- 2,124
5
votes
0
answers
145
views
Find $\quad\cot^2(2^{\circ})+\cot^2(6^{\circ})+\cot^2({10}^{\circ})++\cdots +\cot^2({86}^{\circ})$
Find
$$\cot^2(2^{\circ})+\cot^2(6^{\circ})+\cot^2({10}^{\circ})++\cdots +\cot^2({86}^{\circ})$$
$\mathbf {My Attempt}$
I tried to write the sum backward like this
$$S=\sum_{n=1}^{22} \cot^2(4n-2) = \...
- 749
2
votes
4
answers
350
views
Evaluating $\operatorname{arccos} \frac{2}{\sqrt5} + \operatorname{arccos} \frac{3}{\sqrt{10}}$
Evaluate:
$$\operatorname{arccos} \frac{2}{\sqrt5} + \operatorname{arccos} \frac{3}{\sqrt{10}}$$
We let
$$\alpha = \operatorname{arccos} \frac{2}{\sqrt5} \qquad \beta = \operatorname{arccos}\frac{...
- 61
1
vote
1
answer
48
views
Systematic way of 'zeroing out' some values of a function?
Given a quasi-periodic trigonometric function $f(x)$, is there a systematic way of 'zeroing out' all values where $f(x)\ne1$?
For example, consider the function
$$\sum_{n=0}^m \frac {1}{2n+1} \sin (...
1
vote
0
answers
280
views
How to make Sum of sinusoidal signals with different frequencies have different absolute values of positive peak and negative peak?
Let me define a signal $$f(t)=\sum_{i=1}^N a_i \cos(\omega_it+\theta_i),$$ where $\omega_i = 2\pi f_i$ with a frequency $f_i$.
I want to make a specific $f(t)$ by determining $a_i$'s and $\theta_i$'s ...
- 3,433
2
votes
2
answers
3k
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Find $x$ if $\frac1{\sin1°\sin2°}+\frac1{\sin2°\sin3°}+\cdots+\frac1{\sin89°\sin90°} = \cot x\cdot\csc x$ [duplicate]
If $$\dfrac1{\sin1°\sin2°}+\dfrac1{\sin2°\sin3°}+\cdots+\dfrac1{\sin89°\sin90°} = \cot x\cdot\csc x$$ and $x\in(0°,90°)$, find $x$.
I tried writing in $\sec$ form but nothing clicked. Any ideas?
2
votes
2
answers
1k
views
Proving $\sum_{k=0}^{n-1} \text{cos}\left(\frac{2\pi k}{n} \right) = \sum_{k=0}^{n-1} \text{sin}\left(\frac{2\pi k}{n} \right) = 0$ [duplicate]
I would like to prove the following, $$\sum_{k=0}^{n-1} \text{cos}\left(\frac{2\pi k}{n} \right) = \sum_{k=0}^{n-1} \text{sin}\left(\frac{2\pi k}{n} \right) = 0$$ This is equivalent to showing that if ...
user445909
2
votes
1
answer
175
views
What is $\cos(\frac{\pi t}{2^n})$ in terms of $\cos(\pi t)$?
If it is easier, you can do it the other way around, by writing $\cos(\pi t)$ in terms of $\cos(\frac{\pi t}{2^n})$. I just wanted to know if there was a nice closed form solution to a problem like ...
- 3,105
2
votes
0
answers
134
views
How to determine whether the sum ${\sum}_{k=1}^{\infty} {\sin(2^k)\over k}$ converges?
I saw a question on quora asking whether or not the sum ${\sum}_{k=0}^{\infty}{sin(2^k)\over n}$ is convergent. My opinion, and that of the other answers, is that Dirichlet's test could be used with {...
- 233
2
votes
0
answers
73
views
Finite Sum of Power of Cosine Function
Suppose I have the following sum: Let $n=qr$
$$S_n=\sum_{j=0}^{r-1}(-1)^{qj}\cos^n{\left(\frac{j\pi}{r}\right)}$$
I am interested in which values of $r>1$ produce an integer $k$ for all $n>0$....
- 1,803
-1
votes
1
answer
284
views
Evaluating $\sum_{n=1}^{N} 2^{-n}\sin(n\theta)$ [duplicate]
Use de Moivre's theorem to find
$$\sum_{n=1}^{N} 2^{-n}\sin(n\theta)$$
How to find sum of imaginary parts of geometric progression, or use exponential form of complex numbers to find the summation ...
4
votes
1
answer
812
views
Summation of $\sum_{r=1}^{n} \frac{\cos (2rx)}{\sin((2r+1)x \sin((2r-1)x}$
Summation of $$S=\sum_{r=1}^{n} \frac{\cos (2rx)}{\sin((2r+1)x) \sin((2r-1)x)}$$
My Try:
$$S=\sum_{r=1}^{n} \frac{\cos (2rx) \sin((2r+1)x-(2r-1)x)}{\sin 2x \:\sin((2r+1)x \sin((2r-1)x}$$
$$S=\sum_{...
- 13.2k
2
votes
2
answers
97
views
How do I prove that $\sum_{i=1}^m \cos^2\left(\frac{2\pi i}{m}\right) = \sum_{i=1}^m \sin^2\left(\frac{2\pi i}{m}\right) = \frac{m}{2}$? [duplicate]
I haven't had any good ideas nor found any helpful identities so far, so I'd appreciate some help.
Also, here $m > 2$.
Update: Thanks to the hints and to this previous post I managed to get to ...
- 6,833
0
votes
1
answer
35
views
Why can I cancel this $\sum_i 2\cos^3(\alpha_i)\sin\alpha_i$ term for pairs 180degrees apart?
I have the equation,
$$\tag{1} K_{11}\sum_i \cos^4\alpha_i+K_{12}\sum_i 2\cos^3\alpha_i\sin\alpha_i+K_{22}\sum_i\sin^2\alpha_i\cos^2\alpha_i=\sum_i\rho_i\cos^2\alpha_i$$
In the paper it states, "For ...
- 535
1
vote
1
answer
163
views
Sine series: angle multipliers add to 1
It is known that in an sine series with angles in arithmetic progression (I refer to this question):
$\sum_{k=0}^{n-1}\sin (a+k \cdot d)=\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times ...
- 347
4
votes
3
answers
376
views
Finite sum $\sum_{k=1}^{p-1} \frac{1-\cos\left(\frac{2\pi k r}{p}\right)}{1-\cos\left(\frac{2\pi k s}{p}\right)}$ for $\gcd(p,rs)=1$.
I am wondering if there is a closed form to the following finite sum:
$$\sum_{k=1}^{p-1} \frac{1-\cos\left(\frac{2\pi k r}{p}\right)}{1-\cos\left(\frac{2\pi k s}{p}\right)},$$
where $\gcd(p,rs)=1$ ...
- 678
2
votes
1
answer
102
views
Formula for summing an arbitrary number $n$ of cos functions?
This post gives
$$\cos A+\cos B+\cos C=1+4\sin \frac {A}{2}\sin \frac {B}{2}\sin \frac {C}{2}$$
Is it possible to derive a generalised formula for
$$\cos A+\cos B+\cos C+...+\cos N$$
i.e., a ...
3
votes
4
answers
70
views
how do I show this :$\sum_{k=0}^{2n }\binom{2n}{k} \sin ((n-k)x)=0$ , for every real $x$ and for every integer $n$?
My attempt fails to show this formula $\sum_{k=0}^{2n }\binom{2n}{k} \sin ((n-k)x)=0$ which I have accrossed in my textbook, using induction proof, but I think by induction seems very hard, I want to ...
user517526