1
vote
1answer
49 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
1
vote
1answer
59 views

Hard Trigonometric Equation

its possible to solve a equation like $$\prod^{45}_{k = 0} \left( 1 + \tan \frac{k \pi }{180} \right) = \left[ \log_{\frac{\sqrt{6}}{3}} \big| \sin(2x)\big| \right]^{\frac{9}{2}\sin(2x) + 20}$$ ...
0
votes
1answer
54 views

Solve for x: sin2 x − cos2 x = sin x, −π ≤ x ≤π

I have to solve for x using the domain of −π ≤ x ≤π sin^2 x − cos^2 x = sin x I tried changing cos^2 x to 1 - sin^2 x so it would be sin^2 x - 1 + sin^2 x = sin x making it, 2sin^2 x - 1 = sin x ...
0
votes
0answers
32 views

Using Algebra with Trig Functions

Using Algebra with Trig Functions I'm trying to find the correct 1 second audio signal I would need to apply to a 1 second known noise signal to have the output signal be a sin wave. The basic ...
1
vote
3answers
68 views

Evaluation of a trigonometric series

$$ \mbox{Question: Evaluate}\quad \tan^{2}\left(\pi \over 16\right) + \tan^{2}\left(2\pi \over 16\right) + \tan^{2}\left(3\pi \over 16\right) + \cdots + \tan^{2}\left(7\pi \over 16\right) $$ What I ...
0
votes
0answers
49 views

Trigonometry - proving a rule

I was given that $\sin a + \sin b + \sin c \cdots$ is equal to: (where $a,b,c$ are in arithmetic progression) $$\frac{\sin\frac{a + c}{2}\sin\frac{nb}{2}}{\sin{b/2}}$$ Here $a$ is the first term of ...
0
votes
0answers
27 views

Help in finding a book

That is my problem: i have this series: $\sum_{k=0}^{\infty} \psi_{k}(h)\cos(k\theta)$. I need to see to what it converges assuming something on the functions $\psi_{k}(h)$. There is a book or a site ...
0
votes
2answers
42 views

Proving that $\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$.

Prove: $$\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$$ Thanks a lot!! I tried: With Euler's identity I can get $\sin x= \dfrac{e^{ix} - e^{-ix}}{2i}$ and the ...
1
vote
1answer
25 views

Closed-form expressions for coefficients $a_k$ in $2^{2n-1}\sinh^{2n}(x)=\sum_{k=0}^n a_k \cosh(2kx)$

It is known that $$2^{2-1}\sinh^{2}(x)=\cosh(2x)-1$$ $$2^{4-1}\sinh^{4}(x)=\cosh(4x)-4\cosh(2x)+3$$ What are the closed-form expression for coefficients $a_k$ ($k=0,1,\cdots,n$) in the expression ...
1
vote
2answers
42 views

$\frac{1}{\sin\theta\cdot \sin2\theta} + \frac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$ [closed]

$$\sum_{k=1}^n \frac{1}{\sin k\theta \sin (k+1)\theta} = \dfrac{1}{\sin\theta\cdot \sin2\theta} + \dfrac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$$ up to ...
2
votes
4answers
59 views

Can somebody explain to me why these terms are equal?

I just read a proof on ProofWiki that proves Euler's formula, but I can't seem to understand what is done in this following step: ...
0
votes
0answers
49 views

Find a short formula for $\sin x+\sin (x+y)+\sin (x+2y)+. . .+\sin (x+(n-1)y)$

The answer is : $$\sin(\frac {x+x+(n-1)y}{2}) \dfrac {\sin \frac{ny}{2}}{\sin \frac {y}{2}}$$ I could've written the question as: Show that..., but then people would try induction. What I did: ...
0
votes
0answers
23 views

trigonometric polynomial using FFT

I am trying to use the FFT to approximate a given function. So i have 10 points xk that are given for example, if i use the FFT that will give me Xk. So now using the inverse FFT we can get the ...
0
votes
0answers
33 views

Binomial series for $2^{n-1}\cos^n\vartheta$ and $2^{n-1}(-1)^{\frac{n}{2}}\sin^n\vartheta$

Can somebody confirm for me whether the following series are correct? $$2^{n-1}\cos^n\vartheta=\cos ...
2
votes
1answer
55 views

Explain how the following is equal to $2\cos x$.

The question was Prove $$\frac{1+\sin2x+\cos2x}{\cos x+\sin x}=2\cos x$$ I simplified it using several trigonometric identities, what I got is this "$\dfrac{2\cos^2 x + 2\cos x \sin x}{\cos x + ...
1
vote
3answers
40 views

Find the value of $\sin(B-A)$.

If $A$ is an acute angle whose tangent is $\frac{15}{8}$ and $B$ is and obtuse angle whose sine is $\frac{12}{13}$, find $\sin (B-A)$. [Without calculators] I suppose I gotta use this formula: $\sin ...
0
votes
1answer
57 views

Why do different trig functions sum differently?

Why does the $\sum_{n=1}^{\infty} \sin (\frac 1 {n^2})$ converge but the $\sum_{n=1}^{\infty} \cos (\frac 1 {n^2})$ diverge?
1
vote
1answer
60 views

Does the expansion of $\sin x$ contradict the normal formula $\sin x = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$?

Lets say I have a right angled triangle with sides $3, 4$ and $5$ units. They form a perfect Pythagorean triplet. One of the angles in the triangle, say $\alpha$ must have the following condition: ...
2
votes
2answers
97 views

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
1
vote
1answer
302 views

How do I find the direction angle of a vector?

v= -3i-10j I found the dot product to be 109 and the magnitude to be the square root of 11881. I divided it out and it came out to be one. I don't know what I'm doing wrong. Please help.
0
votes
0answers
50 views

is there any other method to solve this limit [duplicate]

If someone prove this $$ \lim_{x \to 0}\frac{\sin{x}}{x} =1$$ other than using L'Hospital or series expansion, it will be really appreciable
1
vote
3answers
34 views

Series representation of $\sin(nu)$ when $n$ is an odd integer?

So, out of boredom and curiosity, today I came up with a series representation for $\sin(nu)$ when $n$ is an even integer: $$\sin(nu) = \sum_{k=1}^\frac n2 ...
1
vote
3answers
41 views

How to prove that $\Sigma_{n=1}^{N-1}{\sin^2\frac{p\pi{}n}{N}} = \frac{N}{2}$

Here $p$ is a known integer constant ($p > 0$). I know that this is true for a fact (checked numerically in Matlab and it holds), but I'm just not able to prove it. I noticed a similar problem ...
0
votes
1answer
58 views

converting a signal back and forth with trigonometric functions

I'm trying to convert a signal back and forth using trigonometric functions. In the example below: 1) start off with a cos signal 2) convert the signal to a secant signal 3) would like to convert ...
2
votes
3answers
75 views

Closed form for a trigonometric partial sum

I know that: $$\sum_{k=1}^n\arctan(2k^2)=\frac{\pi n}{2}-\frac{1}{2}\arctan(\frac{2n(n+1)}{2n+1})$$ Can a similar closed form expressions be given for $\sum_{k=1}^n \arctan(k^2)$? I was able to ...
3
votes
7answers
361 views

Prove $ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $

$$ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $$ Wolframalpha shows that it is a correct identity, ...
0
votes
1answer
29 views

Use Osborn's Rule to suggest an identity involving cothx^2 and cosechx^2 then to solve

i know 1+cothx^2=cschx^2 but how to suggest the identity then use to solve part b?
0
votes
3answers
373 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
2
votes
2answers
223 views

$\sum_{k=0}^{n-1}\sin(k\frac{\pi}{n}+\theta)$

I'm trying to find the closed form of the above formula. This link shows the solution of tan version. Sum of tangent functions where arguments are in specific arithmetic series Though I'm trying to ...
0
votes
1answer
46 views

Trigonometry and complex numbers

Suppose $z_0=e^{i\theta_0}$ a complexe number as $\theta_0\in ]-\pi,\pi[ \setminus\{0\}$. For $n\in \mathbb{N}$, we pose $z_{n+1}=\dfrac{|z_n|+z_n}{2}$ and $z_n=r_ne^{i\theta_n}$ with ...
3
votes
0answers
65 views

Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
0
votes
2answers
2k views

How to simplify these trigonometry terms? (involving sin, cos, tan, cot)

Please, simplify these trigonometry term: $$ \frac{\sin 2x + \sin 4x + \sin 6x}{\cos 2x + \cos 4x + \cos 6x}=? $$
3
votes
2answers
237 views

Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$

For any real $x$ and positive integer $n$, is it true that: $$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$ Please justify.
4
votes
2answers
389 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
4
votes
1answer
102 views

Solve $x\sqrt{10} = \prod\limits_{k = 1}^{90} \sin(k), x\in \mathbb Q$.

Can someone help me with this question? I've found a solution but it's not a very nice one. I used 6 times the relation $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. There's got to be a better way. ...
4
votes
1answer
130 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 ...
1
vote
2answers
57 views

Verifying the trigonometric identity $\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$

I have the following trigonometric identity $$\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$$ I've been trying to verify it for almost 20 minutes but coming up ...
11
votes
1answer
194 views

Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$

I need help with calculating this sum: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
4
votes
1answer
107 views

finding $\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$

I want to find $$L=\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$$ I already know that I need to split the expression $\frac{2r}{1-r^2+r^4}$ of the form ...
2
votes
2answers
660 views

Power Reducing Formula

I need some how with the power reducing formula. I'm having trouble understanding how to apply it. Here's an equation that utilizes it: $\cos^2 (\theta x) - 1 = 0$ How do I solve this on the ...
1
vote
1answer
89 views

Trigonometry Identities

Consider a collection of five points evenly spaced around a circle to form a regular pentagon. Assume the figure is scaled so that the sides of the pentagon have length 1. Question: Use Ptolemy’s ...
0
votes
1answer
77 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 ...
2
votes
2answers
272 views

Evaluation of a trigonometric partial sum

I just wanted to evaluate $$ \sum_{k=0}^n \cos k\theta $$ and I know that it should give $$ \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)} $$ ...
7
votes
2answers
265 views

Calculate: $\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$

Calculate the following sum for integers $n\ge2$: $$\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$$ I'm trying to obtain a closed form if that is possible.
6
votes
6answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...