# Questions tagged [trigonometric-series]

For questions about or related to trigonometric series, i.e. series of the form $a_0 + \sum_{n = 1}^{\infty} (a_n \cos{nx} + b_n \sin{nx})$ or $\sum_n c_n e^{inx}$.

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### Prove that , $\sin \theta + \cos \theta > 1$ [closed]

Prove that $\sin \theta + \cos \theta > 1$. You can prove it with geometrical terms and others.
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### Counting roots of oscillatory functions

Consider the functions $f_m : [0,1] \to \mathbb{R}$ defined by $f_m(x) = \displaystyle\sum_{n=1}^{m} \frac{\sin(nx)}{1+\sin^2(n)}$ for every natural number $m$. My task is to find the number of ...
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### $\int_0^\pi t^2\sin(t)\sin(n(\sin(t)-t\cos(t)))dt$ for alternate Goat problem Fourier sine series solution

The goat problem has the following transcendental equation: $$\sin(a)-a\cos(a)=\frac\pi2,a=1.905695729\dots$$ If $f^{-1}(x)$ is odd, its Fourier sine series of period $\left[-\frac L2,\frac L2\right]$ ...
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### Proving some trigonometric series are convergent

I have to prove that if $(a_k)_k$ is decreasing and tending to 0, then, for all $\epsilon>0$, the series $\sum a_k \sin(2\pi kx), \sum a_k \cos(2\pi kx)$ and $\sum a_k e^{2\pi ikx}$ are all ...
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### $\sum_{k=1}^{n} \arctan(k)=?$ Using argument of complex number

Question: \begin{align} \sum_{k=1}^{n} \arctan(k)=? \end{align} My approach: \begin{align} \prod_{k=1}^{n} (1+ik)=a+ib\qquad(1)\end{align} \begin{align} \Rightarrow arg \Bigg(\prod_{k=1}^{n} (1+ik)\... 273 views

### Proving $\left|\sin x\right|\leq\frac{(2m+1)!}{2^{4m}(m!)^2}\left[\binom{2m}{m} - \sum_{k=1}^{m} \frac{2}{4k^2-1} \binom{2m}{m+k} \cos(2kx) \right]$

Update. Based on @Exodd's answer, it turns out that the upper bound is equal to $$T_{2m}(\cos x) = \sum_{k=0}^{m} (-1)^k \binom{1/2}{k}\cos^{2k} x,$$ where $T_{2m}(x)$ is the degree $2m$ Taylor ...
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### Showing that $R= \frac{1}{n}\sum_{k=1}^n\cos(\theta_k - \overline{\theta})$ given a system of equations

Let angles $\theta_1,\dots,\theta_n$ be given and define $C = \frac{1}{n}\sum_{k=1}^n\cos(\theta_k)$ and $S = \frac{1}{n}\sum_{k=1}^n\sin(\theta_k)$. Then, there exists a value $\overline{\theta}$ ...
### Is it possible to express $\sin(\frac{k\pi}{4})$ as something of the form $(-1)^{f(k)}$?
For example $\cos(k\pi)=(-1)^k$ similarily, $\sin(\frac{k\pi}{2})=(-1)^\frac{k-1}{2}$ when k is odd, and 0 otherwise. Is there a similar representation for $\sin(\frac{k\pi}{4})$