All Questions
Tagged with chebyshev-polynomials trigonometry
30
questions
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62
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Writing the $sin$ $cos$ power sum as a sum of multiple angles.
Writing the $sin$ $cos$ power sum as a sum of multiple angles.
Trying to answer the se question, which did not specify the multiple angle solutions, I started to look for a generalization and arrived ...
- 175
-1
votes
2
answers
78
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Prove $\prod_{u=0}^{v-1}y\cos\frac{u\pi}v-x\sin\frac{u\pi}v=(-2)^{-v+1}\sum_{w=0}^{\lfloor\frac{v-1}2\rfloor}(-1)^w\binom{v}{2w+1}x^{v-2w-1}y^{2w+1}$ [closed]
Prove the following trigonometric equation
$$\prod_{u=0}^{v-1}y\cos{\frac{u\pi}{v}}-x\sin{\frac{u\pi}{v}}=(-2)^{-v+1}\sum_{w=0}^{\lfloor\frac{v-1}{2}\rfloor}(-1)^{w}\binom{v}{2w+1}x^{v-2w-1}y^{2w+1}$$
...
- 131
1
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1
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170
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How to derive the sum $ \sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi k}{n}\right)}$
\begin{align*}
\sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi k}{n}\right)}\end{align*}
I tried to solve with mathematica that shows
Does anyone know how to derive this and does it is possible for ...
1
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1
answer
52
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Solving product of two cosine terms [closed]
I have an equation of $\cos Ax \cos Bx = c$ where $A$,$B$ and $c$ are known constants - how to solve for unknown $x$?
0
votes
1
answer
44
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Show that for any positive integers $i$ and $j$ with $ i > j$, we have $T_i (x)T_j(x)= \frac{1}{2}[T_{i+j}(x)T_{i-j}(x)]$
Guys can you explain this demo to me step by step, I don't understand it at all.
Show that for any positive integers $i$ and $j$ with $ i > j$, we have
$$T_i (x)T_j(x)= \frac{1}{2}[T_{i+j}(x)T_{i-j}...
0
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0
answers
233
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Sum of Chebyshev Polynomials of the 2nd kind
I am attempting to simplify the following summation and would appreciate being pointed in the correct direction on how to do this, or if even possible. When I say simplify, I specifically mean ...
- 1
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votes
2
answers
359
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Chebyshev Nodes with Interval Endpoints
The "classic" Chebyshev nodes on interval $[-1, 1]$ are given by
$$ x_k = \cos \left( \frac{2k - 1}{2n} \pi \right), k = 1, \dots, n. $$
These are the roots of the Chebyshev polynomials of ...
- 2,274
2
votes
0
answers
58
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Fractional exponent elementary symmetric polynomials.
I am wondering if there is any literature on relations between fractional power symmetric polynomials. For a particular example, with the variables $\textbf{x} = (x_1,x_2,\dots x_n),$, can we ...
- 14.1k
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1
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82
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How we generalize the cartesian form of epicycloids?
I have the following parametric form of epicycloids:
$x(t)=\frac{a\cdot\cos t+\cos(a\cdot t)}{1+a}$
$y(t)=\frac{a\cdot\sin t+\sin(a\cdot t)}{1+a}$
where $a=2,3,4,\ldots$ is a variable that ...
user736865
1
vote
2
answers
107
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Knowing $\cos\theta$, can $\cos(n\theta)=\cos(\pi k)$?
$\cos(\pi k)=1$ or $-1$. After expressing $\cos(n\theta)$ in terms of $\cos\theta$, I have found that $\cos(n\theta)=\sum^{\lfloor{\frac{n}{2}}{\rfloor}}_{l=0}{n\choose2l}(-\frac{8}{9})^{l}(-\frac{1}{...
- 81
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1
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Divisibility of Chebyshev Polynomials
I was trying to solve a problem involving an Insect crawling on the Cartesian/Coordinate Plane.
We have an insect on the origin of the coordinate plane, who remembers a particular angle $\theta.$ We ...
- 1,373
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1
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159
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Deriving the recurrence relation for Chebyshev polynomials using law of cosines?
I am trying to derive the recurrence relation in the Chebyshev polynomial using the following recurrence relation:
$\cos((n+1)\cos^{-1}x)$ $= x\cos(n\cos^{-1}x) $ - $\sin(n\cos^{-1}x)\sin(\cos^{-1}x)$...
user709452
1
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1
answer
120
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Proof of irrationality of $\arcsin(\frac{1}{4})$
I was working to find a different approach to Niven's theorem from the one in my textbook taking a route via Chebyshev polynomials. It all comes to proving the irrationality of $\arcsin(\frac{1}{4})$ ...
- 650
1
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1
answer
308
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Chebyshev polynomial generalization for non-integer degrees
I am trying to generalize the Chebyshev polynomials (especially of first kind) for non-integer degree.
The properties I would like to keep is
$$2 T_m(x) T_n(x) = T_{m+n}(x) + T_{|m-n|}(x)$$
and
$$T_m(...
- 1,818
3
votes
3
answers
121
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Proving that $\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$
I recently saw on this site, the identity
$$\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$$
which I instantly wanted to prove.
I know that I can "reduce" the problem to the evaluation of $\...
- 17.3k