Mixing
The function $f(x)=arcsin(asinh(sin(x)))$
$begingroup$
While playing around with Desmos, I came across the function $$f(x)=arcsin(asinh(sin(x)))$$
where $ainmathbb{R}$ is a constant (whose value changes the function drastically; if you see the graph, you will understand).
So I came up with the following question (which I don't know how to extract an answer other than an approximation):
We have the function $f:Atomathbb{R}$, given by $$f(x)=arcsin(asinh(sin(x)))$$ with $ain B=[-m,m]subsetmathbb{R}$, and $minmathbb{R}$.
What is the biggest value of $m$ such that $f$ is continuous for every $x=(2k+1)dfracpi{2}$ (with $k$ an integer), and thus $A=mathbb{R}$? Is the number $m$ rational or irrational?
(A little hint: $mapprox 0.850918$.)
functions trigonometry recreational-mathematics
edited Jan 6 at 23:20
Blue
48k870153
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
$endgroup$
add a comment |
$begingroup$
While playing around with Desmos, I came across the function $$f(x)=arcsin(asinh(sin(x)))$$
where $ainmathbb{R}$ is a constant (whose value changes the function drastically; if you see the graph, you will understand).
So I came up with the following question (which I don't know how to extract an answer other than an approximation):
We have the function $f:Atomathbb{R}$, given by $$f(x)=arcsin(asinh(sin(x)))$$ with $ain B=[-m,m]subsetmathbb{R}$, and $minmathbb{R}$.
What is the biggest value of $m$ such that $f$ is continuous for every $x=(2k+1)dfracpi{2}$ (with $k$ an integer), and thus $A=mathbb{R}$? Is the number $m$ rational or irrational?
(A little hint: $mapprox 0.850918$.)
functions trigonometry recreational-mathematics
edited Jan 6 at 23:20
Blue
48k870153
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
$endgroup$
add a comment |
$begingroup$
While playing around with Desmos, I came across the function $$f(x)=arcsin(asinh(sin(x)))$$
where $ainmathbb{R}$ is a constant (whose value changes the function drastically; if you see the graph, you will understand).
So I came up with the following question (which I don't know how to extract an answer other than an approximation):
We have the function $f:Atomathbb{R}$, given by $$f(x)=arcsin(asinh(sin(x)))$$ with $ain B=[-m,m]subsetmathbb{R}$, and $minmathbb{R}$.
What is the biggest value of $m$ such that $f$ is continuous for every $x=(2k+1)dfracpi{2}$ (with $k$ an integer), and thus $A=mathbb{R}$? Is the number $m$ rational or irrational?
(A little hint: $mapprox 0.850918$.)
functions trigonometry recreational-mathematics
edited Jan 6 at 23:20
Blue
48k870153
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
$endgroup$
While playing around with Desmos, I came across the function $$f(x)=arcsin(asinh(sin(x)))$$
where $ainmathbb{R}$ is a constant (whose value changes the function drastically; if you see the graph, you will understand).
So I came up with the following question (which I don't know how to extract an answer other than an approximation):
We have the function $f:Atomathbb{R}$, given by $$f(x)=arcsin(asinh(sin(x)))$$ with $ain B=[-m,m]subsetmathbb{R}$, and $minmathbb{R}$.
What is the biggest value of $m$ such that $f$ is continuous for every $x=(2k+1)dfracpi{2}$ (with $k$ an integer), and thus $A=mathbb{R}$? Is the number $m$ rational or irrational?
(A little hint: $mapprox 0.850918$.)
functions trigonometry recreational-mathematics
functions trigonometry recreational-mathematics
edited Jan 6 at 23:20
Blue
48k870153
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
edited Jan 6 at 23:20
Blue
48k870153
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
edited Jan 6 at 23:20
Blue
48k870153
edited Jan 6 at 23:20
Blue
48k870153
edited Jan 6 at 23:20
Blue
48k870153
48k870153
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
asked Jan 6 at 22:47
Negafilms OriginsNegafilms Origins
164
164
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The exact value of $m$ must be such that:
$$
msinh(1)=1
$$
and thus:
$$
m=frac{1}{sinh(1)}approx 0.8509181282
$$
which I would suspect is irrational. For greater values of $m$ we will have $msinh(sin(x))$ outside the domain $[-1,1]$ of $arcsin$ which is why the graph breaks.
answered Jan 6 at 23:07
StringString
13.7k32756
$endgroup$
3
$begingroup$
By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $frac{e^1+e^{-1}}{2}=sinh 1$ cannot be rational.
$endgroup$
– vadim123
Jan 6 at 23:15
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The exact value of $m$ must be such that:
$$
msinh(1)=1
$$
and thus:
$$
m=frac{1}{sinh(1)}approx 0.8509181282
$$
which I would suspect is irrational. For greater values of $m$ we will have $msinh(sin(x))$ outside the domain $[-1,1]$ of $arcsin$ which is why the graph breaks.
answered Jan 6 at 23:07
StringString
13.7k32756
$endgroup$
3
$begingroup$
By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $frac{e^1+e^{-1}}{2}=sinh 1$ cannot be rational.
$endgroup$
– vadim123
Jan 6 at 23:15
add a comment |
$begingroup$
The exact value of $m$ must be such that:
$$
msinh(1)=1
$$
and thus:
$$
m=frac{1}{sinh(1)}approx 0.8509181282
$$
which I would suspect is irrational. For greater values of $m$ we will have $msinh(sin(x))$ outside the domain $[-1,1]$ of $arcsin$ which is why the graph breaks.
answered Jan 6 at 23:07
StringString
13.7k32756
$endgroup$
3
$begingroup$
By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $frac{e^1+e^{-1}}{2}=sinh 1$ cannot be rational.
$endgroup$
– vadim123
Jan 6 at 23:15
add a comment |
$begingroup$
The exact value of $m$ must be such that:
$$
msinh(1)=1
$$
and thus:
$$
m=frac{1}{sinh(1)}approx 0.8509181282
$$
which I would suspect is irrational. For greater values of $m$ we will have $msinh(sin(x))$ outside the domain $[-1,1]$ of $arcsin$ which is why the graph breaks.
answered Jan 6 at 23:07
StringString
13.7k32756
$endgroup$
The exact value of $m$ must be such that:
$$
msinh(1)=1
$$
and thus:
$$
m=frac{1}{sinh(1)}approx 0.8509181282
$$
which I would suspect is irrational. For greater values of $m$ we will have $msinh(sin(x))$ outside the domain $[-1,1]$ of $arcsin$ which is why the graph breaks.
answered Jan 6 at 23:07
StringString
13.7k32756
answered Jan 6 at 23:07
StringString
13.7k32756
answered Jan 6 at 23:07
StringString
13.7k32756
answered Jan 6 at 23:07
StringString
13.7k32756
13.7k32756
3
$begingroup$
By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $frac{e^1+e^{-1}}{2}=sinh 1$ cannot be rational.
$endgroup$
– vadim123
Jan 6 at 23:15
add a comment |
3
$begingroup$
By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $frac{e^1+e^{-1}}{2}=sinh 1$ cannot be rational.
$endgroup$
– vadim123
Jan 6 at 23:15
3
3
$begingroup$
By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $frac{e^1+e^{-1}}{2}=sinh 1$ cannot be rational.
$endgroup$
– vadim123
Jan 6 at 23:15
$begingroup$
By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $frac{e^1+e^{-1}}{2}=sinh 1$ cannot be rational.
$endgroup$
– vadim123
Jan 6 at 23:15
add a comment |
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