The smallest positive solution to a trigonometric equation
$begingroup$
I'm trying to determine the order of growth of the function
begin{align*}
k mapsto min lbrace s > 0 : 2cdot cos(tfrac{s}{2})^{2(k-2)} - sin(tfrac{s}{2}) - 1 = 0 rbrace, quad k geqslant 3 text{ an integer.}
end{align*}
However, I must say that I'm not really sure how to even begin... Any thoughts on this?
Motivation: In relation to a project I'm working on, I'm trying to understand the images $Omega_k$ of a certain family of maps
begin{align*}
J_k : (0, pi)^{{k}choose{2}} rightarrow (0, pi)^{{k}choose{2}},
end{align*}
mapping a bunch of angles to another bunch of angles. (I'll spare you the details as they're not much to look at...) The question above turns out to be related to the existence of some nice sets (i.e. cubes) inside the quite strange-looking $Omega_k$.
trigonometry
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
$endgroup$
add a comment |
$begingroup$
I'm trying to determine the order of growth of the function
begin{align*}
k mapsto min lbrace s > 0 : 2cdot cos(tfrac{s}{2})^{2(k-2)} - sin(tfrac{s}{2}) - 1 = 0 rbrace, quad k geqslant 3 text{ an integer.}
end{align*}
However, I must say that I'm not really sure how to even begin... Any thoughts on this?
Motivation: In relation to a project I'm working on, I'm trying to understand the images $Omega_k$ of a certain family of maps
begin{align*}
J_k : (0, pi)^{{k}choose{2}} rightarrow (0, pi)^{{k}choose{2}},
end{align*}
mapping a bunch of angles to another bunch of angles. (I'll spare you the details as they're not much to look at...) The question above turns out to be related to the existence of some nice sets (i.e. cubes) inside the quite strange-looking $Omega_k$.
trigonometry
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
$endgroup$
$begingroup$
Notation is not clear. Is it $cos(a^b)$ or $(cos(a))^b$?
$endgroup$
– Andrei
Jan 24 at 16:43
$begingroup$
It's $(cos(a))^b$.
$endgroup$
– Teddan the Terran
Jan 24 at 16:55
add a comment |
$begingroup$
I'm trying to determine the order of growth of the function
begin{align*}
k mapsto min lbrace s > 0 : 2cdot cos(tfrac{s}{2})^{2(k-2)} - sin(tfrac{s}{2}) - 1 = 0 rbrace, quad k geqslant 3 text{ an integer.}
end{align*}
However, I must say that I'm not really sure how to even begin... Any thoughts on this?
Motivation: In relation to a project I'm working on, I'm trying to understand the images $Omega_k$ of a certain family of maps
begin{align*}
J_k : (0, pi)^{{k}choose{2}} rightarrow (0, pi)^{{k}choose{2}},
end{align*}
mapping a bunch of angles to another bunch of angles. (I'll spare you the details as they're not much to look at...) The question above turns out to be related to the existence of some nice sets (i.e. cubes) inside the quite strange-looking $Omega_k$.
trigonometry
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
$endgroup$
I'm trying to determine the order of growth of the function
begin{align*}
k mapsto min lbrace s > 0 : 2cdot cos(tfrac{s}{2})^{2(k-2)} - sin(tfrac{s}{2}) - 1 = 0 rbrace, quad k geqslant 3 text{ an integer.}
end{align*}
However, I must say that I'm not really sure how to even begin... Any thoughts on this?
Motivation: In relation to a project I'm working on, I'm trying to understand the images $Omega_k$ of a certain family of maps
begin{align*}
J_k : (0, pi)^{{k}choose{2}} rightarrow (0, pi)^{{k}choose{2}},
end{align*}
mapping a bunch of angles to another bunch of angles. (I'll spare you the details as they're not much to look at...) The question above turns out to be related to the existence of some nice sets (i.e. cubes) inside the quite strange-looking $Omega_k$.
trigonometry
trigonometry
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
asked Jan 24 at 16:30
Teddan the TerranTeddan the Terran
1,206210
1,206210
$begingroup$
Notation is not clear. Is it $cos(a^b)$ or $(cos(a))^b$?
$endgroup$
– Andrei
Jan 24 at 16:43
$begingroup$
It's $(cos(a))^b$.
$endgroup$
– Teddan the Terran
Jan 24 at 16:55
add a comment |
$begingroup$
Notation is not clear. Is it $cos(a^b)$ or $(cos(a))^b$?
$endgroup$
– Andrei
Jan 24 at 16:43
$begingroup$
It's $(cos(a))^b$.
$endgroup$
– Teddan the Terran
Jan 24 at 16:55
$begingroup$
Notation is not clear. Is it $cos(a^b)$ or $(cos(a))^b$?
$endgroup$
– Andrei
Jan 24 at 16:43
$begingroup$
Notation is not clear. Is it $cos(a^b)$ or $(cos(a))^b$?
$endgroup$
– Andrei
Jan 24 at 16:43
$begingroup$
It's $(cos(a))^b$.
$endgroup$
– Teddan the Terran
Jan 24 at 16:55
$begingroup$
It's $(cos(a))^b$.
$endgroup$
– Teddan the Terran
Jan 24 at 16:55
add a comment |
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$begingroup$
Notation is not clear. Is it $cos(a^b)$ or $(cos(a))^b$?
$endgroup$
– Andrei
Jan 24 at 16:43
$begingroup$
It's $(cos(a))^b$.
$endgroup$
– Teddan the Terran
Jan 24 at 16:55