The smallest positive solution to a trigonometric equation
Multi tool use
$begingroup$
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
$endgroup$
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
$endgroup$
$begingroup$
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
$endgroup$
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
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3086079%2fthe-smallest-positive-solution-to-a-trigonometric-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3086079%2fthe-smallest-positive-solution-to-a-trigonometric-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
NJ,kN7Q8gWHm,sgf3
$begingroup$
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