Trigonometric system of 3 equations with many solutions in radical form.
$begingroup$
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin{x}sin{y})^{1/4}+(cos{x} cos{y})^{1/4}=sqrt{1+sqrt{2}(sin{2x} sin {2y})^{1/4}} tag1$
$(sin{y}sin{z})^{1/4}+(cos{y} cos{z})^{1/4}=sqrt{1+sqrt{2}(sin{2y} sin {2z})^{1/4}} tag2$
then
$(sin{x} cos{z})^{1/4}+(cos{x} sin{z})^{1/4}=(8sin{2y})^{1/12}tag3$
One solution is:
$sin{2y}=sqrt{5}-2;$
$sin{2x}=(sqrt{5}-2)^3 (4+sqrt{15})^2;$
$sin{2z}=(sqrt{5}-2)^3 (4-sqrt{15})^2.$
systems-of-equations radicals
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
$endgroup$
|
show 2 more comments
$begingroup$
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin{x}sin{y})^{1/4}+(cos{x} cos{y})^{1/4}=sqrt{1+sqrt{2}(sin{2x} sin {2y})^{1/4}} tag1$
$(sin{y}sin{z})^{1/4}+(cos{y} cos{z})^{1/4}=sqrt{1+sqrt{2}(sin{2y} sin {2z})^{1/4}} tag2$
then
$(sin{x} cos{z})^{1/4}+(cos{x} sin{z})^{1/4}=(8sin{2y})^{1/12}tag3$
One solution is:
$sin{2y}=sqrt{5}-2;$
$sin{2x}=(sqrt{5}-2)^3 (4+sqrt{15})^2;$
$sin{2z}=(sqrt{5}-2)^3 (4-sqrt{15})^2.$
systems-of-equations radicals
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
$endgroup$
1
$begingroup$
What is the question ?
$endgroup$
– Claude Leibovici
Sep 22 '18 at 9:11
$begingroup$
To find almost another solution.
$endgroup$
– giuseppe mancò
Sep 22 '18 at 9:15
1
$begingroup$
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
$endgroup$
– Somos
Sep 22 '18 at 11:18
1
$begingroup$
The last sentence of your answer to MSE question 310026 is "I calculated several solutions for $n=1,1/2,2,3,1/3,4,1/4,5,1/5,6,1/6,7,1/7,8,1/8,9,1/9,10,1/10,15,1/15$." Why don't you tell us more about these solutions?
$endgroup$
– Somos
Sep 22 '18 at 15:53
$begingroup$
From equation $(3)$ it appears that $sin x, sin y, sin z$ are elliptic moduli of first third and ninth degree. But equations $(1),(2)$ don't seem to be related to third degree modular equation. Are you sure there is no typo?
$endgroup$
– Paramanand Singh
Nov 5 '18 at 10:07
|
show 2 more comments
$begingroup$
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin{x}sin{y})^{1/4}+(cos{x} cos{y})^{1/4}=sqrt{1+sqrt{2}(sin{2x} sin {2y})^{1/4}} tag1$
$(sin{y}sin{z})^{1/4}+(cos{y} cos{z})^{1/4}=sqrt{1+sqrt{2}(sin{2y} sin {2z})^{1/4}} tag2$
then
$(sin{x} cos{z})^{1/4}+(cos{x} sin{z})^{1/4}=(8sin{2y})^{1/12}tag3$
One solution is:
$sin{2y}=sqrt{5}-2;$
$sin{2x}=(sqrt{5}-2)^3 (4+sqrt{15})^2;$
$sin{2z}=(sqrt{5}-2)^3 (4-sqrt{15})^2.$
systems-of-equations radicals
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
$endgroup$
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin{x}sin{y})^{1/4}+(cos{x} cos{y})^{1/4}=sqrt{1+sqrt{2}(sin{2x} sin {2y})^{1/4}} tag1$
$(sin{y}sin{z})^{1/4}+(cos{y} cos{z})^{1/4}=sqrt{1+sqrt{2}(sin{2y} sin {2z})^{1/4}} tag2$
then
$(sin{x} cos{z})^{1/4}+(cos{x} sin{z})^{1/4}=(8sin{2y})^{1/12}tag3$
One solution is:
$sin{2y}=sqrt{5}-2;$
$sin{2x}=(sqrt{5}-2)^3 (4+sqrt{15})^2;$
$sin{2z}=(sqrt{5}-2)^3 (4-sqrt{15})^2.$
systems-of-equations radicals
systems-of-equations radicals
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
edited Jan 9 at 10:15
giuseppe mancò
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
asked Sep 22 '18 at 9:05
giuseppe mancògiuseppe mancò
346210
346210
1
$begingroup$
What is the question ?
$endgroup$
– Claude Leibovici
Sep 22 '18 at 9:11
$begingroup$
To find almost another solution.
$endgroup$
– giuseppe mancò
Sep 22 '18 at 9:15
1
$begingroup$
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
$endgroup$
– Somos
Sep 22 '18 at 11:18
1
$begingroup$
The last sentence of your answer to MSE question 310026 is "I calculated several solutions for $n=1,1/2,2,3,1/3,4,1/4,5,1/5,6,1/6,7,1/7,8,1/8,9,1/9,10,1/10,15,1/15$." Why don't you tell us more about these solutions?
$endgroup$
– Somos
Sep 22 '18 at 15:53
$begingroup$
From equation $(3)$ it appears that $sin x, sin y, sin z$ are elliptic moduli of first third and ninth degree. But equations $(1),(2)$ don't seem to be related to third degree modular equation. Are you sure there is no typo?
$endgroup$
– Paramanand Singh
Nov 5 '18 at 10:07
|
show 2 more comments
1
$begingroup$
What is the question ?
$endgroup$
– Claude Leibovici
Sep 22 '18 at 9:11
$begingroup$
To find almost another solution.
$endgroup$
– giuseppe mancò
Sep 22 '18 at 9:15
1
$begingroup$
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
$endgroup$
– Somos
Sep 22 '18 at 11:18
1
$begingroup$
The last sentence of your answer to MSE question 310026 is "I calculated several solutions for $n=1,1/2,2,3,1/3,4,1/4,5,1/5,6,1/6,7,1/7,8,1/8,9,1/9,10,1/10,15,1/15$." Why don't you tell us more about these solutions?
$endgroup$
– Somos
Sep 22 '18 at 15:53
$begingroup$
From equation $(3)$ it appears that $sin x, sin y, sin z$ are elliptic moduli of first third and ninth degree. But equations $(1),(2)$ don't seem to be related to third degree modular equation. Are you sure there is no typo?
$endgroup$
– Paramanand Singh
Nov 5 '18 at 10:07
1
1
$begingroup$
What is the question ?
$endgroup$
– Claude Leibovici
Sep 22 '18 at 9:11
$begingroup$
What is the question ?
$endgroup$
– Claude Leibovici
Sep 22 '18 at 9:11
$begingroup$
To find almost another solution.
$endgroup$
– giuseppe mancò
Sep 22 '18 at 9:15
$begingroup$
To find almost another solution.
$endgroup$
– giuseppe mancò
Sep 22 '18 at 9:15
1
1
$begingroup$
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
$endgroup$
– Somos
Sep 22 '18 at 11:18
$begingroup$
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
$endgroup$
– Somos
Sep 22 '18 at 11:18
1
1
$begingroup$
The last sentence of your answer to MSE question 310026 is "I calculated several solutions for $n=1,1/2,2,3,1/3,4,1/4,5,1/5,6,1/6,7,1/7,8,1/8,9,1/9,10,1/10,15,1/15$." Why don't you tell us more about these solutions?
$endgroup$
– Somos
Sep 22 '18 at 15:53
$begingroup$
The last sentence of your answer to MSE question 310026 is "I calculated several solutions for $n=1,1/2,2,3,1/3,4,1/4,5,1/5,6,1/6,7,1/7,8,1/8,9,1/9,10,1/10,15,1/15$." Why don't you tell us more about these solutions?
$endgroup$
– Somos
Sep 22 '18 at 15:53
$begingroup$
From equation $(3)$ it appears that $sin x, sin y, sin z$ are elliptic moduli of first third and ninth degree. But equations $(1),(2)$ don't seem to be related to third degree modular equation. Are you sure there is no typo?
$endgroup$
– Paramanand Singh
Nov 5 '18 at 10:07
$begingroup$
From equation $(3)$ it appears that $sin x, sin y, sin z$ are elliptic moduli of first third and ninth degree. But equations $(1),(2)$ don't seem to be related to third degree modular equation. Are you sure there is no typo?
$endgroup$
– Paramanand Singh
Nov 5 '18 at 10:07
|
show 2 more comments
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%2f2926337%2ftrigonometric-system-of-3-equations-with-many-solutions-in-radical-form%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%2f2926337%2ftrigonometric-system-of-3-equations-with-many-solutions-in-radical-form%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
1
$begingroup$
What is the question ?
$endgroup$
– Claude Leibovici
Sep 22 '18 at 9:11
$begingroup$
To find almost another solution.
$endgroup$
– giuseppe mancò
Sep 22 '18 at 9:15
1
$begingroup$
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
$endgroup$
– Somos
Sep 22 '18 at 11:18
1
$begingroup$
The last sentence of your answer to MSE question 310026 is "I calculated several solutions for $n=1,1/2,2,3,1/3,4,1/4,5,1/5,6,1/6,7,1/7,8,1/8,9,1/9,10,1/10,15,1/15$." Why don't you tell us more about these solutions?
$endgroup$
– Somos
Sep 22 '18 at 15:53
$begingroup$
From equation $(3)$ it appears that $sin x, sin y, sin z$ are elliptic moduli of first third and ninth degree. But equations $(1),(2)$ don't seem to be related to third degree modular equation. Are you sure there is no typo?
$endgroup$
– Paramanand Singh
Nov 5 '18 at 10:07