Trigonometric system of 3 equations with many solutions in radical form.
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin xsin y)^{1/4}+(cos x cos y)^{1/4}=sqrt{1+sqrt{2}(sin 2x sin 2y)^{1/4}} tag1$
$(sin ysin z)^{1/4}+(cos y cos z)^{1/4}=sqrt{1+sqrt{2}(sin 2y sin 2z)^{1/4}} tag2$
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 mancò
18719
|
show 2 more comments
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin xsin y)^{1/4}+(cos x cos y)^{1/4}=sqrt{1+sqrt{2}(sin 2x sin 2y)^{1/4}} tag1$
$(sin ysin z)^{1/4}+(cos y cos z)^{1/4}=sqrt{1+sqrt{2}(sin 2y sin 2z)^{1/4}} tag2$
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 mancò
18719
1
What is the question ?
– Claude Leibovici
Sep 22 '18 at 9:11
To find almost another solution.
– giuseppe mancò
Sep 22 '18 at 9:15
1
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
– Somos
Sep 22 '18 at 11:18
1
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?
– Somos
Sep 22 '18 at 15:53
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?
– Paramanand Singh
Nov 5 '18 at 10:07
|
show 2 more comments
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin xsin y)^{1/4}+(cos x cos y)^{1/4}=sqrt{1+sqrt{2}(sin 2x sin 2y)^{1/4}} tag1$
$(sin ysin z)^{1/4}+(cos y cos z)^{1/4}=sqrt{1+sqrt{2}(sin 2y sin 2z)^{1/4}} tag2$
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 mancò
18719
Here is another (Ramanujan's type) trigonometric system of three equations.
Question:
"
If
$(sin xsin y)^{1/4}+(cos x cos y)^{1/4}=sqrt{1+sqrt{2}(sin 2x sin 2y)^{1/4}} tag1$
$(sin ysin z)^{1/4}+(cos y cos z)^{1/4}=sqrt{1+sqrt{2}(sin 2y sin 2z)^{1/4}} tag2$
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 mancò
18719
asked Sep 22 '18 at 9:05
giuseppe mancò
18719
edited 2 days ago
asked Sep 22 '18 at 9:05
giuseppe mancò
18719
asked Sep 22 '18 at 9:05
giuseppe mancò
18719
asked Sep 22 '18 at 9:05
giuseppe mancò
18719
18719
1
What is the question ?
– Claude Leibovici
Sep 22 '18 at 9:11
To find almost another solution.
– giuseppe mancò
Sep 22 '18 at 9:15
1
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
– Somos
Sep 22 '18 at 11:18
1
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?
– Somos
Sep 22 '18 at 15:53
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?
– Paramanand Singh
Nov 5 '18 at 10:07
|
show 2 more comments
1
What is the question ?
– Claude Leibovici
Sep 22 '18 at 9:11
To find almost another solution.
– giuseppe mancò
Sep 22 '18 at 9:15
1
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
– Somos
Sep 22 '18 at 11:18
1
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?
– Somos
Sep 22 '18 at 15:53
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?
– Paramanand Singh
Nov 5 '18 at 10:07
1
1
What is the question ?
– Claude Leibovici
Sep 22 '18 at 9:11
What is the question ?
– Claude Leibovici
Sep 22 '18 at 9:11
To find almost another solution.
– giuseppe mancò
Sep 22 '18 at 9:15
To find almost another solution.
– giuseppe mancò
Sep 22 '18 at 9:15
1
1
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
– Somos
Sep 22 '18 at 11:18
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
– Somos
Sep 22 '18 at 11:18
1
1
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?
– Somos
Sep 22 '18 at 15:53
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?
– Somos
Sep 22 '18 at 15:53
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?
– Paramanand Singh
Nov 5 '18 at 10:07
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?
– Paramanand Singh
Nov 5 '18 at 10:07
|
show 2 more comments
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1
What is the question ?
– Claude Leibovici
Sep 22 '18 at 9:11
To find almost another solution.
– giuseppe mancò
Sep 22 '18 at 9:15
1
An answer to MSE question 310026 "On Ramanujan's Question 359" by proposer is very related.
– Somos
Sep 22 '18 at 11:18
1
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?
– Somos
Sep 22 '18 at 15:53
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?
– Paramanand Singh
Nov 5 '18 at 10:07