Trigonometric system of 3 equations with many solutions in radical form.












1














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







share|cite|improve this question




















  • 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














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







share|cite|improve this question




















  • 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








1







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







share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago

























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














  • 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










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