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[Explore Stack Internal](https://stackoverflow.co/internal/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams-compact-popover) # [How to find the degree of \$\\sin(2\\pi /n)\$ over \$\\mathbb{Q}\$](https://math.stackexchange.com/questions/2319458/how-to-find-the-degree-of-sin2-pi-n-over-mathbbq) [Ask Question](https://math.stackexchange.com/questions/ask) Asked 8 years, 6 months ago Modified [8 years, 6 months ago](https://math.stackexchange.com/questions/2319458/how-to-find-the-degree-of-sin2-pi-n-over-mathbbq?lastactivity "2017-06-12 08:46:45Z") Viewed 1k times 1 \$\\begingroup\$ Let \$\\zeta\$ be the n-th primitive unit root. with not too much effort I have proved that \$\\cos(2\\pi /n)\$ has degree \$\\varphi (n)/2\$ over \$\\mathbb{Q}\$, but failed to the degree of \$\\sin(2\\pi /n)\$ over \$\\mathbb{Q}\$. > I can myself only proof that \$\\sin(2\\pi /n)=(\\zeta -{{\\zeta }^{-1}})/2i\$, but don't know how to proceed . I am waiting for your help, thank you\! I have searched this website and find a [related question](https://math.stackexchange.com/questions/1352274/a-question-about-the-degree-of-sin2%CF%80-n-over-the-rationals), but that seems doesn't solve this problem, I need some detail about this quesion. - [field-theory](https://math.stackexchange.com/questions/tagged/field-theory "show questions tagged 'field-theory'") - [extension-field](https://math.stackexchange.com/questions/tagged/extension-field "show questions tagged 'extension-field'") - [cyclotomic-fields](https://math.stackexchange.com/questions/tagged/cyclotomic-fields "show questions tagged 'cyclotomic-fields'") [Share](https://math.stackexchange.com/q/2319458 "Short permalink to this question") Cite Follow [edited Jun 12, 2020 at 10:38](https://math.stackexchange.com/posts/2319458/revisions "show all edits to this post") [](https://math.stackexchange.com/users/-1/community) [Community](https://math.stackexchange.com/users/-1/community)Bot 1 asked Jun 12, 2017 at 8:46 [](https://math.stackexchange.com/users/334260/%E7%8E%8B%E6%9D%8E%E8%BF%9C) [王李远](https://math.stackexchange.com/users/334260/%E7%8E%8B%E6%9D%8E%E8%BF%9C) 89211 gold badge66 silver badges1919 bronze badges \$\\endgroup\$ 7 - \$\\begingroup\$ What do you mean by "it doesn't solve this problem"? It does. \$\\endgroup\$ user23365 – user23365 2017-06-12 08:50:17 +00:00 Commented Jun 12, 2017 at 8:50 - \$\\begingroup\$ Not "the" but "**a**" primitive root. \$\\endgroup\$ DonAntonio – [DonAntonio](https://math.stackexchange.com/users/31254/donantonio "215,485 reputation") 2017-06-12 08:50:22 +00:00 Commented Jun 12, 2017 at 8:50 - \$\\begingroup\$ yes , I should say "a". I don't know what he means by saying :\$Sin(2\\pi /n)=(\\zeta -{{\\zeta }^{-1}})/i\$ so the matching extension field is \$\\mathbb{Q}(\\zeta -{{\\zeta }^{-1}})\$ \$\\endgroup\$ 王李远 – [王李远](https://math.stackexchange.com/users/334260/%E7%8E%8B%E6%9D%8E%E8%BF%9C "892 reputation") 2017-06-12 08:56:47 +00:00 Commented Jun 12, 2017 at 8:56 - \$\\begingroup\$ Would you please show me some details and post as an answer?@DonAntonio \$\\endgroup\$ 王李远 – [王李远](https://math.stackexchange.com/users/334260/%E7%8E%8B%E6%9D%8E%E8%BF%9C "892 reputation") 2017-06-12 08:58:50 +00:00 Commented Jun 12, 2017 at 8:58 - \$\\begingroup\$ I present two approaches for solving a specific case of the \$\\cos(2\\pi/n)\$ problem in the link below. The first approach might be tough to adapt given the generality of "\$n\$", but the Galois-theoretic approach generalizes nicely if you can determine the elements of \$\\text{Gal}(\\mathbb{Q}(\\zeta)/\\mathbb{Q})\$ in terms of their action on \$\\zeta\$. [math.stackexchange.com/questions/2230280/…](https://math.stackexchange.com/questions/2230280/find-the-degree-of-extension-bbb-q%CE%B6-9-%CE%B6%E2%88%921-9-over-bbb-q/2230302#2230302 "find the degree of extension bbb q%25CE%25B6 9 %25CE%25B6%25E2%2588%25921 9 over bbb q") \$\\endgroup\$ Kaj Hansen – [Kaj Hansen](https://math.stackexchange.com/users/138538/kaj-hansen "33,711 reputation") 2017-06-12 09:06:41 +00:00 Commented Jun 12, 2017 at 9:06 \| [Show **2** more comments](https://math.stackexchange.com/questions/2319458/how-to-find-the-degree-of-sin2-pi-n-over-mathbbq "Expand to show all comments on this post") ## 0 Sorted by: [Reset to default](https://math.stackexchange.com/questions/2319458/how-to-find-the-degree-of-sin2-pi-n-over-mathbbq?answertab=scoredesc#tab-top) ## You must [log in](https://math.stackexchange.com/users/login?ssrc=question_page&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F2319458) to answer this question. 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