when 7 is degree and when it is radian what will be the behaviour?Is in both the cases it is algebraic over Q or not?
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3What do you think? What have you tried? – Arthur Aug 09 '19 at 20:19
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Define $\theta:=7^\circ$ so $\exp(i180\theta)=-1$, implying $\exp(\pm i\theta)$ are algebraic, as is $\sin\theta=\frac{\exp(i\theta)-\exp(-i\theta)}{2i}$. Next define $\phi=7\operatorname{rad}$ so $\sin\phi$ is transcendental, because otherwise$$\sin\phi\in\Bbb A\implies\cos\phi\in\Bbb A\implies\exp i\phi\in\Bbb A$$would contradict the Lindemann–Weierstrass theorem.
J.G.
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