Most if the the values of the sine function that I am familiar with are irrational, like $\sin(\pi/3)$ or $\sin(\pi/6)$, or even rational like $\sin(\pi)$ or $\sin(0)$. Surely the sine function must give transcendental values somewhere, so my question is this: is there a way to determine whether the sine function will be transcendental or not? And if so, how?
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For a cheap example of $\alpha$ so that $\sin(\alpha)$ is transcendental you can take $\alpha = \sin^{-1}(1/\pi)$. In fact, there are lots of examples. Note that the range of $\sin(x)$ is $[-1,1]$, and there are infinitely many transcendental numbers in $[-1,1]$, so there are infinitely many $\alpha$ so that $\sin(\alpha)$ is transcendental. In fact, a number picked at random from $[-1,1]$ is transcendental with probability $1$. So "most" of the time $\sin(\alpha)$ is transcendental.
To find more "natural" examples, we can use the Lindemann-Weierstrass theorem, which tells us that $e^\alpha$ is transcendental whenever $\alpha$ is an algebraic number (meaning $\alpha$ is a root of a polynomial with rational coefficients). According to Wikipedia, a variant of this argument will also show that $\sin(\alpha)$ is transcendental when $\alpha$ is algebraic. So $\sin(1), \sin(\sqrt{2}), \sin(\phi)$ ($\phi$ the golden ratio) are all known to be transcendental. But there is no easy way to check in general if $\sin(\alpha)$ is transcendental (since then there would be an easy way to check if any number is transcendental, and this is notoriously hard.) I am hardly an expert in this area, but I imagine it is unknown whether $\sin(e)$ is transcendental, for example.
Jair Taylor
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