# What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental.

My question is what are the transcendentals produced? Are there known $\alpha \in \mathbb{Q}$ where $sin(\alpha) = r\pi$, with $r \in \mathbb{Q}$? What about $r \in \mathbb{E}$(constructible)? What about $r \in \mathbb{\overline{Q}}$(algebraic)? What about $\sin(\alpha) = re$? I'm not trying to ask multiple questions, I just want to know if some or all values with algebraic (or especially rational or constructible) input for trig functions produce a certain type or types of transcendental number.

I'm afraid theory is not my strongest suit, as my background is engineering. I'm investigating differential equations involving springs and exploring what kinds of numbers to expect for equations involving simple harmonic motion. Sometimes my angular frequency $\omega$ is rational or constructible. I find the proof of LW on Wikipedia a little dense and hard to follow, so please try to keep your answers approachable to a broader audience if you can.

• If $\sin(\alpha) = k\pi$ then $\sin(\sin(\alpha)) = 0$ and I personally don't think that happens for rational or algebraic $\alpha$. Probably it's not even EL. Constructible numbers are just algebraics of degree $2^k$. – Balarka Sen Jul 8 '14 at 16:03
• A promising way of thought : $\sin(\sin(\alpha))$ is the imaginary part of $\exp(i\sin(\alpha))$ which is $\exp(\exp(i\alpha)/2 - \exp(-i\alpha)/2)$ so we are really asking whether $\exp(\exp(\alpha))$ and $\exp(\exp(-\alpha))$ are linearly independent. – Balarka Sen Jul 8 '14 at 16:11
• You are assuming $k$ is an integer, correct? What if $k$ is rational or algebraic? That would make $\sin(\sin(\alpha))$ an algebraic number (for rational $k$, I'm not sure about algebraic $k$). So is there a way to test if $\exp(\exp(\alpha))$ and $\exp(\exp(-\alpha))$ are linearly independent? – hatch22 Jul 8 '14 at 16:19
• Also, multiples of $\pi$ are only one possible set of transcendental numbers. Are there others that $\sin(\alpha)$ might produce? – hatch22 Jul 8 '14 at 16:28

## 1 Answer

Consider the following:$$\arccos(x)=\frac{\ln(x\pm\sqrt{x^2-1})}{i}+\frac{\pi}2\pm2\pi n, n=0,1,2,3,\cdots$$Assume C is algebraic. $\pi C=\frac{\ln(x\pm\sqrt{x^2-1})}{i}$$\arccos(x)=\pi[C+\frac12\pm2n]$$$$x=\cos(\pi[C+\frac12\pm2n])=\cos(\pi)\cos(C+\frac12\pm2n)-\sin(\pi)\sin(C+\frac12\pm2n)=-\cos(C+\frac12\pm2n)$$$$x=-\cos(C+\frac12\pm2n)$$I have assumed C to be algebraic. But for C to be algebraic, the logarithm at the beginning must have had a non-algebraic$x\$. I don't know how to do anything beyond this.