rational points at rational inputs of x+ sinx

Question

Let $f(x)=x+\sin x$

The question: Is there an $x\in Q$ for which output of the function is also rational? (Apart from the trivial case of $x=0$ where $f(0)=0$)

My attempt: Clearly the range of $f(x)$ is set of real numbers and from graph of the function it is very clear that rational outputs do come for many real numbers, but except for x$=0$ I couldn't find any other value.

Kindly help me. Any help would be greatly appreciated.

There are many questions of this type - did you have a look already? I found this one first. — Dietrich Burde, Apr 15 '20 at 13:55
Isn't this the same as asking for which rational values of $x$ the value of $\sin x$ is rational, which is well known? — Dave L. Renfro, Apr 15 '20 at 13:55
Does this answer your question? Is sin(x) necessarily irrational where x is rational? — KReiser, Apr 15 '20 at 20:23

score 2 · Accepted Answer · answered Apr 15 '20 at 13:58

2

A sum of a rational number and a irrational number is irrational but $sin(x)$ is irrational (except 0) for all rational (also transcendental but not matter) then $x+sin(x)$ is irrational(also transcendental but not matter) for all rational

answered Apr 15 '20 at 13:58

I wanted a rigorous proof on this so that it is established well beyond any doubt. – Ginger bread Apr 15 '20 at 14:18
I think that my proof is perfectly rigorous – Apr 15 '20 at 14:21
It's ok don't downvote – Ginger bread Apr 15 '20 at 14:39
I don't down vote . – Apr 15 '20 at 14:40

rational points at rational inputs of x+ sinx

1 Answers1