Is $\sin(x)$ an irrational number?

Question

I was wondering, is $\sin(x)$ an irrational number?

I know that for $0$ it is $0$, so rational, but from basic view.

Answer 1

Well, it depends on the value of $x$. It could be either rational or irrational. $\sin(x)$ takes all real values between $-1$ and $+1$. There are both rational and irrational values between $-1$ and $+1$.

For example, consider the following graph:

The graph of $\sin(x)$ is represented by the red curve. The blue line is $y=1/\sqrt 2$. $1/\sqrt 2$ is an irrational number and $\sin(x)$ attains this value at infinitely many points. The green line in the graph represent $y=1/2$ which is an rational number. Indeed, due to the periodic nature of sine, the value is attained at infinitely many points.

Similarly you could consider any rational or irrational values between $-1$ and $+1$ and show that sine takes these values.

Moral of the story: $\sin(x)$ is a function which can take real values between $-1$ and $+1$. It can be rational or irrational depending upon the value of $x$ it takes.

Answer 2

In general $sin(x)$ can be irrational for $x \in \mathbb{R}$ (and will be in most cases). For example $sin(\pi \cdot \frac{1}{4}) = \frac{1}{\sqrt{2}}$ which is irrational.

Answer 3

Ok, thx, well...

If I have to choose irrational number between some numbers and there are eg. π, i, 5/2 and sin(x).

Would you choose only π or π and sin(x)?