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

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

Vishnu
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naruto25
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    well, clearly that depends on the value of $x$. For some values, as you pointed out, it's rational, while some other values, for ex. $x=\frac{\pi}{4}$ it's irrational. What more do you want now? – Matti P. Oct 08 '19 at 10:02
  • Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers – The Demonix _ Hermit Oct 08 '19 at 10:03
  • What is the value of $x$? – 5xum Oct 08 '19 at 10:21

3 Answers3

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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:

enter image description here

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.

Vishnu
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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.

Kaligule
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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)?

naruto25
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  • Asking a new question as an answer to your old question is not a good use of this site; no one will see it, except for people like me who might scroll down to the bottom of your old question. If you can formulate a good mathematical question along these lines, just ask a new question. – Lee Mosher Oct 08 '19 at 16:25